--- a/src/HOL/Decision_Procs/Approximation.thy Wed Apr 26 13:41:32 2017 +0200
+++ b/src/HOL/Decision_Procs/Approximation.thy Wed Apr 26 17:01:10 2017 +0200
@@ -1,2806 +1,14 @@
(* Author: Johannes Hoelzl, TU Muenchen
Coercions removed by Dmitriy Traytel *)
-section \<open>Prove Real Valued Inequalities by Computation\<close>
-
theory Approximation
imports
Complex_Main
- "~~/src/HOL/Library/Float"
- Dense_Linear_Order
"~~/src/HOL/Library/Code_Target_Numeral"
+ Approximation_Bounds
keywords "approximate" :: diag
begin
-declare powr_neg_one [simp]
-declare powr_neg_numeral [simp]
-
-section "Horner Scheme"
-
-subsection \<open>Define auxiliary helper \<open>horner\<close> function\<close>
-
-primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
-"horner F G 0 i k x = 0" |
-"horner F G (Suc n) i k x = 1 / k - x * horner F G n (F i) (G i k) x"
-
-lemma horner_schema':
- fixes x :: real and a :: "nat \<Rightarrow> real"
- shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)"
-proof -
- have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
- by auto
- show ?thesis
- unfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric]
- sum_head_upt_Suc[OF zero_less_Suc]
- sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto
-qed
-
-lemma horner_schema:
- fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
- assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
- shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)"
-proof (induct n arbitrary: j')
- case 0
- then show ?case by auto
-next
- case (Suc n)
- show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
- using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto
-qed
-
-lemma horner_bounds':
- fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
- assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
- and lb_0: "\<And> i k x. lb 0 i k x = 0"
- and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
- (lapprox_rat prec 1 k)
- (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
- and ub_0: "\<And> i k x. ub 0 i k x = 0"
- and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
- (rapprox_rat prec 1 k)
- (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
- shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and>
- horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)"
- (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
-proof (induct n arbitrary: j')
- case 0
- thus ?case unfolding lb_0 ub_0 horner.simps by auto
-next
- case (Suc n)
- thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
- Suc[where j'="Suc j'"] \<open>0 \<le> real_of_float x\<close>
- by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le
- order_trans[OF add_mono[OF _ float_plus_down_le]]
- order_trans[OF _ add_mono[OF _ float_plus_up_le]]
- simp add: lb_Suc ub_Suc field_simps f_Suc)
-qed
-
-subsection "Theorems for floating point functions implementing the horner scheme"
-
-text \<open>
-
-Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
-all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
-
-\<close>
-
-lemma horner_bounds:
- fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
- and lb_0: "\<And> i k x. lb 0 i k x = 0"
- and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
- (lapprox_rat prec 1 k)
- (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
- and ub_0: "\<And> i k x. ub 0 i k x = 0"
- and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
- (rapprox_rat prec 1 k)
- (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
- shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))"
- (is "?lb")
- and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)"
- (is "?ub")
-proof -
- have "?lb \<and> ?ub"
- using horner_bounds'[where lb=lb, OF \<open>0 \<le> real_of_float x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc]
- unfolding horner_schema[where f=f, OF f_Suc] by simp
- thus "?lb" and "?ub" by auto
-qed
-
-lemma horner_bounds_nonpos:
- fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "real_of_float x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
- and lb_0: "\<And> i k x. lb 0 i k x = 0"
- and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
- (lapprox_rat prec 1 k)
- (float_round_down prec (x * (ub n (F i) (G i k) x)))"
- and ub_0: "\<And> i k x. ub 0 i k x = 0"
- and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
- (rapprox_rat prec 1 k)
- (float_round_up prec (x * (lb n (F i) (G i k) x)))"
- shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j)" (is "?lb")
- and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
-proof -
- have diff_mult_minus: "x - y * z = x + - y * z" for x y z :: float by simp
- have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) =
- (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)"
- by (auto simp add: field_simps power_mult_distrib[symmetric])
- have "0 \<le> real_of_float (-x)" using assms by auto
- from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
- and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)",
- unfolded lb_Suc ub_Suc diff_mult_minus,
- OF this f_Suc lb_0 _ ub_0 _]
- show "?lb" and "?ub" unfolding minus_minus sum_eq
- by (auto simp: minus_float_round_up_eq minus_float_round_down_eq)
-qed
-
-
-subsection \<open>Selectors for next even or odd number\<close>
-
-text \<open>
-The horner scheme computes alternating series. To get the upper and lower bounds we need to
-guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
-\<close>
-
-definition get_odd :: "nat \<Rightarrow> nat" where
- "get_odd n = (if odd n then n else (Suc n))"
-
-definition get_even :: "nat \<Rightarrow> nat" where
- "get_even n = (if even n then n else (Suc n))"
-
-lemma get_odd[simp]: "odd (get_odd n)"
- unfolding get_odd_def by (cases "odd n") auto
-
-lemma get_even[simp]: "even (get_even n)"
- unfolding get_even_def by (cases "even n") auto
-
-lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
- by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
-
-lemma get_even_double: "\<exists>i. get_even n = 2 * i"
- using get_even by (blast elim: evenE)
-
-lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1"
- using get_odd by (blast elim: oddE)
-
-
-section "Power function"
-
-definition float_power_bnds :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
-"float_power_bnds prec n l u =
- (if 0 < l then (power_down_fl prec l n, power_up_fl prec u n)
- else if odd n then
- (- power_up_fl prec \<bar>l\<bar> n,
- if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n)
- else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n)
- else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))"
-
-lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"
- by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd)
-
-lemma float_power_bnds:
- "(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
- by (auto
- simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff
- split: if_split_asm
- intro!: power_up_le power_down_le le_minus_power_downI
- intro: power_mono_odd power_mono power_mono_even zero_le_even_power)
-
-lemma bnds_power:
- "\<forall>(x::real) l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {l .. u} \<longrightarrow>
- l1 \<le> x ^ n \<and> x ^ n \<le> u1"
- using float_power_bnds by auto
-
-section \<open>Approximation utility functions\<close>
-
-definition bnds_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<times> float" where
- "bnds_mult prec a1 a2 b1 b2 =
- (float_plus_down prec (nprt a1 * pprt b2)
- (float_plus_down prec (nprt a2 * nprt b2)
- (float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))),
- float_plus_up prec (pprt a2 * pprt b2)
- (float_plus_up prec (pprt a1 * nprt b2)
- (float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1))))"
-
-lemma bnds_mult:
- fixes prec :: nat and a1 aa2 b1 b2 :: float
- assumes "(l, u) = bnds_mult prec a1 a2 b1 b2"
- assumes "a \<in> {real_of_float a1..real_of_float a2}"
- assumes "b \<in> {real_of_float b1..real_of_float b2}"
- shows "a * b \<in> {real_of_float l..real_of_float u}"
-proof -
- from assms have "real_of_float l \<le> a * b"
- by (intro order.trans[OF _ mult_ge_prts[of a1 a a2 b1 b b2]])
- (auto simp: bnds_mult_def intro!: float_plus_down_le)
- moreover from assms have "real_of_float u \<ge> a * b"
- by (intro order.trans[OF mult_le_prts[of a1 a a2 b1 b b2]])
- (auto simp: bnds_mult_def intro!: float_plus_up_le)
- ultimately show ?thesis by simp
-qed
-
-definition map_bnds :: "(nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow> (nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow>
- nat \<Rightarrow> (float \<times> float) \<Rightarrow> (float \<times> float)" where
- "map_bnds lb ub prec = (\<lambda>(l,u). (lb prec l, ub prec u))"
-
-lemma map_bnds:
- assumes "(lf, uf) = map_bnds lb ub prec (l, u)"
- assumes "mono f"
- assumes "x \<in> {real_of_float l..real_of_float u}"
- assumes "real_of_float (lb prec l) \<le> f (real_of_float l)"
- assumes "real_of_float (ub prec u) \<ge> f (real_of_float u)"
- shows "f x \<in> {real_of_float lf..real_of_float uf}"
-proof -
- from assms have "real_of_float lf = real_of_float (lb prec l)"
- by (simp add: map_bnds_def)
- also have "real_of_float (lb prec l) \<le> f (real_of_float l)" by fact
- also from assms have "\<dots> \<le> f x"
- by (intro monoD[OF \<open>mono f\<close>]) auto
- finally have lf: "real_of_float lf \<le> f x" .
-
- from assms have "f x \<le> f (real_of_float u)"
- by (intro monoD[OF \<open>mono f\<close>]) auto
- also have "\<dots> \<le> real_of_float (ub prec u)" by fact
- also from assms have "\<dots> = real_of_float uf"
- by (simp add: map_bnds_def)
- finally have uf: "f x \<le> real_of_float uf" .
-
- from lf uf show ?thesis by simp
-qed
-
-
-section "Square root"
-
-text \<open>
-The square root computation is implemented as newton iteration. As first first step we use the
-nearest power of two greater than the square root.
-\<close>
-
-fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-"sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" |
-"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
- in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))"
-
-lemma compute_sqrt_iteration_base[code]:
- shows "sqrt_iteration prec n (Float m e) =
- (if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1)
- else (let y = sqrt_iteration prec (n - 1) (Float m e) in
- Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))"
- using bitlen_Float by (cases n) simp_all
-
-function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
- else if x < 0 then - lb_sqrt prec (- x)
- else 0)" |
-"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
- else if x < 0 then - ub_sqrt prec (- x)
- else 0)"
-by pat_completeness auto
-termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
-
-declare lb_sqrt.simps[simp del]
-declare ub_sqrt.simps[simp del]
-
-lemma sqrt_ub_pos_pos_1:
- assumes "sqrt x < b" and "0 < b" and "0 < x"
- shows "sqrt x < (b + x / b)/2"
-proof -
- from assms have "0 < (b - sqrt x)\<^sup>2 " by simp
- also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra
- also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp
- finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" .
- hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
- by (simp add: field_simps power2_eq_square)
- thus ?thesis by (simp add: field_simps)
-qed
-
-lemma sqrt_iteration_bound:
- assumes "0 < real_of_float x"
- shows "sqrt x < sqrt_iteration prec n x"
-proof (induct n)
- case 0
- show ?case
- proof (cases x)
- case (Float m e)
- hence "0 < m"
- using assms
- apply (auto simp: sign_simps)
- by (meson not_less powr_ge_pzero)
- hence "0 < sqrt m" by auto
-
- have int_nat_bl: "(nat (bitlen m)) = bitlen m"
- using bitlen_nonneg by auto
-
- have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))"
- unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add)
- also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))"
- proof (rule mult_strict_right_mono, auto)
- show "m < 2^nat (bitlen m)"
- using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
- unfolding of_int_less_iff[of m, symmetric] by auto
- qed
- finally have "sqrt x < sqrt (2 powr (e + bitlen m))"
- unfolding int_nat_bl by auto
- also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)"
- proof -
- let ?E = "e + bitlen m"
- have E_mod_pow: "2 powr (?E mod 2) < 4"
- proof (cases "?E mod 2 = 1")
- case True
- thus ?thesis by auto
- next
- case False
- have "0 \<le> ?E mod 2" by auto
- have "?E mod 2 < 2" by auto
- from this[THEN zless_imp_add1_zle]
- have "?E mod 2 \<le> 0" using False by auto
- from xt1(5)[OF \<open>0 \<le> ?E mod 2\<close> this]
- show ?thesis by auto
- qed
- hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
- by (auto simp del: real_sqrt_four)
- hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto
-
- have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)"
- by auto
- have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
- unfolding E_eq unfolding powr_add[symmetric] by (metis of_int_add)
- also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
- unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
- also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
- by (rule mult_strict_left_mono) (auto intro: E_mod_pow)
- also have "\<dots> = 2 powr (?E div 2 + 1)"
- unfolding add.commute[of _ 1] powr_add[symmetric] by simp
- finally show ?thesis by auto
- qed
- finally show ?thesis using \<open>0 < m\<close>
- unfolding Float
- by (subst compute_sqrt_iteration_base) (simp add: ac_simps)
- qed
-next
- case (Suc n)
- let ?b = "sqrt_iteration prec n x"
- have "0 < sqrt x"
- using \<open>0 < real_of_float x\<close> by auto
- also have "\<dots> < real_of_float ?b"
- using Suc .
- finally have "sqrt x < (?b + x / ?b)/2"
- using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto
- also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2"
- by (rule divide_right_mono, auto simp add: float_divr)
- also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))"
- by simp
- also have "\<dots> \<le> (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))"
- by (auto simp add: algebra_simps float_plus_up_le)
- finally show ?case
- unfolding sqrt_iteration.simps Let_def distrib_left .
-qed
-
-lemma sqrt_iteration_lower_bound:
- assumes "0 < real_of_float x"
- shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt")
-proof -
- have "0 < sqrt x" using assms by auto
- also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
- finally show ?thesis .
-qed
-
-lemma lb_sqrt_lower_bound:
- assumes "0 \<le> real_of_float x"
- shows "0 \<le> real_of_float (lb_sqrt prec x)"
-proof (cases "0 < x")
- case True
- hence "0 < real_of_float x" and "0 \<le> x"
- using \<open>0 \<le> real_of_float x\<close> by auto
- hence "0 < sqrt_iteration prec prec x"
- using sqrt_iteration_lower_bound by auto
- hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))"
- using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto
- thus ?thesis
- unfolding lb_sqrt.simps using True by auto
-next
- case False
- with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto
- thus ?thesis
- unfolding lb_sqrt.simps by auto
-qed
-
-lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}"
-proof -
- have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float
- proof -
- from that have "0 < real_of_float x" and "0 \<le> real_of_float x" by auto
- hence sqrt_gt0: "0 < sqrt x" by auto
- hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x"
- using sqrt_iteration_bound by auto
- have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
- x / (sqrt_iteration prec prec x)" by (rule float_divl)
- also have "\<dots> < x / sqrt x"
- by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close>
- mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
- also have "\<dots> = sqrt x"
- unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
- sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto
- finally show ?thesis
- unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
- qed
- have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float
- proof -
- from that have "0 < real_of_float x" by auto
- hence "0 < sqrt x" by auto
- hence "sqrt x < sqrt_iteration prec prec x"
- using sqrt_iteration_bound by auto
- then show ?thesis
- unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
- qed
- show ?thesis
- using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
- by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
-qed
-
-lemma bnds_sqrt: "\<forall>(x::real) lx ux.
- (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u"
-proof ((rule allI) +, rule impI, erule conjE, rule conjI)
- fix x :: real
- fix lx ux
- assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
- and x: "x \<in> {lx .. ux}"
- hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
-
- have "sqrt lx \<le> sqrt x" using x by auto
- from order_trans[OF _ this]
- show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
-
- have "sqrt x \<le> sqrt ux" using x by auto
- from order_trans[OF this]
- show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto
-qed
-
-
-section "Arcus tangens and \<pi>"
-
-subsection "Compute arcus tangens series"
-
-text \<open>
-As first step we implement the computation of the arcus tangens series. This is only valid in the range
-@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
-\<close>
-
-fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
- "ub_arctan_horner prec 0 k x = 0"
-| "ub_arctan_horner prec (Suc n) k x = float_plus_up prec
- (rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))"
-| "lb_arctan_horner prec 0 k x = 0"
-| "lb_arctan_horner prec (Suc n) k x = float_plus_down prec
- (lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))"
-
-lemma arctan_0_1_bounds':
- assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
- and "even n"
- shows "arctan (sqrt y) \<in>
- {(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}"
-proof -
- let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))"
- let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"
-
- have "0 \<le> sqrt y" using assms by auto
- have "sqrt y \<le> 1" using assms by auto
- from \<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE)
-
- have "arctan (sqrt y) \<in> { ?S n .. ?S (Suc n) }"
- proof (cases "sqrt y = 0")
- case True
- then show ?thesis by simp
- next
- case False
- hence "0 < sqrt y" using \<open>0 \<le> sqrt y\<close> by auto
- hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)" by auto
-
- have "\<bar> sqrt y \<bar> \<le> 1" using \<open>0 \<le> sqrt y\<close> \<open>sqrt y \<le> 1\<close> by auto
- from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this]
- monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded \<open>2 * m = n\<close>]
- show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan .
- qed
- note arctan_bounds = this[unfolded atLeastAtMost_iff]
-
- have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
-
- note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
- and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
- and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
- OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]
-
- have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)"
- proof -
- have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n"
- using bounds(1) \<open>0 \<le> sqrt y\<close>
- apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
- apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
- apply (auto intro!: mult_left_mono)
- done
- also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds ..
- finally show ?thesis .
- qed
- moreover
- have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
- proof -
- have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds ..
- also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
- using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close>
- apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
- apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
- apply (auto intro!: mult_left_mono)
- done
- finally show ?thesis .
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma arctan_0_1_bounds:
- assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
- shows "arctan (sqrt y) \<in>
- {(sqrt y * lb_arctan_horner prec (get_even n) 1 y) ..
- (sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}"
- using
- arctan_0_1_bounds'[OF assms, of n prec]
- arctan_0_1_bounds'[OF assms, of "n + 1" prec]
- arctan_0_1_bounds'[OF assms, of "n - 1" prec]
- by (auto simp: get_even_def get_odd_def odd_pos
- simp del: ub_arctan_horner.simps lb_arctan_horner.simps)
-
-lemma arctan_lower_bound:
- assumes "0 \<le> x"
- shows "x / (1 + x\<^sup>2) \<le> arctan x" (is "?l x \<le> _")
-proof -
- have "?l x - arctan x \<le> ?l 0 - arctan 0"
- using assms
- by (intro DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. ?l x - arctan x"])
- (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps)
- thus ?thesis by simp
-qed
-
-lemma arctan_divide_mono: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> arctan y / y \<le> arctan x / x"
- by (rule DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. arctan x / x"])
- (auto intro!: derivative_eq_intros divide_nonpos_nonneg
- simp: inverse_eq_divide arctan_lower_bound)
-
-lemma arctan_mult_mono: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x * arctan y \<le> y * arctan x"
- using arctan_divide_mono[of x y] by (cases "x = 0") (simp_all add: field_simps)
-
-lemma arctan_mult_le:
- assumes "0 \<le> x" "x \<le> y" "y * z \<le> arctan y"
- shows "x * z \<le> arctan x"
-proof (cases "x = 0")
- case True
- then show ?thesis by simp
-next
- case False
- with assms have "z \<le> arctan y / y" by (simp add: field_simps)
- also have "\<dots> \<le> arctan x / x" using assms \<open>x \<noteq> 0\<close> by (auto intro!: arctan_divide_mono)
- finally show ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps)
-qed
-
-lemma arctan_le_mult:
- assumes "0 < x" "x \<le> y" "arctan x \<le> x * z"
- shows "arctan y \<le> y * z"
-proof -
- from assms have "arctan y / y \<le> arctan x / x" by (auto intro!: arctan_divide_mono)
- also have "\<dots> \<le> z" using assms by (auto simp: field_simps)
- finally show ?thesis using assms by (simp add: field_simps)
-qed
-
-lemma arctan_0_1_bounds_le:
- assumes "0 \<le> x" "x \<le> 1" "0 < real_of_float xl" "real_of_float xl \<le> x * x" "x * x \<le> real_of_float xu" "real_of_float xu \<le> 1"
- shows "arctan x \<in>
- {x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}"
-proof -
- from assms have "real_of_float xl \<le> 1" "sqrt (real_of_float xl) \<le> x" "x \<le> sqrt (real_of_float xu)" "0 \<le> real_of_float xu"
- "0 \<le> real_of_float xl" "0 < sqrt (real_of_float xl)"
- by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square)
- from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xu\<close> \<open>real_of_float xu \<le> 1\<close>]
- have "sqrt (real_of_float xu) * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real_of_float xu))"
- by simp
- from arctan_mult_le[OF \<open>0 \<le> x\<close> \<open>x \<le> sqrt _\<close> this]
- have "x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
- moreover
- from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xl\<close> \<open>real_of_float xl \<le> 1\<close>]
- have "arctan (sqrt (real_of_float xl)) \<le> sqrt (real_of_float xl) * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)"
- by simp
- from arctan_le_mult[OF \<open>0 < sqrt xl\<close> \<open>sqrt xl \<le> x\<close> this]
- have "arctan x \<le> x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" .
- ultimately show ?thesis by simp
-qed
-
-lemma arctan_0_1_bounds_round:
- assumes "0 \<le> real_of_float x" "real_of_float x \<le> 1"
- shows "arctan x \<in>
- {real_of_float x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
- real_of_float x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
- using assms
- apply (cases "x > 0")
- apply (intro arctan_0_1_bounds_le)
- apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq
- intro!: truncate_up_le1 mult_le_one truncate_down_le truncate_up_le truncate_down_pos
- mult_pos_pos)
- done
-
-
-subsection "Compute \<pi>"
-
-definition ub_pi :: "nat \<Rightarrow> float" where
- "ub_pi prec =
- (let
- A = rapprox_rat prec 1 5 ;
- B = lapprox_rat prec 1 239
- in ((Float 1 2) * float_plus_up prec
- ((Float 1 2) * float_round_up prec (A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1
- (float_round_down (Suc prec) (A * A)))))
- (- float_round_down prec (B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1
- (float_round_up (Suc prec) (B * B)))))))"
-
-definition lb_pi :: "nat \<Rightarrow> float" where
- "lb_pi prec =
- (let
- A = lapprox_rat prec 1 5 ;
- B = rapprox_rat prec 1 239
- in ((Float 1 2) * float_plus_down prec
- ((Float 1 2) * float_round_down prec (A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1
- (float_round_up (Suc prec) (A * A)))))
- (- float_round_up prec (B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1
- (float_round_down (Suc prec) (B * B)))))))"
-
-lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}"
-proof -
- have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))"
- unfolding machin[symmetric] by auto
-
- {
- fix prec n :: nat
- fix k :: int
- assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
- let ?k = "rapprox_rat prec 1 k"
- let ?kl = "float_round_down (Suc prec) (?k * ?k)"
- have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
-
- have "0 \<le> real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>)
- have "real_of_float ?k \<le> 1"
- by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le
- intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
- have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
- hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
- also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)"
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
- by auto
- finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" .
- } note ub_arctan = this
-
- {
- fix prec n :: nat
- fix k :: int
- assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
- let ?k = "lapprox_rat prec 1 k"
- let ?ku = "float_round_up (Suc prec) (?k * ?k)"
- have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
- have "1 / k \<le> 1" using \<open>1 < k\<close> by auto
- have "0 \<le> real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>]
- by (auto simp add: \<open>1 div k = 0\<close>)
- have "0 \<le> real_of_float (?k * ?k)" by simp
- have "real_of_float ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>)
- hence "real_of_float (?k * ?k) \<le> 1" using \<open>0 \<le> real_of_float ?k\<close> by (auto intro!: mult_le_one)
-
- have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto
-
- have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k"
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
- by auto
- also have "\<dots> \<le> arctan (1 / k)" using \<open>?k \<le> 1 / k\<close> by (rule arctan_monotone')
- finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" .
- } note lb_arctan = this
-
- have "pi \<le> ub_pi n "
- unfolding ub_pi_def machin_pi Let_def times_float.rep_eq Float_num
- using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
- by (intro mult_left_mono float_plus_up_le float_plus_down_le)
- (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
- moreover have "lb_pi n \<le> pi"
- unfolding lb_pi_def machin_pi Let_def times_float.rep_eq Float_num
- using lb_arctan[of 5] ub_arctan[of 239]
- by (intro mult_left_mono float_plus_up_le float_plus_down_le)
- (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
- ultimately show ?thesis by auto
-qed
-
-
-subsection "Compute arcus tangens in the entire domain"
-
-function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
- "lb_arctan prec x =
- (let
- ub_horner = \<lambda> x. float_round_up prec
- (x *
- ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)));
- lb_horner = \<lambda> x. float_round_down prec
- (x *
- lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))
- in
- if x < 0 then - ub_arctan prec (-x)
- else if x \<le> Float 1 (- 1) then lb_horner x
- else if x \<le> Float 1 1 then
- Float 1 1 *
- lb_horner
- (float_divl prec x
- (float_plus_up prec 1
- (ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x))))))
- else let inv = float_divr prec 1 x in
- if inv > 1 then 0
- else float_plus_down prec (lb_pi prec * Float 1 (- 1)) ( - ub_horner inv))"
-
-| "ub_arctan prec x =
- (let
- lb_horner = \<lambda> x. float_round_down prec
- (x *
- lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) ;
- ub_horner = \<lambda> x. float_round_up prec
- (x *
- ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))
- in if x < 0 then - lb_arctan prec (-x)
- else if x \<le> Float 1 (- 1) then ub_horner x
- else if x \<le> Float 1 1 then
- let y = float_divr prec x
- (float_plus_down
- (Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x)))))
- in if y > 1 then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y
- else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))"
-by pat_completeness auto
-termination
-by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
-
-declare ub_arctan_horner.simps[simp del]
-declare lb_arctan_horner.simps[simp del]
-
-lemma lb_arctan_bound':
- assumes "0 \<le> real_of_float x"
- shows "lb_arctan prec x \<le> arctan x"
-proof -
- have "\<not> x < 0" and "0 \<le> x"
- using \<open>0 \<le> real_of_float x\<close> by (auto intro!: truncate_up_le )
-
- let "?ub_horner x" =
- "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))"
- and "?lb_horner x" =
- "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))"
-
- show ?thesis
- proof (cases "x \<le> Float 1 (- 1)")
- case True
- hence "real_of_float x \<le> 1" by simp
- from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
- show ?thesis
- unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using \<open>0 \<le> x\<close>
- by (auto intro!: float_round_down_le)
- next
- case False
- hence "0 < real_of_float x" by auto
- let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
- let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))"
- let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)"
- let ?DIV = "float_divl prec x ?fR"
-
- have divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
-
- have "sqrt (1 + x*x) \<le> sqrt ?sxx"
- by (auto simp: float_plus_up.rep_eq plus_up_def float_round_up.rep_eq intro!: truncate_up_le)
- also have "\<dots> \<le> ub_sqrt prec ?sxx"
- using bnds_sqrt'[of ?sxx prec] by auto
- finally
- have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" .
- hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
- hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto
-
- have monotone: "?DIV \<le> x / ?R"
- proof -
- have "?DIV \<le> real_of_float x / ?fR" by (rule float_divl)
- also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real_of_float ?fR\<close>] divisor_gt0]])
- finally show ?thesis .
- qed
-
- show ?thesis
- proof (cases "x \<le> Float 1 1")
- case True
- have "x \<le> sqrt (1 + x * x)"
- using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
- also note \<open>\<dots> \<le> (ub_sqrt prec ?sxx)\<close>
- finally have "real_of_float x \<le> ?fR"
- by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
- moreover have "?DIV \<le> real_of_float x / ?fR"
- by (rule float_divl)
- ultimately have "real_of_float ?DIV \<le> 1"
- unfolding divide_le_eq_1_pos[OF \<open>0 < real_of_float ?fR\<close>, symmetric] by auto
-
- have "0 \<le> real_of_float ?DIV"
- using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close>
- unfolding less_eq_float_def by auto
-
- from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>]
- have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV"
- by simp
- also have "\<dots> \<le> 2 * arctan (x / ?R)"
- using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone')
- also have "2 * arctan (x / ?R) = arctan x"
- using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
- finally show ?thesis
- unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF True]
- by (auto simp: float_round_down.rep_eq
- intro!: order_trans[OF mult_left_mono[OF truncate_down]])
- next
- case False
- hence "2 < real_of_float x" by auto
- hence "1 \<le> real_of_float x" by auto
-
- let "?invx" = "float_divr prec 1 x"
- have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>]
- using arctan_tan[of 0, unfolded tan_zero] by auto
-
- show ?thesis
- proof (cases "1 < ?invx")
- case True
- show ?thesis
- unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] if_P[OF True]
- using \<open>0 \<le> arctan x\<close> by auto
- next
- case False
- hence "real_of_float ?invx \<le> 1" by auto
- have "0 \<le> real_of_float ?invx"
- by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>)
-
- have "1 / x \<noteq> 0" and "0 < 1 / x"
- using \<open>0 < real_of_float x\<close> by auto
-
- have "arctan (1 / x) \<le> arctan ?invx"
- unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
- also have "\<dots> \<le> ?ub_horner ?invx"
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
- by (auto intro!: float_round_up_le)
- also note float_round_up
- finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x"
- using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
- unfolding sgn_pos[OF \<open>0 < 1 / real_of_float x\<close>] le_diff_eq by auto
- moreover
- have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
- unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
- ultimately
- show ?thesis
- unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x \<le> Float 1 1\<close>] if_not_P[OF False]
- by (auto intro!: float_plus_down_le)
- qed
- qed
- qed
-qed
-
-lemma ub_arctan_bound':
- assumes "0 \<le> real_of_float x"
- shows "arctan x \<le> ub_arctan prec x"
-proof -
- have "\<not> x < 0" and "0 \<le> x"
- using \<open>0 \<le> real_of_float x\<close> by auto
-
- let "?ub_horner x" =
- "float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))"
- let "?lb_horner x" =
- "float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))"
-
- show ?thesis
- proof (cases "x \<le> Float 1 (- 1)")
- case True
- hence "real_of_float x \<le> 1" by auto
- show ?thesis
- unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True]
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
- by (auto intro!: float_round_up_le)
- next
- case False
- hence "0 < real_of_float x" by auto
- let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
- let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))"
- let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)"
- let ?DIV = "float_divr prec x ?fR"
-
- have sqr_ge0: "0 \<le> 1 + real_of_float x * real_of_float x"
- using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto
- hence "0 \<le> real_of_float (1 + x*x)" by auto
-
- hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
-
- have "lb_sqrt prec ?sxx \<le> sqrt ?sxx"
- using bnds_sqrt'[of ?sxx] by auto
- also have "\<dots> \<le> sqrt (1 + x*x)"
- by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le)
- finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" .
- hence "?fR \<le> ?R"
- by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le)
- have "0 < real_of_float ?fR"
- by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq
- intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one]
- truncate_down_nonneg add_nonneg_nonneg)
- have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
- proof -
- from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]]
- have "x / ?R \<le> x / ?fR" .
- also have "\<dots> \<le> ?DIV" by (rule float_divr)
- finally show ?thesis .
- qed
-
- show ?thesis
- proof (cases "x \<le> Float 1 1")
- case True
- show ?thesis
- proof (cases "?DIV > 1")
- case True
- have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
- unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto
- from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
- show ?thesis
- unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] .
- next
- case False
- hence "real_of_float ?DIV \<le> 1" by auto
-
- have "0 \<le> x / ?R"
- using \<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto
- hence "0 \<le> real_of_float ?DIV"
- using monotone by (rule order_trans)
-
- have "arctan x = 2 * arctan (x / ?R)"
- using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
- also have "\<dots> \<le> 2 * arctan (?DIV)"
- using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
- also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>]
- by (auto intro!: float_round_up_le)
- finally show ?thesis
- unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_not_P[OF False] .
- qed
- next
- case False
- hence "2 < real_of_float x" by auto
- hence "1 \<le> real_of_float x" by auto
- hence "0 < real_of_float x" by auto
- hence "0 < x" by auto
-
- let "?invx" = "float_divl prec 1 x"
- have "0 \<le> arctan x"
- using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto
-
- have "real_of_float ?invx \<le> 1"
- unfolding less_float_def
- by (rule order_trans[OF float_divl])
- (auto simp add: \<open>1 \<le> real_of_float x\<close> divide_le_eq_1_pos[OF \<open>0 < real_of_float x\<close>])
- have "0 \<le> real_of_float ?invx"
- using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto
-
- have "1 / x \<noteq> 0" and "0 < 1 / x"
- using \<open>0 < real_of_float x\<close> by auto
-
- have "(?lb_horner ?invx) \<le> arctan (?invx)"
- using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
- by (auto intro!: float_round_down_le)
- also have "\<dots> \<le> arctan (1 / x)"
- unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone') (rule float_divl)
- finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)"
- using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
- unfolding sgn_pos[OF \<open>0 < 1 / x\<close>] le_diff_eq by auto
- moreover
- have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
- unfolding Float_num times_divide_eq_right mult_1_right
- using pi_boundaries by auto
- ultimately
- show ?thesis
- unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False]
- by (auto intro!: float_round_up_le float_plus_up_le)
- qed
- qed
-qed
-
-lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
-proof (cases "0 \<le> x")
- case True
- hence "0 \<le> real_of_float x" by auto
- show ?thesis
- using ub_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>]
- unfolding atLeastAtMost_iff by auto
-next
- case False
- let ?mx = "-x"
- from False have "x < 0" and "0 \<le> real_of_float ?mx"
- by auto
- hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx"
- using ub_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] by auto
- show ?thesis
- unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x]
- ub_arctan.simps[where x=x] Let_def if_P[OF \<open>x < 0\<close>]
- unfolding atLeastAtMost_iff using bounds[unfolded minus_float.rep_eq arctan_minus]
- by (simp add: arctan_minus)
-qed
-
-lemma bnds_arctan: "\<forall> (x::real) lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> arctan x \<and> arctan x \<le> u"
-proof (rule allI, rule allI, rule allI, rule impI)
- fix x :: real
- fix lx ux
- assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}"
- hence l: "lb_arctan prec lx = l "
- and u: "ub_arctan prec ux = u"
- and x: "x \<in> {lx .. ux}"
- by auto
- show "l \<le> arctan x \<and> arctan x \<le> u"
- proof
- show "l \<le> arctan x"
- proof -
- from arctan_boundaries[of lx prec, unfolded l]
- have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps)
- also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
- finally show ?thesis .
- qed
- show "arctan x \<le> u"
- proof -
- have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone')
- also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
- finally show ?thesis .
- qed
- qed
-qed
-
-
-section "Sinus and Cosinus"
-
-subsection "Compute the cosinus and sinus series"
-
-fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
- "ub_sin_cos_aux prec 0 i k x = 0"
-| "ub_sin_cos_aux prec (Suc n) i k x = float_plus_up prec
- (rapprox_rat prec 1 k) (-
- float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
-| "lb_sin_cos_aux prec 0 i k x = 0"
-| "lb_sin_cos_aux prec (Suc n) i k x = float_plus_down prec
- (lapprox_rat prec 1 k) (-
- float_round_up prec (x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
-
-lemma cos_aux:
- shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb")
- and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
-proof -
- have "0 \<le> real_of_float (x * x)" by auto
- let "?f n" = "fact (2 * n) :: nat"
- have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n
- proof -
- have "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
- then show ?thesis by auto
- qed
- from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
- OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
- show ?lb and ?ub
- by (auto simp add: power_mult power2_eq_square[of "real_of_float x"])
-qed
-
-lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1"
- by (cases j n rule: nat.exhaust[case_product nat.exhaust])
- (auto intro!: float_plus_down_le order_trans[OF lapprox_rat])
-
-lemma one_le_ub_sin_cos_aux: "odd n \<Longrightarrow> 1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0"
- by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat])
-
-lemma cos_boundaries:
- assumes "0 \<le> real_of_float x" and "x \<le> pi / 2"
- shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
-proof (cases "real_of_float x = 0")
- case False
- hence "real_of_float x \<noteq> 0" by auto
- hence "0 < x" and "0 < real_of_float x"
- using \<open>0 \<le> real_of_float x\<close> by auto
- have "0 < x * x"
- using \<open>0 < x\<close> by simp
-
- have morph_to_if_power: "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i)) =
- (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)"
- (is "?sum = ?ifsum") for x n
- proof -
- have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
- also have "\<dots> =
- (\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
- also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)"
- unfolding sum_split_even_odd atLeast0LessThan ..
- also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)"
- by (rule sum.cong) auto
- finally show ?thesis .
- qed
-
- { fix n :: nat assume "0 < n"
- hence "0 < 2 * n" by auto
- obtain t where "0 < t" and "t < real_of_float x" and
- cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i)
- + (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)"
- (is "_ = ?SUM + ?rest / ?fact * ?pow")
- using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>]
- unfolding cos_coeff_def atLeast0LessThan by auto
-
- have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
- also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add)
- also have "\<dots> = ?rest" by auto
- finally have "cos t * (- 1) ^ n = ?rest" .
- moreover
- have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
- hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto
- ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
-
- have "0 < ?fact" by auto
- have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto
-
- {
- assume "even n"
- have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
- unfolding morph_to_if_power[symmetric] using cos_aux by auto
- also have "\<dots> \<le> cos x"
- proof -
- from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
- have "0 \<le> (?rest / ?fact) * ?pow" by simp
- thus ?thesis unfolding cos_eq by auto
- qed
- finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" .
- } note lb = this
-
- {
- assume "odd n"
- have "cos x \<le> ?SUM"
- proof -
- from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
- have "0 \<le> (- ?rest) / ?fact * ?pow"
- by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
- thus ?thesis unfolding cos_eq by auto
- qed
- also have "\<dots> \<le> (ub_sin_cos_aux prec n 1 1 (x * x))"
- unfolding morph_to_if_power[symmetric] using cos_aux by auto
- finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" .
- } note ub = this and lb
- } note ub = this(1) and lb = this(2)
-
- have "cos x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))"
- using ub[OF odd_pos[OF get_odd] get_odd] .
- moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x"
- proof (cases "0 < get_even n")
- case True
- show ?thesis using lb[OF True get_even] .
- next
- case False
- hence "get_even n = 0" by auto
- have "- (pi / 2) \<le> x"
- by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto
- with \<open>x \<le> pi / 2\<close> show ?thesis
- unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq
- using cos_ge_zero by auto
- qed
- ultimately show ?thesis by auto
-next
- case True
- hence "x = 0"
- by transfer
- thus ?thesis
- using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux
- by simp
-qed
-
-lemma sin_aux:
- assumes "0 \<le> real_of_float x"
- shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
- (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
- and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le>
- (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
-proof -
- have "0 \<le> real_of_float (x * x)" by auto
- let "?f n" = "fact (2 * n + 1) :: nat"
- have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n
- proof -
- have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
- show ?thesis
- unfolding F by auto
- qed
- from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
- OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
- show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close>
- apply (simp_all only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
- apply (simp_all only: mult.commute[where 'a=real] of_nat_fact)
- apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"])
- done
-qed
-
-lemma sin_boundaries:
- assumes "0 \<le> real_of_float x"
- and "x \<le> pi / 2"
- shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
-proof (cases "real_of_float x = 0")
- case False
- hence "real_of_float x \<noteq> 0" by auto
- hence "0 < x" and "0 < real_of_float x"
- using \<open>0 \<le> real_of_float x\<close> by auto
- have "0 < x * x"
- using \<open>0 < x\<close> by simp
-
- have sum_morph: "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((fact (2 * j + 1))) * x ^(2 * j + 1)) =
- (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * x ^ i)"
- (is "?SUM = _") for x :: real and n
- proof -
- have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)"
- by auto
- have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM"
- by auto
- also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)"
- unfolding sum_split_even_odd atLeast0LessThan ..
- also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)"
- by (rule sum.cong) auto
- finally show ?thesis .
- qed
-
- { fix n :: nat assume "0 < n"
- hence "0 < 2 * n + 1" by auto
- obtain t where "0 < t" and "t < real_of_float x" and
- sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)
- + (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)"
- (is "_ = ?SUM + ?rest / ?fact * ?pow")
- using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>]
- unfolding sin_coeff_def atLeast0LessThan by auto
-
- have "?rest = cos t * (- 1) ^ n"
- unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto
- moreover
- have "t \<le> pi / 2"
- using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
- hence "0 \<le> cos t"
- using \<open>0 < t\<close> and cos_ge_zero by auto
- ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest"
- by auto
-
- have "0 < ?fact"
- by (simp del: fact_Suc)
- have "0 < ?pow"
- using \<open>0 < real_of_float x\<close> by (rule zero_less_power)
-
- {
- assume "even n"
- have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
- (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
- using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
- also have "\<dots> \<le> ?SUM" by auto
- also have "\<dots> \<le> sin x"
- proof -
- from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
- have "0 \<le> (?rest / ?fact) * ?pow" by simp
- thus ?thesis unfolding sin_eq by auto
- qed
- finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" .
- } note lb = this
-
- {
- assume "odd n"
- have "sin x \<le> ?SUM"
- proof -
- from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
- have "0 \<le> (- ?rest) / ?fact * ?pow"
- by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
- thus ?thesis unfolding sin_eq by auto
- qed
- also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
- by auto
- also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
- using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
- finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
- } note ub = this and lb
- } note ub = this(1) and lb = this(2)
-
- have "sin x \<le> (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))"
- using ub[OF odd_pos[OF get_odd] get_odd] .
- moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x"
- proof (cases "0 < get_even n")
- case True
- show ?thesis
- using lb[OF True get_even] .
- next
- case False
- hence "get_even n = 0" by auto
- with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close>
- show ?thesis
- unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq
- using sin_ge_zero by auto
- qed
- ultimately show ?thesis by auto
-next
- case True
- show ?thesis
- proof (cases "n = 0")
- case True
- thus ?thesis
- unfolding \<open>n = 0\<close> get_even_def get_odd_def
- using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto
- next
- case False
- with not0_implies_Suc obtain m where "n = Suc m" by blast
- thus ?thesis
- unfolding \<open>n = Suc m\<close> get_even_def get_odd_def
- using \<open>real_of_float x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1]
- by (cases "even (Suc m)") auto
- qed
-qed
-
-
-subsection "Compute the cosinus in the entire domain"
-
-definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"lb_cos prec x = (let
- horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
- half = \<lambda> x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)
- in if x < Float 1 (- 1) then horner x
-else if x < 1 then half (horner (x * Float 1 (- 1)))
- else half (half (horner (x * Float 1 (- 2)))))"
-
-definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_cos prec x = (let
- horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
- half = \<lambda> x. float_plus_up prec (Float 1 1 * x * x) (- 1)
- in if x < Float 1 (- 1) then horner x
-else if x < 1 then half (horner (x * Float 1 (- 1)))
- else half (half (horner (x * Float 1 (- 2)))))"
-
-lemma lb_cos:
- assumes "0 \<le> real_of_float x" and "x \<le> pi"
- shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
-proof -
- have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real
- proof -
- have "cos x = cos (x / 2 + x / 2)"
- by auto
- also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
- unfolding cos_add by auto
- also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1"
- by algebra
- finally show ?thesis .
- qed
-
- have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto
- let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
- let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
- let "?ub_half x" = "float_plus_up prec (Float 1 1 * x * x) (- 1)"
- let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)"
-
- show ?thesis
- proof (cases "x < Float 1 (- 1)")
- case True
- hence "x \<le> pi / 2"
- using pi_ge_two by auto
- show ?thesis
- unfolding lb_cos_def[where x=x] ub_cos_def[where x=x]
- if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF \<open>x < Float 1 (- 1)\<close>] Let_def
- using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] .
- next
- case False
- { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
- assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi"
- hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
- using pi_ge_two unfolding Float_num by auto
- hence "0 \<le> cos ?x2"
- by (rule cos_ge_zero)
-
- have "(?lb_half y) \<le> cos x"
- proof (cases "y < 0")
- case True
- show ?thesis
- using cos_ge_minus_one unfolding if_P[OF True] by auto
- next
- case False
- hence "0 \<le> real_of_float y" by auto
- from mult_mono[OF \<open>y \<le> cos ?x2\<close> \<open>y \<le> cos ?x2\<close> \<open>0 \<le> cos ?x2\<close> this]
- have "real_of_float y * real_of_float y \<le> cos ?x2 * cos ?x2" .
- hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2"
- by auto
- hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1"
- unfolding Float_num by auto
- thus ?thesis
- unfolding if_not_P[OF False] x_half Float_num
- by (auto intro!: float_plus_down_le)
- qed
- } note lb_half = this
-
- { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
- assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi"
- hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
- using pi_ge_two unfolding Float_num by auto
- hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
-
- have "cos x \<le> (?ub_half y)"
- proof -
- have "0 \<le> real_of_float y"
- using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans)
- from mult_mono[OF ub ub this \<open>0 \<le> cos ?x2\<close>]
- have "cos ?x2 * cos ?x2 \<le> real_of_float y * real_of_float y" .
- hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y"
- by auto
- hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1"
- unfolding Float_num by auto
- thus ?thesis
- unfolding x_half Float_num
- by (auto intro!: float_plus_up_le)
- qed
- } note ub_half = this
-
- let ?x2 = "x * Float 1 (- 1)"
- let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"
-
- have "-pi \<le> x"
- using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close>
- by (rule order_trans)
-
- show ?thesis
- proof (cases "x < 1")
- case True
- hence "real_of_float x \<le> 1" by auto
- have "0 \<le> real_of_float ?x2" and "?x2 \<le> pi / 2"
- using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto
- from cos_boundaries[OF this]
- have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)"
- by auto
-
- have "(?lb x) \<le> ?cos x"
- proof -
- from lb_half[OF lb \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
- show ?thesis
- unfolding lb_cos_def[where x=x] Let_def
- using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
- qed
- moreover have "?cos x \<le> (?ub x)"
- proof -
- from ub_half[OF ub \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
- show ?thesis
- unfolding ub_cos_def[where x=x] Let_def
- using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
- qed
- ultimately show ?thesis by auto
- next
- case False
- have "0 \<le> real_of_float ?x4" and "?x4 \<le> pi / 2"
- using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto
- from cos_boundaries[OF this]
- have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)"
- by auto
-
- have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)"
- by transfer simp
-
- have "(?lb x) \<le> ?cos x"
- proof -
- have "-pi \<le> ?x2" and "?x2 \<le> pi"
- using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto
- from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
- show ?thesis
- unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
- qed
- moreover have "?cos x \<le> (?ub x)"
- proof -
- have "-pi \<le> ?x2" and "?x2 \<le> pi"
- using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto
- from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
- show ?thesis
- unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
- if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
- qed
- ultimately show ?thesis by auto
- qed
- qed
-qed
-
-lemma lb_cos_minus:
- assumes "-pi \<le> x"
- and "real_of_float x \<le> 0"
- shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
-proof -
- have "0 \<le> real_of_float (-x)" and "(-x) \<le> pi"
- using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto
- from lb_cos[OF this] show ?thesis .
-qed
-
-definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
-"bnds_cos prec lx ux = (let
- lpi = float_round_down prec (lb_pi prec) ;
- upi = float_round_up prec (ub_pi prec) ;
- k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
- lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)) ;
- ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi))
- in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux))
- else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx)
- else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
- else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
- else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
- else (Float (- 1) 0, Float 1 0))"
-
-lemma floor_int: obtains k :: int where "real_of_int k = (floor_fl f)"
- by (simp add: floor_fl_def)
-
-lemma cos_periodic_nat[simp]:
- fixes n :: nat
- shows "cos (x + n * (2 * pi)) = cos x"
-proof (induct n arbitrary: x)
- case 0
- then show ?case by simp
-next
- case (Suc n)
- have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
- unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto
- show ?case
- unfolding split_pi_off using Suc by auto
-qed
-
-lemma cos_periodic_int[simp]:
- fixes i :: int
- shows "cos (x + i * (2 * pi)) = cos x"
-proof (cases "0 \<le> i")
- case True
- hence i_nat: "real_of_int i = nat i" by auto
- show ?thesis
- unfolding i_nat by auto
-next
- case False
- hence i_nat: "i = - real (nat (-i))" by auto
- have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))"
- by auto
- also have "\<dots> = cos (x + i * (2 * pi))"
- unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
- finally show ?thesis by auto
-qed
-
-lemma bnds_cos: "\<forall>(x::real) lx ux. (l, u) =
- bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u"
-proof (rule allI | rule impI | erule conjE)+
- fix x :: real
- fix lx ux
- assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}"
-
- let ?lpi = "float_round_down prec (lb_pi prec)"
- let ?upi = "float_round_up prec (ub_pi prec)"
- let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
- let ?lx2 = "(- ?k * 2 * (if ?k < 0 then ?lpi else ?upi))"
- let ?ux2 = "(- ?k * 2 * (if ?k < 0 then ?upi else ?lpi))"
- let ?lx = "float_plus_down prec lx ?lx2"
- let ?ux = "float_plus_up prec ux ?ux2"
-
- obtain k :: int where k: "k = real_of_float ?k"
- by (rule floor_int)
-
- have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
- using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
- float_round_down[of prec "lb_pi prec"]
- by auto
- hence "lx + ?lx2 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ux + ?ux2"
- using x
- by (cases "k = 0")
- (auto intro!: add_mono
- simp add: k [symmetric] uminus_add_conv_diff [symmetric]
- simp del: float_of_numeral uminus_add_conv_diff)
- hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
- by (auto intro!: float_plus_down_le float_plus_up_le)
- note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
- hence lx_less_ux: "?lx \<le> real_of_float ?ux" by (rule order_trans)
-
- { assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
- with lpi[THEN le_imp_neg_le] lx
- have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0"
- by simp_all
-
- have "(lb_cos prec (- ?lx)) \<le> cos (real_of_float (- ?lx))"
- using lb_cos_minus[OF pi_lx lx_0] by simp
- also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
- using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
- by (simp only: uminus_float.rep_eq of_int_minus
- cos_minus mult_minus_left) simp
- finally have "(lb_cos prec (- ?lx)) \<le> cos x"
- unfolding cos_periodic_int . }
- note negative_lx = this
-
- { assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
- with lx
- have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx"
- by auto
-
- have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
- using cos_monotone_0_pi_le[OF lx_0 lx pi_x]
- by (simp only: of_int_minus
- cos_minus mult_minus_left) simp
- also have "\<dots> \<le> (ub_cos prec ?lx)"
- using lb_cos[OF lx_0 pi_lx] by simp
- finally have "cos x \<le> (ub_cos prec ?lx)"
- unfolding cos_periodic_int . }
- note positive_lx = this
-
- { assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
- with ux
- have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0"
- by simp_all
-
- have "cos (x + (-k) * (2 * pi)) \<le> cos (real_of_float (- ?ux))"
- using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
- by (simp only: uminus_float.rep_eq of_int_minus
- cos_minus mult_minus_left) simp
- also have "\<dots> \<le> (ub_cos prec (- ?ux))"
- using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
- finally have "cos x \<le> (ub_cos prec (- ?ux))"
- unfolding cos_periodic_int . }
- note negative_ux = this
-
- { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
- with lpi ux
- have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux"
- by simp_all
-
- have "(lb_cos prec ?ux) \<le> cos ?ux"
- using lb_cos[OF ux_0 pi_ux] by simp
- also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
- using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux]
- by (simp only: of_int_minus
- cos_minus mult_minus_left) simp
- finally have "(lb_cos prec ?ux) \<le> cos x"
- unfolding cos_periodic_int . }
- note positive_ux = this
-
- show "l \<le> cos x \<and> cos x \<le> u"
- proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
- case True
- with bnds have l: "l = lb_cos prec (-?lx)" and u: "u = ub_cos prec (-?ux)"
- by (auto simp add: bnds_cos_def Let_def)
- from True lpi[THEN le_imp_neg_le] lx ux
- have "- pi \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> 0"
- by auto
- with True negative_ux negative_lx show ?thesis
- unfolding l u by simp
- next
- case 1: False
- show ?thesis
- proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
- case True with bnds 1
- have l: "l = lb_cos prec ?ux"
- and u: "u = ub_cos prec ?lx"
- by (auto simp add: bnds_cos_def Let_def)
- from True lpi lx ux
- have "0 \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> pi"
- by auto
- with True positive_ux positive_lx show ?thesis
- unfolding l u by simp
- next
- case 2: False
- show ?thesis
- proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
- case Cond: True
- with bnds 1 2 have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
- and u: "u = Float 1 0"
- by (auto simp add: bnds_cos_def Let_def)
- show ?thesis
- unfolding u l using negative_lx positive_ux Cond
- by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min)
- next
- case 3: False
- show ?thesis
- proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
- case Cond: True
- with bnds 1 2 3
- have l: "l = Float (- 1) 0"
- and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
- by (auto simp add: bnds_cos_def Let_def)
-
- have "cos x \<le> real_of_float u"
- proof (cases "x - k * (2 * pi) < pi")
- case True
- hence "x - k * (2 * pi) \<le> pi" by simp
- from positive_lx[OF Cond[THEN conjunct1] this] show ?thesis
- unfolding u by (simp add: real_of_float_max)
- next
- case False
- hence "pi \<le> x - k * (2 * pi)" by simp
- hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp
-
- have "?ux \<le> 2 * pi"
- using Cond lpi by auto
- hence "x - k * (2 * pi) - 2 * pi \<le> 0"
- using ux by simp
-
- have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0"
- using Cond by auto
-
- from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
- hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto
- hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)"
- using lpi[THEN le_imp_neg_le] by auto
-
- have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)"
- using ux lpi by auto
- have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))"
- unfolding cos_periodic_int ..
- also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
- using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
- by (simp only: minus_float.rep_eq of_int_minus of_int_1
- mult_minus_left mult_1_left) simp
- also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
- unfolding uminus_float.rep_eq cos_minus ..
- also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
- using lb_cos_minus[OF pi_ux ux_0] by simp
- finally show ?thesis unfolding u by (simp add: real_of_float_max)
- qed
- thus ?thesis unfolding l by auto
- next
- case 4: False
- show ?thesis
- proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
- case Cond: True
- with bnds 1 2 3 4 have l: "l = Float (- 1) 0"
- and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
- by (auto simp add: bnds_cos_def Let_def)
-
- have "cos x \<le> u"
- proof (cases "-pi < x - k * (2 * pi)")
- case True
- hence "-pi \<le> x - k * (2 * pi)" by simp
- from negative_ux[OF this Cond[THEN conjunct2]] show ?thesis
- unfolding u by (simp add: real_of_float_max)
- next
- case False
- hence "x - k * (2 * pi) \<le> -pi" by simp
- hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp
-
- have "-2 * pi \<le> ?lx" using Cond lpi by auto
-
- hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp
-
- have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)"
- using Cond lpi by auto
-
- from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
- hence "?lx + 2 * ?lpi \<le> ?lpi" by auto
- hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi"
- using lpi[THEN le_imp_neg_le] by auto
-
- have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi"
- using lx lpi by auto
-
- have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))"
- unfolding cos_periodic_int ..
- also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
- using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x]
- by (simp only: minus_float.rep_eq of_int_minus of_int_1
- mult_minus_left mult_1_left) simp
- also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
- using lb_cos[OF lx_0 pi_lx] by simp
- finally show ?thesis unfolding u by (simp add: real_of_float_max)
- qed
- thus ?thesis unfolding l by auto
- next
- case False
- with bnds 1 2 3 4 show ?thesis
- by (auto simp add: bnds_cos_def Let_def)
- qed
- qed
- qed
- qed
- qed
-qed
-
-
-section "Exponential function"
-
-subsection "Compute the series of the exponential function"
-
-fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
- and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-where
-"ub_exp_horner prec 0 i k x = 0" |
-"ub_exp_horner prec (Suc n) i k x = float_plus_up prec
- (rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" |
-"lb_exp_horner prec 0 i k x = 0" |
-"lb_exp_horner prec (Suc n) i k x = float_plus_down prec
- (lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))"
-
-lemma bnds_exp_horner:
- assumes "real_of_float x \<le> 0"
- shows "exp x \<in> {lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x}"
-proof -
- have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n
- proof -
- have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m"
- by (induct n) auto
- show ?thesis
- unfolding F by auto
- qed
-
- note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
- OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
-
- have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x"
- proof -
- have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real_of_float x ^ j)"
- using bounds(1) by auto
- also have "\<dots> \<le> exp x"
- proof -
- obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real_of_float x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
- using Maclaurin_exp_le unfolding atLeast0LessThan by blast
- moreover have "0 \<le> exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
- by (auto simp: zero_le_even_power)
- ultimately show ?thesis using get_odd exp_gt_zero by auto
- qed
- finally show ?thesis .
- qed
- moreover
- have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x"
- proof -
- have x_less_zero: "real_of_float x ^ get_odd n \<le> 0"
- proof (cases "real_of_float x = 0")
- case True
- have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
- thus ?thesis unfolding True power_0_left by auto
- next
- case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 0\<close> by auto
- show ?thesis by (rule less_imp_le, auto simp add: \<open>real_of_float x < 0\<close>)
- qed
- obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>"
- and "exp x = (\<Sum>m = 0..<get_odd n. (real_of_float x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n)"
- using Maclaurin_exp_le unfolding atLeast0LessThan by blast
- moreover have "exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n) \<le> 0"
- by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero)
- ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)"
- using get_odd exp_gt_zero by auto
- also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
- using bounds(2) by auto
- finally show ?thesis .
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma ub_exp_horner_nonneg: "real_of_float x \<le> 0 \<Longrightarrow>
- 0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
- using bnds_exp_horner[of x prec n]
- by (intro order_trans[OF exp_ge_zero]) auto
-
-
-subsection "Compute the exponential function on the entire domain"
-
-function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
-"lb_exp prec x =
- (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
- else
- let
- horner = (\<lambda> x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in
- if y \<le> 0 then Float 1 (- 2) else y)
- in
- if x < - 1 then
- power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x))
- else horner x)" |
-"ub_exp prec x =
- (if 0 < x then float_divr prec 1 (lb_exp prec (-x))
- else if x < - 1 then
- power_up_fl prec
- (ub_exp_horner prec (get_odd (prec + 2)) 1 1
- (float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x))
- else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
- by pat_completeness auto
-termination
- by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))") auto
-
-lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
-proof -
- have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
- have "1 / 4 = (Float 1 (- 2))"
- unfolding Float_num by auto
- also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)"
- by (subst less_eq_float.rep_eq [symmetric]) code_simp
- also have "\<dots> \<le> exp (- 1 :: float)"
- using bnds_exp_horner[where x="- 1"] by auto
- finally show ?thesis
- by simp
-qed
-
-lemma lb_exp_pos:
- assumes "\<not> 0 < x"
- shows "0 < lb_exp prec x"
-proof -
- let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
- let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y"
- have pos_horner: "0 < ?horner x" for x
- unfolding Let_def by (cases "?lb_horner x \<le> 0") auto
- moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat
- proof -
- have "0 < real_of_float (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp
- also have "\<dots> = (?horner x) ^ num" by auto
- finally show ?thesis .
- qed
- ultimately show ?thesis
- unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] Let_def
- by (cases "floor_fl x", cases "x < - 1")
- (auto simp: real_power_up_fl real_power_down_fl intro!: power_up_less power_down_pos)
-qed
-
-lemma exp_boundaries':
- assumes "x \<le> 0"
- shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}"
-proof -
- let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
- let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
-
- have "real_of_float x \<le> 0" and "\<not> x > 0"
- using \<open>x \<le> 0\<close> by auto
- show ?thesis
- proof (cases "x < - 1")
- case False
- hence "- 1 \<le> real_of_float x" by auto
- show ?thesis
- proof (cases "?lb_exp_horner x \<le> 0")
- case True
- from \<open>\<not> x < - 1\<close>
- have "- 1 \<le> real_of_float x" by auto
- hence "exp (- 1) \<le> exp x"
- unfolding exp_le_cancel_iff .
- from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x"
- unfolding Float_num .
- with True show ?thesis
- using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto
- next
- case False
- thus ?thesis
- using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def)
- qed
- next
- case True
- let ?num = "nat (- int_floor_fl x)"
-
- have "real_of_int (int_floor_fl x) < - 1"
- using int_floor_fl[of x] \<open>x < - 1\<close> by simp
- hence "real_of_int (int_floor_fl x) < 0" by simp
- hence "int_floor_fl x < 0" by auto
- hence "1 \<le> - int_floor_fl x" by auto
- hence "0 < nat (- int_floor_fl x)" by auto
- hence "0 < ?num" by auto
- hence "real ?num \<noteq> 0" by auto
- have num_eq: "real ?num = - int_floor_fl x"
- using \<open>0 < nat (- int_floor_fl x)\<close> by auto
- have "0 < - int_floor_fl x"
- using \<open>0 < ?num\<close>[unfolded of_nat_less_iff[symmetric]] by simp
- hence "real_of_int (int_floor_fl x) < 0"
- unfolding less_float_def by auto
- have fl_eq: "real_of_int (- int_floor_fl x) = real_of_float (- floor_fl x)"
- by (simp add: floor_fl_def int_floor_fl_def)
- from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real_of_float (- floor_fl x)"
- by (simp add: floor_fl_def int_floor_fl_def)
- from \<open>real_of_int (int_floor_fl x) < 0\<close> have "real_of_float (floor_fl x) < 0"
- by (simp add: floor_fl_def int_floor_fl_def)
- have "exp x \<le> ub_exp prec x"
- proof -
- have div_less_zero: "real_of_float (float_divr prec x (- floor_fl x)) \<le> 0"
- using float_divr_nonpos_pos_upper_bound[OF \<open>real_of_float x \<le> 0\<close> \<open>0 \<le> real_of_float (- floor_fl x)\<close>]
- unfolding less_eq_float_def zero_float.rep_eq .
-
- have "exp x = exp (?num * (x / ?num))"
- using \<open>real ?num \<noteq> 0\<close> by auto
- also have "\<dots> = exp (x / ?num) ^ ?num"
- unfolding exp_of_nat_mult ..
- also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num"
- unfolding num_eq fl_eq
- by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
- also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
- unfolding real_of_float_power
- by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
- also have "\<dots> \<le> real_of_float (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
- by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero)
- finally show ?thesis
- unfolding ub_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] floor_fl_def Let_def .
- qed
- moreover
- have "lb_exp prec x \<le> exp x"
- proof -
- let ?divl = "float_divl prec x (- floor_fl x)"
- let ?horner = "?lb_exp_horner ?divl"
-
- show ?thesis
- proof (cases "?horner \<le> 0")
- case False
- hence "0 \<le> real_of_float ?horner" by auto
-
- have div_less_zero: "real_of_float (float_divl prec x (- floor_fl x)) \<le> 0"
- using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close>
- by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
-
- have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
- exp (float_divl prec x (- floor_fl x)) ^ ?num"
- using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]]
- bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1]
- by (auto intro!: power_mono)
- also have "\<dots> \<le> exp (x / ?num) ^ ?num"
- unfolding num_eq fl_eq
- using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq)
- also have "\<dots> = exp (?num * (x / ?num))"
- unfolding exp_of_nat_mult ..
- also have "\<dots> = exp x"
- using \<open>real ?num \<noteq> 0\<close> by auto
- finally show ?thesis
- using False
- unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
- int_floor_fl_def Let_def if_not_P[OF False]
- by (auto simp: real_power_down_fl intro!: power_down_le)
- next
- case True
- have "power_down_fl prec (Float 1 (- 2)) ?num \<le> (Float 1 (- 2)) ^ ?num"
- by (metis Float_le_zero_iff less_imp_le linorder_not_less
- not_numeral_le_zero numeral_One power_down_fl)
- then have "power_down_fl prec (Float 1 (- 2)) ?num \<le> real_of_float (Float 1 (- 2)) ^ ?num"
- by simp
- also
- have "real_of_float (floor_fl x) \<noteq> 0" and "real_of_float (floor_fl x) \<le> 0"
- using \<open>real_of_float (floor_fl x) < 0\<close> by auto
- from divide_right_mono_neg[OF floor_fl[of x] \<open>real_of_float (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real_of_float (floor_fl x) \<noteq> 0\<close>]]
- have "- 1 \<le> x / (- floor_fl x)"
- unfolding minus_float.rep_eq by auto
- from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
- have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))"
- unfolding Float_num .
- hence "real_of_float (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
- by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral)
- also have "\<dots> = exp x"
- unfolding num_eq fl_eq exp_of_nat_mult[symmetric]
- using \<open>real_of_float (floor_fl x) \<noteq> 0\<close> by auto
- finally show ?thesis
- unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
- int_floor_fl_def Let_def if_P[OF True] real_of_float_power .
- qed
- qed
- ultimately show ?thesis by auto
- qed
-qed
-
-lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }"
-proof -
- show ?thesis
- proof (cases "0 < x")
- case False
- hence "x \<le> 0" by auto
- from exp_boundaries'[OF this] show ?thesis .
- next
- case True
- hence "-x \<le> 0" by auto
-
- have "lb_exp prec x \<le> exp x"
- proof -
- from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
- have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)"
- unfolding atLeastAtMost_iff minus_float.rep_eq by auto
-
- have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)"
- using float_divl[where x=1] by auto
- also have "\<dots> \<le> exp x"
- using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp]
- exp_gt_zero, symmetric]]
- unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide
- by auto
- finally show ?thesis
- unfolding lb_exp.simps if_P[OF True] .
- qed
- moreover
- have "exp x \<le> ub_exp prec x"
- proof -
- have "\<not> 0 < -x" using \<open>0 < x\<close> by auto
-
- from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
- have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)"
- unfolding atLeastAtMost_iff minus_float.rep_eq by auto
-
- have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
- using lb_exp lb_exp_pos[OF \<open>\<not> 0 < -x\<close>, of prec]
- by (simp del: lb_exp.simps add: exp_minus field_simps)
- also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))"
- using float_divr .
- finally show ?thesis
- unfolding ub_exp.simps if_P[OF True] .
- qed
- ultimately show ?thesis
- by auto
- qed
-qed
-
-lemma bnds_exp: "\<forall>(x::real) lx ux. (l, u) =
- (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u"
-proof (rule allI, rule allI, rule allI, rule impI)
- fix x :: real and lx ux
- assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}"
- hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}"
- by auto
- show "l \<le> exp x \<and> exp x \<le> u"
- proof
- show "l \<le> exp x"
- proof -
- from exp_boundaries[of lx prec, unfolded l]
- have "l \<le> exp lx" by (auto simp del: lb_exp.simps)
- also have "\<dots> \<le> exp x" using x by auto
- finally show ?thesis .
- qed
- show "exp x \<le> u"
- proof -
- have "exp x \<le> exp ux" using x by auto
- also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
- finally show ?thesis .
- qed
- qed
-qed
-
-
-section "Logarithm"
-
-subsection "Compute the logarithm series"
-
-fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
-and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
-"ub_ln_horner prec 0 i x = 0" |
-"ub_ln_horner prec (Suc n) i x = float_plus_up prec
- (rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" |
-"lb_ln_horner prec 0 i x = 0" |
-"lb_ln_horner prec (Suc n) i x = float_plus_down prec
- (lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))"
-
-lemma ln_bounds:
- assumes "0 \<le> x"
- and "x < 1"
- shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
- and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
-proof -
- let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
-
- have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)"
- using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto
-
- have "norm x < 1" using assms by auto
- have "?a \<longlonglongrightarrow> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
- using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto
- have "0 \<le> ?a n" for n
- by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>)
- have "?a (Suc n) \<le> ?a n" for n
- unfolding inverse_eq_divide[symmetric]
- proof (rule mult_mono)
- show "0 \<le> x ^ Suc (Suc n)"
- by (auto simp add: \<open>0 \<le> x\<close>)
- have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1"
- unfolding power_Suc2 mult.assoc[symmetric]
- by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>)
- thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
- qed auto
- from summable_Leibniz'(2,4)[OF \<open>?a \<longlonglongrightarrow> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq]
- show ?lb and ?ub
- unfolding atLeast0LessThan by auto
-qed
-
-lemma ln_float_bounds:
- assumes "0 \<le> real_of_float x"
- and "real_of_float x < 1"
- shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
- and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub")
-proof -
- obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
- obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
-
- let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)"
-
- have "?lb \<le> sum ?s {0 ..< 2 * ev}"
- unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
- unfolding mult.commute[of "real_of_float x"] ev
- using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x"
- and lb="\<lambda>n i k x. lb_ln_horner prec n k x"
- and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
- OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
- unfolding real_of_float_power
- by (rule mult_right_mono)
- also have "\<dots> \<le> ?ln"
- using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
- finally show "?lb \<le> ?ln" .
-
- have "?ln \<le> sum ?s {0 ..< 2 * od + 1}"
- using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
- also have "\<dots> \<le> ?ub"
- unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
- unfolding mult.commute[of "real_of_float x"] od
- using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
- OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
- unfolding real_of_float_power
- by (rule mult_right_mono)
- finally show "?ln \<le> ?ub" .
-qed
-
-lemma ln_add:
- fixes x :: real
- assumes "0 < x" and "0 < y"
- shows "ln (x + y) = ln x + ln (1 + y / x)"
-proof -
- have "x \<noteq> 0" using assms by auto
- have "x + y = x * (1 + y / x)"
- unfolding distrib_left times_divide_eq_right nonzero_mult_div_cancel_left[OF \<open>x \<noteq> 0\<close>]
- by auto
- moreover
- have "0 < y / x" using assms by auto
- hence "0 < 1 + y / x" by auto
- ultimately show ?thesis
- using ln_mult assms by auto
-qed
-
-
-subsection "Compute the logarithm of 2"
-
-definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
- in float_plus_up prec
- ((Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))))
- (float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))"
-definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
- in float_plus_down prec
- ((Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))))
- (float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))"
-
-lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2")
- and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2")
-proof -
- let ?uthird = "rapprox_rat (max prec 1) 1 3"
- let ?lthird = "lapprox_rat prec 1 3"
-
- have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)"
- using ln_add[of "3 / 2" "1 / 2"] by auto
- have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
- hence lb3_ub: "real_of_float ?lthird < 1" by auto
- have lb3_lb: "0 \<le> real_of_float ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
- have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
- hence ub3_lb: "0 \<le> real_of_float ?uthird" by auto
-
- have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1"
- unfolding Float_num by auto
-
- have "0 \<le> (1::int)" and "0 < (3::int)" by auto
- have ub3_ub: "real_of_float ?uthird < 1"
- by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1)
-
- have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
- have uthird_gt0: "0 < real_of_float ?uthird + 1" using ub3_lb by auto
- have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto
-
- show ?ub_ln2
- unfolding ub_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
- proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
- have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)"
- unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
- also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
- using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
- also note float_round_up
- finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
- qed
- show ?lb_ln2
- unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
- proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
- have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)"
- using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
- note float_round_down_le[OF this]
- also have "\<dots> \<le> ln (1 / 3 + 1)"
- unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0]
- using lb3 by auto
- finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le>
- ln (1 / 3 + 1)" .
- qed
-qed
-
-
-subsection "Compute the logarithm in the entire domain"
-
-function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
-"ub_ln prec x = (if x \<le> 0 then None
- else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
- else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
- if x \<le> Float 3 (- 1) then Some (horner (x - 1))
- else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
- else let l = bitlen (mantissa x) - 1 in
- Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
-"lb_ln prec x = (if x \<le> 0 then None
- else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
- else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
- if x \<le> Float 3 (- 1) then Some (horner (x - 1))
- else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
- horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
- else let l = bitlen (mantissa x) - 1 in
- Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))"
- by pat_completeness auto
-
-termination
-proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
- fix prec and x :: float
- assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1"
- hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1"
- by auto
- from float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>]
- show False
- using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto
-next
- fix prec x
- assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divr prec 1 x) < 1"
- hence "0 < x" by auto
- from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real_of_float x < 1\<close> show False
- using \<open>real_of_float (float_divr prec 1 x) < 1\<close> by auto
-qed
-
-lemma float_pos_eq_mantissa_pos: "x > 0 \<longleftrightarrow> mantissa x > 0"
- apply (subst Float_mantissa_exponent[of x, symmetric])
- apply (auto simp add: zero_less_mult_iff zero_float_def dest: less_zeroE)
- apply (metis not_le powr_ge_pzero)
- done
-
-lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0"
- using powr_gt_zero[of 2 "e"]
- by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)
-
-lemma Float_representation_aux:
- fixes m e
- defines "x \<equiv> Float m e"
- assumes "x > 0"
- shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1)
- and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))" (is ?th2)
-proof -
- from assms have mantissa_pos: "m > 0" "mantissa x > 0"
- using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all
- thus ?th1
- using bitlen_Float[of m e] assms
- by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float])
- have "x \<noteq> float_of 0"
- unfolding zero_float_def[symmetric] using \<open>0 < x\<close> by auto
- from denormalize_shift[OF assms(1) this] guess i . note i = this
-
- have "2 powr (1 - (real_of_int (bitlen (mantissa x)) + real_of_int i)) =
- 2 powr (1 - (real_of_int (bitlen (mantissa x)))) * inverse (2 powr (real i))"
- by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
- hence "real_of_int (mantissa x) * 2 powr (1 - real_of_int (bitlen (mantissa x))) =
- (real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))"
- using \<open>mantissa x > 0\<close> by (simp add: powr_realpow)
- then show ?th2
- unfolding i by transfer auto
-qed
-
-lemma compute_ln[code]:
- fixes m e
- defines "x \<equiv> Float m e"
- shows "ub_ln prec x = (if x \<le> 0 then None
- else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
- else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
- if x \<le> Float 3 (- 1) then Some (horner (x - 1))
- else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
- else let l = bitlen m - 1 in
- Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
- (is ?th1)
- and "lb_ln prec x = (if x \<le> 0 then None
- else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
- else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
- if x \<le> Float 3 (- 1) then Some (horner (x - 1))
- else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
- horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
- else let l = bitlen m - 1 in
- Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
- (is ?th2)
-proof -
- from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0"
- by simp
- thus ?th1 ?th2
- using Float_representation_aux[of m e]
- unfolding x_def[symmetric]
- by (auto dest: not_le_imp_less)
-qed
-
-lemma ln_shifted_float:
- assumes "0 < m"
- shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))"
-proof -
- let ?B = "2^nat (bitlen m - 1)"
- define bl where "bl = bitlen m - 1"
- have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0"
- using assms by auto
- hence "0 \<le> bl" by (simp add: bitlen_alt_def bl_def)
- show ?thesis
- proof (cases "0 \<le> e")
- case True
- thus ?thesis
- unfolding bl_def[symmetric] using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
- apply (simp add: ln_mult)
- apply (cases "e=0")
- apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
- apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps)
- done
- next
- case False
- hence "0 < -e" by auto
- have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))"
- by (simp add: powr_minus)
- hence pow_gt0: "(0::real) < 2^nat (-e)"
- by auto
- hence inv_gt0: "(0::real) < inverse (2^nat (-e))"
- by auto
- show ?thesis
- using False unfolding bl_def[symmetric]
- using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
- by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
- qed
-qed
-
-lemma ub_ln_lb_ln_bounds':
- assumes "1 \<le> x"
- shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
- (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
-proof (cases "x < Float 1 1")
- case True
- hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto
- have "\<not> x \<le> 0" and "\<not> x < 1" using \<open>1 \<le> x\<close> by auto
- hence "0 \<le> real_of_float (x - 1)" using \<open>1 \<le> x\<close> by auto
-
- have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp
-
- show ?thesis
- proof (cases "x \<le> Float 3 (- 1)")
- case True
- show ?thesis
- unfolding lb_ln.simps
- unfolding ub_ln.simps Let_def
- using ln_float_bounds[OF \<open>0 \<le> real_of_float (x - 1)\<close> \<open>real_of_float (x - 1) < 1\<close>, of prec]
- \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True
- by (auto intro!: float_round_down_le float_round_up_le)
- next
- case False
- hence *: "3 / 2 < x" by auto
-
- with ln_add[of "3 / 2" "x - 3 / 2"]
- have add: "ln x = ln (3 / 2) + ln (real_of_float x * 2 / 3)"
- by (auto simp add: algebra_simps diff_divide_distrib)
-
- let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)"
- let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)"
-
- { have up: "real_of_float (rapprox_rat prec 2 3) \<le> 1"
- by (rule rapprox_rat_le1) simp_all
- have low: "2 / 3 \<le> rapprox_rat prec 2 3"
- by (rule order_trans[OF _ rapprox_rat]) simp
- from mult_less_le_imp_less[OF * low] *
- have pos: "0 < real_of_float (x * rapprox_rat prec 2 3 - 1)" by auto
-
- have "ln (real_of_float x * 2/3)
- \<le> ln (real_of_float (x * rapprox_rat prec 2 3 - 1) + 1)"
- proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
- show "real_of_float x * 2 / 3 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1) + 1"
- using * low by auto
- show "0 < real_of_float x * 2 / 3" using * by simp
- show "0 < real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
- qed
- also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
- proof (rule float_round_up_le, rule ln_float_bounds(2))
- from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] low *
- show "real_of_float (x * rapprox_rat prec 2 3 - 1) < 1" by auto
- show "0 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1)" using pos by auto
- qed
- finally have "ln x \<le> ?ub_horner (Float 1 (-1))
- + ?ub_horner ((x * rapprox_rat prec 2 3 - 1))"
- using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add
- by (auto intro!: add_mono float_round_up_le)
- note float_round_up_le[OF this, of prec]
- }
- moreover
- { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
-
- have up: "lapprox_rat prec 2 3 \<le> 2/3"
- by (rule order_trans[OF lapprox_rat], simp)
-
- have low: "0 \<le> real_of_float (lapprox_rat prec 2 3)"
- using lapprox_rat_nonneg[of 2 3 prec] by simp
-
- have "?lb_horner ?max
- \<le> ln (real_of_float ?max + 1)"
- proof (rule float_round_down_le, rule ln_float_bounds(1))
- from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] * low
- show "real_of_float ?max < 1" by (cases "real_of_float (lapprox_rat prec 2 3) = 0",
- auto simp add: real_of_float_max)
- show "0 \<le> real_of_float ?max" by (auto simp add: real_of_float_max)
- qed
- also have "\<dots> \<le> ln (real_of_float x * 2/3)"
- proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
- show "0 < real_of_float ?max + 1" by (auto simp add: real_of_float_max)
- show "0 < real_of_float x * 2/3" using * by auto
- show "real_of_float ?max + 1 \<le> real_of_float x * 2/3" using * up
- by (cases "0 < real_of_float x * real_of_float (lapprox_posrat prec 2 3) - 1",
- auto simp add: max_def)
- qed
- finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x"
- using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add
- by (auto intro!: add_mono float_round_down_le)
- note float_round_down_le[OF this, of prec]
- }
- ultimately
- show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
- using \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True False by auto
- qed
-next
- case False
- hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)"
- using \<open>1 \<le> x\<close> by auto
- show ?thesis
- proof -
- define m where "m = mantissa x"
- define e where "e = exponent x"
- from Float_mantissa_exponent[of x] have Float: "x = Float m e"
- by (simp add: m_def e_def)
- let ?s = "Float (e + (bitlen m - 1)) 0"
- let ?x = "Float m (- (bitlen m - 1))"
-
- have "0 < m" and "m \<noteq> 0" using \<open>0 < x\<close> Float powr_gt_zero[of 2 e]
- apply (auto simp add: zero_less_mult_iff)
- using not_le powr_ge_pzero apply blast
- done
- define bl where "bl = bitlen m - 1"
- hence "bl \<ge> 0"
- using \<open>m > 0\<close> by (simp add: bitlen_alt_def)
- have "1 \<le> Float m e"
- using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto
- from bitlen_div[OF \<open>0 < m\<close>] float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] \<open>bl \<ge> 0\<close>
- have x_bnds: "0 \<le> real_of_float (?x - 1)" "real_of_float (?x - 1) < 1"
- unfolding bl_def[symmetric]
- by (auto simp: powr_realpow[symmetric] field_simps)
- (auto simp : powr_minus field_simps)
-
- {
- have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))"
- (is "real_of_float ?lb2 \<le> _")
- apply (rule float_round_down_le)
- unfolding nat_0 power_0 mult_1_right times_float.rep_eq
- using lb_ln2[of prec]
- proof (rule mult_mono)
- from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
- show "0 \<le> real_of_float (Float (e + (bitlen m - 1)) 0)" by simp
- qed auto
- moreover
- from ln_float_bounds(1)[OF x_bnds]
- have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real_of_float ?lb_horner \<le> _")
- by (auto intro!: float_round_down_le)
- ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x"
- unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] by (auto intro!: float_plus_down_le)
- }
- moreover
- {
- from ln_float_bounds(2)[OF x_bnds]
- have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))"
- (is "_ \<le> real_of_float ?ub_horner")
- by (auto intro!: float_round_up_le)
- moreover
- have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)"
- (is "_ \<le> real_of_float ?ub2")
- apply (rule float_round_up_le)
- unfolding nat_0 power_0 mult_1_right times_float.rep_eq
- using ub_ln2[of prec]
- proof (rule mult_mono)
- from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
- show "0 \<le> real_of_int (e + (bitlen m - 1))" by auto
- have "0 \<le> ln (2 :: real)" by simp
- thus "0 \<le> real_of_float (ub_ln2 prec)" using ub_ln2[of prec] by arith
- qed auto
- ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner"
- unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e]
- by (auto intro!: float_plus_up_le)
- }
- ultimately show ?thesis
- unfolding lb_ln.simps
- unfolding ub_ln.simps
- unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_not_P[OF \<open>\<not> x < 1\<close>]
- if_not_P[OF False] if_not_P[OF \<open>\<not> x \<le> Float 3 (- 1)\<close>] Let_def
- unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric]
- by simp
- qed
-qed
-
-lemma ub_ln_lb_ln_bounds:
- assumes "0 < x"
- shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
- (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
-proof (cases "x < 1")
- case False
- hence "1 \<le> x"
- unfolding less_float_def less_eq_float_def by auto
- show ?thesis
- using ub_ln_lb_ln_bounds'[OF \<open>1 \<le> x\<close>] .
-next
- case True
- have "\<not> x \<le> 0" using \<open>0 < x\<close> by auto
- from True have "real_of_float x \<le> 1" "x \<le> 1"
- by simp_all
- have "0 < real_of_float x" and "real_of_float x \<noteq> 0"
- using \<open>0 < x\<close> by auto
- hence A: "0 < 1 / real_of_float x" by auto
-
- {
- let ?divl = "float_divl (max prec 1) 1 x"
- have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by auto
- hence B: "0 < real_of_float ?divl" by auto
-
- have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
- hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
- from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
- have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans)
- } moreover
- {
- let ?divr = "float_divr prec 1 x"
- have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close> \<open>x \<le> 1\<close>] unfolding less_eq_float_def less_float_def by auto
- hence B: "0 < real_of_float ?divr" by auto
-
- have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
- hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
- from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
- have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans)
- }
- ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x]
- unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_P[OF True] by auto
-qed
-
-lemma lb_ln:
- assumes "Some y = lb_ln prec x"
- shows "y \<le> ln x" and "0 < real_of_float x"
-proof -
- have "0 < x"
- proof (rule ccontr)
- assume "\<not> 0 < x"
- hence "x \<le> 0"
- unfolding less_eq_float_def less_float_def by auto
- thus False
- using assms by auto
- qed
- thus "0 < real_of_float x" by auto
- have "the (lb_ln prec x) \<le> ln x"
- using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
- thus "y \<le> ln x"
- unfolding assms[symmetric] by auto
-qed
-
-lemma ub_ln:
- assumes "Some y = ub_ln prec x"
- shows "ln x \<le> y" and "0 < real_of_float x"
-proof -
- have "0 < x"
- proof (rule ccontr)
- assume "\<not> 0 < x"
- hence "x \<le> 0" by auto
- thus False
- using assms by auto
- qed
- thus "0 < real_of_float x" by auto
- have "ln x \<le> the (ub_ln prec x)"
- using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
- thus "ln x \<le> y"
- unfolding assms[symmetric] by auto
-qed
-
-lemma bnds_ln: "\<forall>(x::real) lx ux. (Some l, Some u) =
- (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u"
-proof (rule allI, rule allI, rule allI, rule impI)
- fix x :: real
- fix lx ux
- assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}"
- hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}"
- by auto
-
- have "ln ux \<le> u" and "0 < real_of_float ux"
- using ub_ln u by auto
- have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x"
- using lb_ln[OF l] x by auto
-
- from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close>
- have "l \<le> ln x"
- using x unfolding atLeastAtMost_iff by auto
- moreover
- from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real_of_float ux\<close>] \<open>ln ux \<le> real_of_float u\<close>
- have "ln x \<le> u"
- using x unfolding atLeastAtMost_iff by auto
- ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
-qed
-
-
-section \<open>Real power function\<close>
-
-definition bnds_powr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float \<times> float) option" where
- "bnds_powr prec l1 u1 l2 u2 = (
- if l1 = 0 \<and> u1 = 0 then
- Some (0, 0)
- else if l1 = 0 \<and> l2 \<ge> 1 then
- let uln = the (ub_ln prec u1)
- in Some (0, ub_exp prec (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2))))
- else if l1 \<le> 0 then
- None
- else
- Some (map_bnds lb_exp ub_exp prec
- (bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2)))"
-
-lemmas [simp del] = lb_exp.simps ub_exp.simps
-
-lemma mono_exp_real: "mono (exp :: real \<Rightarrow> real)"
- by (auto simp: mono_def)
-
-lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0"
-proof -
- have "0 \<le> exp (real_of_float x)" by simp
- also from exp_boundaries[of x prec]
- have "\<dots> \<le> real_of_float (ub_exp prec x)" by simp
- finally show ?thesis .
-qed
-
-lemma bnds_powr:
- assumes lu: "Some (l, u) = bnds_powr prec l1 u1 l2 u2"
- assumes x: "x \<in> {real_of_float l1..real_of_float u1}"
- assumes y: "y \<in> {real_of_float l2..real_of_float u2}"
- shows "x powr y \<in> {real_of_float l..real_of_float u}"
-proof -
- consider "l1 = 0" "u1 = 0" | "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1" |
- "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))" | "l1 > 0" by force
- thus ?thesis
- proof cases
- assume "l1 = 0" "u1 = 0"
- with x lu show ?thesis by (auto simp: bnds_powr_def)
- next
- assume A: "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1"
- define uln where "uln = the (ub_ln prec u1)"
- show ?thesis
- proof (cases "x = 0")
- case False
- with A x y have "x powr y = exp (ln x * y)" by (simp add: powr_def)
- also {
- from A x False have "ln x \<le> ln (real_of_float u1)" by simp
- also from ub_ln_lb_ln_bounds[of u1 prec] A y x False
- have "ln (real_of_float u1) \<le> real_of_float uln" by (simp add: uln_def del: lb_ln.simps)
- also from A x y have "\<dots> * y \<le> real_of_float uln * (if uln \<ge> 0 then u2 else l2)"
- by (auto intro: mult_left_mono mult_left_mono_neg)
- also have "\<dots> \<le> real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))"
- by (simp add: float_round_up_le)
- finally have "ln x * y \<le> \<dots>" using A y by - simp
- }
- also have "exp (real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))) \<le>
- real_of_float (ub_exp prec (float_round_up prec
- (uln * (if uln \<ge> 0 then u2 else l2))))"
- using exp_boundaries by simp
- finally show ?thesis using A x y lu
- by (simp add: bnds_powr_def uln_def Let_def del: lb_ln.simps ub_ln.simps)
- qed (insert x y lu A, simp_all add: bnds_powr_def Let_def ub_exp_nonneg
- del: lb_ln.simps ub_ln.simps)
- next
- assume "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))"
- with lu show ?thesis by (simp add: bnds_powr_def split: if_split_asm)
- next
- assume l1: "l1 > 0"
- obtain lm um where lmum:
- "(lm, um) = bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2"
- by (cases "bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2") simp
- with l1 have "(l, u) = map_bnds lb_exp ub_exp prec (lm, um)"
- using lu by (simp add: bnds_powr_def del: lb_ln.simps ub_ln.simps split: if_split_asm)
- hence "exp (ln x * y) \<in> {real_of_float l..real_of_float u}"
- proof (rule map_bnds[OF _ mono_exp_real], goal_cases)
- case 1
- let ?lln = "the (lb_ln prec l1)" and ?uln = "the (ub_ln prec u1)"
- from ub_ln_lb_ln_bounds[of l1 prec] ub_ln_lb_ln_bounds[of u1 prec] x l1
- have "real_of_float ?lln \<le> ln (real_of_float l1) \<and>
- ln (real_of_float u1) \<le> real_of_float ?uln"
- by (auto simp del: lb_ln.simps ub_ln.simps)
- moreover from l1 x have "ln (real_of_float l1) \<le> ln x \<and> ln x \<le> ln (real_of_float u1)"
- by auto
- ultimately have ln: "real_of_float ?lln \<le> ln x \<and> ln x \<le> real_of_float ?uln" by simp
- from lmum show ?case
- by (rule bnds_mult) (insert y ln, simp_all)
- qed (insert exp_boundaries[of lm prec] exp_boundaries[of um prec], simp_all)
- with x l1 show ?thesis
- by (simp add: powr_def mult_ac)
- qed
-qed
-
-
section "Implement floatarith"
subsection "Define syntax and semantics"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Approximation_Bounds.thy Wed Apr 26 17:01:10 2017 +0200
@@ -0,0 +1,2806 @@
+(*
+ Author: Johannes Hoelzl, TU Muenchen
+ Coercions removed by Dmitriy Traytel
+
+ This file contains only general material about computing lower/upper bounds
+ on real functions. Approximation.thy contains the actual approximation algorithm
+ and the approximation oracle. This is in order to make a clear separation between
+ "morally immaculate" material about upper/lower bounds and the trusted oracle/reflection.
+*)
+
+theory Approximation_Bounds
+imports
+ Complex_Main
+ "~~/src/HOL/Library/Float"
+ Dense_Linear_Order
+begin
+
+declare powr_neg_one [simp]
+declare powr_neg_numeral [simp]
+
+section "Horner Scheme"
+
+subsection \<open>Define auxiliary helper \<open>horner\<close> function\<close>
+
+primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
+"horner F G 0 i k x = 0" |
+"horner F G (Suc n) i k x = 1 / k - x * horner F G n (F i) (G i k) x"
+
+lemma horner_schema':
+ fixes x :: real and a :: "nat \<Rightarrow> real"
+ shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)"
+proof -
+ have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
+ by auto
+ show ?thesis
+ unfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric]
+ sum_head_upt_Suc[OF zero_less_Suc]
+ sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto
+qed
+
+lemma horner_schema:
+ fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
+ assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)"
+proof (induct n arbitrary: j')
+ case 0
+ then show ?case by auto
+next
+ case (Suc n)
+ show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
+ using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto
+qed
+
+lemma horner_bounds':
+ fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+ assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ and lb_0: "\<And> i k x. lb 0 i k x = 0"
+ and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
+ (lapprox_rat prec 1 k)
+ (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
+ and ub_0: "\<And> i k x. ub 0 i k x = 0"
+ and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
+ (rapprox_rat prec 1 k)
+ (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
+ shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and>
+ horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)"
+ (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
+proof (induct n arbitrary: j')
+ case 0
+ thus ?case unfolding lb_0 ub_0 horner.simps by auto
+next
+ case (Suc n)
+ thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
+ Suc[where j'="Suc j'"] \<open>0 \<le> real_of_float x\<close>
+ by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le
+ order_trans[OF add_mono[OF _ float_plus_down_le]]
+ order_trans[OF _ add_mono[OF _ float_plus_up_le]]
+ simp add: lb_Suc ub_Suc field_simps f_Suc)
+qed
+
+subsection "Theorems for floating point functions implementing the horner scheme"
+
+text \<open>
+
+Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
+all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
+
+\<close>
+
+lemma horner_bounds:
+ fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+ assumes "0 \<le> real_of_float x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ and lb_0: "\<And> i k x. lb 0 i k x = 0"
+ and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
+ (lapprox_rat prec 1 k)
+ (- float_round_up prec (x * (ub n (F i) (G i k) x)))"
+ and ub_0: "\<And> i k x. ub 0 i k x = 0"
+ and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
+ (rapprox_rat prec 1 k)
+ (- float_round_down prec (x * (lb n (F i) (G i k) x)))"
+ shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))"
+ (is "?lb")
+ and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)"
+ (is "?ub")
+proof -
+ have "?lb \<and> ?ub"
+ using horner_bounds'[where lb=lb, OF \<open>0 \<le> real_of_float x\<close> f_Suc lb_0 lb_Suc ub_0 ub_Suc]
+ unfolding horner_schema[where f=f, OF f_Suc] by simp
+ thus "?lb" and "?ub" by auto
+qed
+
+lemma horner_bounds_nonpos:
+ fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+ assumes "real_of_float x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ and lb_0: "\<And> i k x. lb 0 i k x = 0"
+ and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = float_plus_down prec
+ (lapprox_rat prec 1 k)
+ (float_round_down prec (x * (ub n (F i) (G i k) x)))"
+ and ub_0: "\<And> i k x. ub 0 i k x = 0"
+ and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = float_plus_up prec
+ (rapprox_rat prec 1 k)
+ (float_round_up prec (x * (lb n (F i) (G i k) x)))"
+ shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j)" (is "?lb")
+ and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
+proof -
+ have diff_mult_minus: "x - y * z = x + - y * z" for x y z :: float by simp
+ have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real_of_float x ^ j) =
+ (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)"
+ by (auto simp add: field_simps power_mult_distrib[symmetric])
+ have "0 \<le> real_of_float (-x)" using assms by auto
+ from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
+ and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)",
+ unfolded lb_Suc ub_Suc diff_mult_minus,
+ OF this f_Suc lb_0 _ ub_0 _]
+ show "?lb" and "?ub" unfolding minus_minus sum_eq
+ by (auto simp: minus_float_round_up_eq minus_float_round_down_eq)
+qed
+
+
+subsection \<open>Selectors for next even or odd number\<close>
+
+text \<open>
+The horner scheme computes alternating series. To get the upper and lower bounds we need to
+guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
+\<close>
+
+definition get_odd :: "nat \<Rightarrow> nat" where
+ "get_odd n = (if odd n then n else (Suc n))"
+
+definition get_even :: "nat \<Rightarrow> nat" where
+ "get_even n = (if even n then n else (Suc n))"
+
+lemma get_odd[simp]: "odd (get_odd n)"
+ unfolding get_odd_def by (cases "odd n") auto
+
+lemma get_even[simp]: "even (get_even n)"
+ unfolding get_even_def by (cases "even n") auto
+
+lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
+ by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
+
+lemma get_even_double: "\<exists>i. get_even n = 2 * i"
+ using get_even by (blast elim: evenE)
+
+lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1"
+ using get_odd by (blast elim: oddE)
+
+
+section "Power function"
+
+definition float_power_bnds :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
+"float_power_bnds prec n l u =
+ (if 0 < l then (power_down_fl prec l n, power_up_fl prec u n)
+ else if odd n then
+ (- power_up_fl prec \<bar>l\<bar> n,
+ if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n)
+ else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n)
+ else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))"
+
+lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"
+ by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd)
+
+lemma float_power_bnds:
+ "(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
+ by (auto
+ simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff
+ split: if_split_asm
+ intro!: power_up_le power_down_le le_minus_power_downI
+ intro: power_mono_odd power_mono power_mono_even zero_le_even_power)
+
+lemma bnds_power:
+ "\<forall>(x::real) l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {l .. u} \<longrightarrow>
+ l1 \<le> x ^ n \<and> x ^ n \<le> u1"
+ using float_power_bnds by auto
+
+section \<open>Approximation utility functions\<close>
+
+definition bnds_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<times> float" where
+ "bnds_mult prec a1 a2 b1 b2 =
+ (float_plus_down prec (nprt a1 * pprt b2)
+ (float_plus_down prec (nprt a2 * nprt b2)
+ (float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))),
+ float_plus_up prec (pprt a2 * pprt b2)
+ (float_plus_up prec (pprt a1 * nprt b2)
+ (float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1))))"
+
+lemma bnds_mult:
+ fixes prec :: nat and a1 aa2 b1 b2 :: float
+ assumes "(l, u) = bnds_mult prec a1 a2 b1 b2"
+ assumes "a \<in> {real_of_float a1..real_of_float a2}"
+ assumes "b \<in> {real_of_float b1..real_of_float b2}"
+ shows "a * b \<in> {real_of_float l..real_of_float u}"
+proof -
+ from assms have "real_of_float l \<le> a * b"
+ by (intro order.trans[OF _ mult_ge_prts[of a1 a a2 b1 b b2]])
+ (auto simp: bnds_mult_def intro!: float_plus_down_le)
+ moreover from assms have "real_of_float u \<ge> a * b"
+ by (intro order.trans[OF mult_le_prts[of a1 a a2 b1 b b2]])
+ (auto simp: bnds_mult_def intro!: float_plus_up_le)
+ ultimately show ?thesis by simp
+qed
+
+definition map_bnds :: "(nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow> (nat \<Rightarrow> float \<Rightarrow> float) \<Rightarrow>
+ nat \<Rightarrow> (float \<times> float) \<Rightarrow> (float \<times> float)" where
+ "map_bnds lb ub prec = (\<lambda>(l,u). (lb prec l, ub prec u))"
+
+lemma map_bnds:
+ assumes "(lf, uf) = map_bnds lb ub prec (l, u)"
+ assumes "mono f"
+ assumes "x \<in> {real_of_float l..real_of_float u}"
+ assumes "real_of_float (lb prec l) \<le> f (real_of_float l)"
+ assumes "real_of_float (ub prec u) \<ge> f (real_of_float u)"
+ shows "f x \<in> {real_of_float lf..real_of_float uf}"
+proof -
+ from assms have "real_of_float lf = real_of_float (lb prec l)"
+ by (simp add: map_bnds_def)
+ also have "real_of_float (lb prec l) \<le> f (real_of_float l)" by fact
+ also from assms have "\<dots> \<le> f x"
+ by (intro monoD[OF \<open>mono f\<close>]) auto
+ finally have lf: "real_of_float lf \<le> f x" .
+
+ from assms have "f x \<le> f (real_of_float u)"
+ by (intro monoD[OF \<open>mono f\<close>]) auto
+ also have "\<dots> \<le> real_of_float (ub prec u)" by fact
+ also from assms have "\<dots> = real_of_float uf"
+ by (simp add: map_bnds_def)
+ finally have uf: "f x \<le> real_of_float uf" .
+
+ from lf uf show ?thesis by simp
+qed
+
+
+section "Square root"
+
+text \<open>
+The square root computation is implemented as newton iteration. As first first step we use the
+nearest power of two greater than the square root.
+\<close>
+
+fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
+"sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" |
+"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
+ in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))"
+
+lemma compute_sqrt_iteration_base[code]:
+ shows "sqrt_iteration prec n (Float m e) =
+ (if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1)
+ else (let y = sqrt_iteration prec (n - 1) (Float m e) in
+ Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))"
+ using bitlen_Float by (cases n) simp_all
+
+function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
+"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
+ else if x < 0 then - lb_sqrt prec (- x)
+ else 0)" |
+"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
+ else if x < 0 then - ub_sqrt prec (- x)
+ else 0)"
+by pat_completeness auto
+termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
+
+declare lb_sqrt.simps[simp del]
+declare ub_sqrt.simps[simp del]
+
+lemma sqrt_ub_pos_pos_1:
+ assumes "sqrt x < b" and "0 < b" and "0 < x"
+ shows "sqrt x < (b + x / b)/2"
+proof -
+ from assms have "0 < (b - sqrt x)\<^sup>2 " by simp
+ also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra
+ also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp
+ finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" .
+ hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
+ by (simp add: field_simps power2_eq_square)
+ thus ?thesis by (simp add: field_simps)
+qed
+
+lemma sqrt_iteration_bound:
+ assumes "0 < real_of_float x"
+ shows "sqrt x < sqrt_iteration prec n x"
+proof (induct n)
+ case 0
+ show ?case
+ proof (cases x)
+ case (Float m e)
+ hence "0 < m"
+ using assms
+ apply (auto simp: sign_simps)
+ by (meson not_less powr_ge_pzero)
+ hence "0 < sqrt m" by auto
+
+ have int_nat_bl: "(nat (bitlen m)) = bitlen m"
+ using bitlen_nonneg by auto
+
+ have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))"
+ unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add)
+ also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))"
+ proof (rule mult_strict_right_mono, auto)
+ show "m < 2^nat (bitlen m)"
+ using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
+ unfolding of_int_less_iff[of m, symmetric] by auto
+ qed
+ finally have "sqrt x < sqrt (2 powr (e + bitlen m))"
+ unfolding int_nat_bl by auto
+ also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)"
+ proof -
+ let ?E = "e + bitlen m"
+ have E_mod_pow: "2 powr (?E mod 2) < 4"
+ proof (cases "?E mod 2 = 1")
+ case True
+ thus ?thesis by auto
+ next
+ case False
+ have "0 \<le> ?E mod 2" by auto
+ have "?E mod 2 < 2" by auto
+ from this[THEN zless_imp_add1_zle]
+ have "?E mod 2 \<le> 0" using False by auto
+ from xt1(5)[OF \<open>0 \<le> ?E mod 2\<close> this]
+ show ?thesis by auto
+ qed
+ hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
+ by (auto simp del: real_sqrt_four)
+ hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto
+
+ have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)"
+ by auto
+ have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
+ unfolding E_eq unfolding powr_add[symmetric] by (metis of_int_add)
+ also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
+ unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
+ also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
+ by (rule mult_strict_left_mono) (auto intro: E_mod_pow)
+ also have "\<dots> = 2 powr (?E div 2 + 1)"
+ unfolding add.commute[of _ 1] powr_add[symmetric] by simp
+ finally show ?thesis by auto
+ qed
+ finally show ?thesis using \<open>0 < m\<close>
+ unfolding Float
+ by (subst compute_sqrt_iteration_base) (simp add: ac_simps)
+ qed
+next
+ case (Suc n)
+ let ?b = "sqrt_iteration prec n x"
+ have "0 < sqrt x"
+ using \<open>0 < real_of_float x\<close> by auto
+ also have "\<dots> < real_of_float ?b"
+ using Suc .
+ finally have "sqrt x < (?b + x / ?b)/2"
+ using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto
+ also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2"
+ by (rule divide_right_mono, auto simp add: float_divr)
+ also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))"
+ by simp
+ also have "\<dots> \<le> (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))"
+ by (auto simp add: algebra_simps float_plus_up_le)
+ finally show ?case
+ unfolding sqrt_iteration.simps Let_def distrib_left .
+qed
+
+lemma sqrt_iteration_lower_bound:
+ assumes "0 < real_of_float x"
+ shows "0 < real_of_float (sqrt_iteration prec n x)" (is "0 < ?sqrt")
+proof -
+ have "0 < sqrt x" using assms by auto
+ also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
+ finally show ?thesis .
+qed
+
+lemma lb_sqrt_lower_bound:
+ assumes "0 \<le> real_of_float x"
+ shows "0 \<le> real_of_float (lb_sqrt prec x)"
+proof (cases "0 < x")
+ case True
+ hence "0 < real_of_float x" and "0 \<le> x"
+ using \<open>0 \<le> real_of_float x\<close> by auto
+ hence "0 < sqrt_iteration prec prec x"
+ using sqrt_iteration_lower_bound by auto
+ hence "0 \<le> real_of_float (float_divl prec x (sqrt_iteration prec prec x))"
+ using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto
+ thus ?thesis
+ unfolding lb_sqrt.simps using True by auto
+next
+ case False
+ with \<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto
+ thus ?thesis
+ unfolding lb_sqrt.simps by auto
+qed
+
+lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}"
+proof -
+ have lb: "lb_sqrt prec x \<le> sqrt x" if "0 < x" for x :: float
+ proof -
+ from that have "0 < real_of_float x" and "0 \<le> real_of_float x" by auto
+ hence sqrt_gt0: "0 < sqrt x" by auto
+ hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x"
+ using sqrt_iteration_bound by auto
+ have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
+ x / (sqrt_iteration prec prec x)" by (rule float_divl)
+ also have "\<dots> < x / sqrt x"
+ by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close>
+ mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
+ also have "\<dots> = sqrt x"
+ unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
+ sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto
+ finally show ?thesis
+ unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
+ qed
+ have ub: "sqrt x \<le> ub_sqrt prec x" if "0 < x" for x :: float
+ proof -
+ from that have "0 < real_of_float x" by auto
+ hence "0 < sqrt x" by auto
+ hence "sqrt x < sqrt_iteration prec prec x"
+ using sqrt_iteration_bound by auto
+ then show ?thesis
+ unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto
+ qed
+ show ?thesis
+ using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
+ by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
+qed
+
+lemma bnds_sqrt: "\<forall>(x::real) lx ux.
+ (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u"
+proof ((rule allI) +, rule impI, erule conjE, rule conjI)
+ fix x :: real
+ fix lx ux
+ assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
+ and x: "x \<in> {lx .. ux}"
+ hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
+
+ have "sqrt lx \<le> sqrt x" using x by auto
+ from order_trans[OF _ this]
+ show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
+
+ have "sqrt x \<le> sqrt ux" using x by auto
+ from order_trans[OF this]
+ show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto
+qed
+
+
+section "Arcus tangens and \<pi>"
+
+subsection "Compute arcus tangens series"
+
+text \<open>
+As first step we implement the computation of the arcus tangens series. This is only valid in the range
+@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
+\<close>
+
+fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
+ "ub_arctan_horner prec 0 k x = 0"
+| "ub_arctan_horner prec (Suc n) k x = float_plus_up prec
+ (rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))"
+| "lb_arctan_horner prec 0 k x = 0"
+| "lb_arctan_horner prec (Suc n) k x = float_plus_down prec
+ (lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))"
+
+lemma arctan_0_1_bounds':
+ assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
+ and "even n"
+ shows "arctan (sqrt y) \<in>
+ {(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}"
+proof -
+ let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))"
+ let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"
+
+ have "0 \<le> sqrt y" using assms by auto
+ have "sqrt y \<le> 1" using assms by auto
+ from \<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE)
+
+ have "arctan (sqrt y) \<in> { ?S n .. ?S (Suc n) }"
+ proof (cases "sqrt y = 0")
+ case True
+ then show ?thesis by simp
+ next
+ case False
+ hence "0 < sqrt y" using \<open>0 \<le> sqrt y\<close> by auto
+ hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)" by auto
+
+ have "\<bar> sqrt y \<bar> \<le> 1" using \<open>0 \<le> sqrt y\<close> \<open>sqrt y \<le> 1\<close> by auto
+ from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this]
+ monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded \<open>2 * m = n\<close>]
+ show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan .
+ qed
+ note arctan_bounds = this[unfolded atLeastAtMost_iff]
+
+ have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
+
+ note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
+ and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
+ and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
+ OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]
+
+ have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)"
+ proof -
+ have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n"
+ using bounds(1) \<open>0 \<le> sqrt y\<close>
+ apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
+ apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
+ apply (auto intro!: mult_left_mono)
+ done
+ also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds ..
+ finally show ?thesis .
+ qed
+ moreover
+ have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
+ proof -
+ have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds ..
+ also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
+ using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close>
+ apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
+ apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
+ apply (auto intro!: mult_left_mono)
+ done
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma arctan_0_1_bounds:
+ assumes "0 \<le> real_of_float y" "real_of_float y \<le> 1"
+ shows "arctan (sqrt y) \<in>
+ {(sqrt y * lb_arctan_horner prec (get_even n) 1 y) ..
+ (sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}"
+ using
+ arctan_0_1_bounds'[OF assms, of n prec]
+ arctan_0_1_bounds'[OF assms, of "n + 1" prec]
+ arctan_0_1_bounds'[OF assms, of "n - 1" prec]
+ by (auto simp: get_even_def get_odd_def odd_pos
+ simp del: ub_arctan_horner.simps lb_arctan_horner.simps)
+
+lemma arctan_lower_bound:
+ assumes "0 \<le> x"
+ shows "x / (1 + x\<^sup>2) \<le> arctan x" (is "?l x \<le> _")
+proof -
+ have "?l x - arctan x \<le> ?l 0 - arctan 0"
+ using assms
+ by (intro DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. ?l x - arctan x"])
+ (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps)
+ thus ?thesis by simp
+qed
+
+lemma arctan_divide_mono: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> arctan y / y \<le> arctan x / x"
+ by (rule DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. arctan x / x"])
+ (auto intro!: derivative_eq_intros divide_nonpos_nonneg
+ simp: inverse_eq_divide arctan_lower_bound)
+
+lemma arctan_mult_mono: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x * arctan y \<le> y * arctan x"
+ using arctan_divide_mono[of x y] by (cases "x = 0") (simp_all add: field_simps)
+
+lemma arctan_mult_le:
+ assumes "0 \<le> x" "x \<le> y" "y * z \<le> arctan y"
+ shows "x * z \<le> arctan x"
+proof (cases "x = 0")
+ case True
+ then show ?thesis by simp
+next
+ case False
+ with assms have "z \<le> arctan y / y" by (simp add: field_simps)
+ also have "\<dots> \<le> arctan x / x" using assms \<open>x \<noteq> 0\<close> by (auto intro!: arctan_divide_mono)
+ finally show ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps)
+qed
+
+lemma arctan_le_mult:
+ assumes "0 < x" "x \<le> y" "arctan x \<le> x * z"
+ shows "arctan y \<le> y * z"
+proof -
+ from assms have "arctan y / y \<le> arctan x / x" by (auto intro!: arctan_divide_mono)
+ also have "\<dots> \<le> z" using assms by (auto simp: field_simps)
+ finally show ?thesis using assms by (simp add: field_simps)
+qed
+
+lemma arctan_0_1_bounds_le:
+ assumes "0 \<le> x" "x \<le> 1" "0 < real_of_float xl" "real_of_float xl \<le> x * x" "x * x \<le> real_of_float xu" "real_of_float xu \<le> 1"
+ shows "arctan x \<in>
+ {x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}"
+proof -
+ from assms have "real_of_float xl \<le> 1" "sqrt (real_of_float xl) \<le> x" "x \<le> sqrt (real_of_float xu)" "0 \<le> real_of_float xu"
+ "0 \<le> real_of_float xl" "0 < sqrt (real_of_float xl)"
+ by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square)
+ from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xu\<close> \<open>real_of_float xu \<le> 1\<close>]
+ have "sqrt (real_of_float xu) * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real_of_float xu))"
+ by simp
+ from arctan_mult_le[OF \<open>0 \<le> x\<close> \<open>x \<le> sqrt _\<close> this]
+ have "x * real_of_float (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
+ moreover
+ from arctan_0_1_bounds[OF \<open>0 \<le> real_of_float xl\<close> \<open>real_of_float xl \<le> 1\<close>]
+ have "arctan (sqrt (real_of_float xl)) \<le> sqrt (real_of_float xl) * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)"
+ by simp
+ from arctan_le_mult[OF \<open>0 < sqrt xl\<close> \<open>sqrt xl \<le> x\<close> this]
+ have "arctan x \<le> x * real_of_float (ub_arctan_horner p2 (get_odd n) 1 xl)" .
+ ultimately show ?thesis by simp
+qed
+
+lemma arctan_0_1_bounds_round:
+ assumes "0 \<le> real_of_float x" "real_of_float x \<le> 1"
+ shows "arctan x \<in>
+ {real_of_float x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
+ real_of_float x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
+ using assms
+ apply (cases "x > 0")
+ apply (intro arctan_0_1_bounds_le)
+ apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq
+ intro!: truncate_up_le1 mult_le_one truncate_down_le truncate_up_le truncate_down_pos
+ mult_pos_pos)
+ done
+
+
+subsection "Compute \<pi>"
+
+definition ub_pi :: "nat \<Rightarrow> float" where
+ "ub_pi prec =
+ (let
+ A = rapprox_rat prec 1 5 ;
+ B = lapprox_rat prec 1 239
+ in ((Float 1 2) * float_plus_up prec
+ ((Float 1 2) * float_round_up prec (A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1
+ (float_round_down (Suc prec) (A * A)))))
+ (- float_round_down prec (B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1
+ (float_round_up (Suc prec) (B * B)))))))"
+
+definition lb_pi :: "nat \<Rightarrow> float" where
+ "lb_pi prec =
+ (let
+ A = lapprox_rat prec 1 5 ;
+ B = rapprox_rat prec 1 239
+ in ((Float 1 2) * float_plus_down prec
+ ((Float 1 2) * float_round_down prec (A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1
+ (float_round_up (Suc prec) (A * A)))))
+ (- float_round_up prec (B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1
+ (float_round_down (Suc prec) (B * B)))))))"
+
+lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}"
+proof -
+ have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))"
+ unfolding machin[symmetric] by auto
+
+ {
+ fix prec n :: nat
+ fix k :: int
+ assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
+ let ?k = "rapprox_rat prec 1 k"
+ let ?kl = "float_round_down (Suc prec) (?k * ?k)"
+ have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
+
+ have "0 \<le> real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \<open>0 \<le> k\<close>)
+ have "real_of_float ?k \<le> 1"
+ by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le
+ intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
+ have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
+ hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
+ also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)"
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
+ by auto
+ finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" .
+ } note ub_arctan = this
+
+ {
+ fix prec n :: nat
+ fix k :: int
+ assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
+ let ?k = "lapprox_rat prec 1 k"
+ let ?ku = "float_round_up (Suc prec) (?k * ?k)"
+ have "1 div k = 0" using div_pos_pos_trivial[OF _ \<open>1 < k\<close>] by auto
+ have "1 / k \<le> 1" using \<open>1 < k\<close> by auto
+ have "0 \<le> real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \<open>0 \<le> k\<close>]
+ by (auto simp add: \<open>1 div k = 0\<close>)
+ have "0 \<le> real_of_float (?k * ?k)" by simp
+ have "real_of_float ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: \<open>1 / k \<le> 1\<close>)
+ hence "real_of_float (?k * ?k) \<le> 1" using \<open>0 \<le> real_of_float ?k\<close> by (auto intro!: mult_le_one)
+
+ have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto
+
+ have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k"
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>]
+ by auto
+ also have "\<dots> \<le> arctan (1 / k)" using \<open>?k \<le> 1 / k\<close> by (rule arctan_monotone')
+ finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" .
+ } note lb_arctan = this
+
+ have "pi \<le> ub_pi n "
+ unfolding ub_pi_def machin_pi Let_def times_float.rep_eq Float_num
+ using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
+ by (intro mult_left_mono float_plus_up_le float_plus_down_le)
+ (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
+ moreover have "lb_pi n \<le> pi"
+ unfolding lb_pi_def machin_pi Let_def times_float.rep_eq Float_num
+ using lb_arctan[of 5] ub_arctan[of 239]
+ by (intro mult_left_mono float_plus_up_le float_plus_down_le)
+ (auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
+ ultimately show ?thesis by auto
+qed
+
+
+subsection "Compute arcus tangens in the entire domain"
+
+function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
+ "lb_arctan prec x =
+ (let
+ ub_horner = \<lambda> x. float_round_up prec
+ (x *
+ ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)));
+ lb_horner = \<lambda> x. float_round_down prec
+ (x *
+ lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))
+ in
+ if x < 0 then - ub_arctan prec (-x)
+ else if x \<le> Float 1 (- 1) then lb_horner x
+ else if x \<le> Float 1 1 then
+ Float 1 1 *
+ lb_horner
+ (float_divl prec x
+ (float_plus_up prec 1
+ (ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x))))))
+ else let inv = float_divr prec 1 x in
+ if inv > 1 then 0
+ else float_plus_down prec (lb_pi prec * Float 1 (- 1)) ( - ub_horner inv))"
+
+| "ub_arctan prec x =
+ (let
+ lb_horner = \<lambda> x. float_round_down prec
+ (x *
+ lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) ;
+ ub_horner = \<lambda> x. float_round_up prec
+ (x *
+ ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))
+ in if x < 0 then - lb_arctan prec (-x)
+ else if x \<le> Float 1 (- 1) then ub_horner x
+ else if x \<le> Float 1 1 then
+ let y = float_divr prec x
+ (float_plus_down
+ (Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x)))))
+ in if y > 1 then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y
+ else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))"
+by pat_completeness auto
+termination
+by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
+
+declare ub_arctan_horner.simps[simp del]
+declare lb_arctan_horner.simps[simp del]
+
+lemma lb_arctan_bound':
+ assumes "0 \<le> real_of_float x"
+ shows "lb_arctan prec x \<le> arctan x"
+proof -
+ have "\<not> x < 0" and "0 \<le> x"
+ using \<open>0 \<le> real_of_float x\<close> by (auto intro!: truncate_up_le )
+
+ let "?ub_horner x" =
+ "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))"
+ and "?lb_horner x" =
+ "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))"
+
+ show ?thesis
+ proof (cases "x \<le> Float 1 (- 1)")
+ case True
+ hence "real_of_float x \<le> 1" by simp
+ from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
+ show ?thesis
+ unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using \<open>0 \<le> x\<close>
+ by (auto intro!: float_round_down_le)
+ next
+ case False
+ hence "0 < real_of_float x" by auto
+ let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
+ let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))"
+ let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)"
+ let ?DIV = "float_divl prec x ?fR"
+
+ have divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
+
+ have "sqrt (1 + x*x) \<le> sqrt ?sxx"
+ by (auto simp: float_plus_up.rep_eq plus_up_def float_round_up.rep_eq intro!: truncate_up_le)
+ also have "\<dots> \<le> ub_sqrt prec ?sxx"
+ using bnds_sqrt'[of ?sxx prec] by auto
+ finally
+ have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" .
+ hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
+ hence "0 < ?fR" and "0 < real_of_float ?fR" using \<open>0 < ?R\<close> by auto
+
+ have monotone: "?DIV \<le> x / ?R"
+ proof -
+ have "?DIV \<le> real_of_float x / ?fR" by (rule float_divl)
+ also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF \<open>?R \<le> ?fR\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 \<open>?R \<le> real_of_float ?fR\<close>] divisor_gt0]])
+ finally show ?thesis .
+ qed
+
+ show ?thesis
+ proof (cases "x \<le> Float 1 1")
+ case True
+ have "x \<le> sqrt (1 + x * x)"
+ using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
+ also note \<open>\<dots> \<le> (ub_sqrt prec ?sxx)\<close>
+ finally have "real_of_float x \<le> ?fR"
+ by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
+ moreover have "?DIV \<le> real_of_float x / ?fR"
+ by (rule float_divl)
+ ultimately have "real_of_float ?DIV \<le> 1"
+ unfolding divide_le_eq_1_pos[OF \<open>0 < real_of_float ?fR\<close>, symmetric] by auto
+
+ have "0 \<le> real_of_float ?DIV"
+ using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] \<open>0 < ?fR\<close>
+ unfolding less_eq_float_def by auto
+
+ from arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float (?DIV)\<close> \<open>real_of_float (?DIV) \<le> 1\<close>]
+ have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV"
+ by simp
+ also have "\<dots> \<le> 2 * arctan (x / ?R)"
+ using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone')
+ also have "2 * arctan (x / ?R) = arctan x"
+ using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
+ finally show ?thesis
+ unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF True]
+ by (auto simp: float_round_down.rep_eq
+ intro!: order_trans[OF mult_left_mono[OF truncate_down]])
+ next
+ case False
+ hence "2 < real_of_float x" by auto
+ hence "1 \<le> real_of_float x" by auto
+
+ let "?invx" = "float_divr prec 1 x"
+ have "0 \<le> arctan x" using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>]
+ using arctan_tan[of 0, unfolded tan_zero] by auto
+
+ show ?thesis
+ proof (cases "1 < ?invx")
+ case True
+ show ?thesis
+ unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False] if_P[OF True]
+ using \<open>0 \<le> arctan x\<close> by auto
+ next
+ case False
+ hence "real_of_float ?invx \<le> 1" by auto
+ have "0 \<le> real_of_float ?invx"
+ by (rule order_trans[OF _ float_divr]) (auto simp add: \<open>0 \<le> real_of_float x\<close>)
+
+ have "1 / x \<noteq> 0" and "0 < 1 / x"
+ using \<open>0 < real_of_float x\<close> by auto
+
+ have "arctan (1 / x) \<le> arctan ?invx"
+ unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
+ also have "\<dots> \<le> ?ub_horner ?invx"
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
+ by (auto intro!: float_round_up_le)
+ also note float_round_up
+ finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x"
+ using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
+ unfolding sgn_pos[OF \<open>0 < 1 / real_of_float x\<close>] le_diff_eq by auto
+ moreover
+ have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
+ unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
+ ultimately
+ show ?thesis
+ unfolding lb_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x \<le> Float 1 1\<close>] if_not_P[OF False]
+ by (auto intro!: float_plus_down_le)
+ qed
+ qed
+ qed
+qed
+
+lemma ub_arctan_bound':
+ assumes "0 \<le> real_of_float x"
+ shows "arctan x \<le> ub_arctan prec x"
+proof -
+ have "\<not> x < 0" and "0 \<le> x"
+ using \<open>0 \<le> real_of_float x\<close> by auto
+
+ let "?ub_horner x" =
+ "float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))"
+ let "?lb_horner x" =
+ "float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))"
+
+ show ?thesis
+ proof (cases "x \<le> Float 1 (- 1)")
+ case True
+ hence "real_of_float x \<le> 1" by auto
+ show ?thesis
+ unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True]
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>]
+ by (auto intro!: float_round_up_le)
+ next
+ case False
+ hence "0 < real_of_float x" by auto
+ let ?R = "1 + sqrt (1 + real_of_float x * real_of_float x)"
+ let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))"
+ let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)"
+ let ?DIV = "float_divr prec x ?fR"
+
+ have sqr_ge0: "0 \<le> 1 + real_of_float x * real_of_float x"
+ using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto
+ hence "0 \<le> real_of_float (1 + x*x)" by auto
+
+ hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
+
+ have "lb_sqrt prec ?sxx \<le> sqrt ?sxx"
+ using bnds_sqrt'[of ?sxx] by auto
+ also have "\<dots> \<le> sqrt (1 + x*x)"
+ by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le)
+ finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" .
+ hence "?fR \<le> ?R"
+ by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le)
+ have "0 < real_of_float ?fR"
+ by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq
+ intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one]
+ truncate_down_nonneg add_nonneg_nonneg)
+ have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
+ proof -
+ from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]]
+ have "x / ?R \<le> x / ?fR" .
+ also have "\<dots> \<le> ?DIV" by (rule float_divr)
+ finally show ?thesis .
+ qed
+
+ show ?thesis
+ proof (cases "x \<le> Float 1 1")
+ case True
+ show ?thesis
+ proof (cases "?DIV > 1")
+ case True
+ have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
+ unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto
+ from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
+ show ?thesis
+ unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] .
+ next
+ case False
+ hence "real_of_float ?DIV \<le> 1" by auto
+
+ have "0 \<le> x / ?R"
+ using \<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto
+ hence "0 \<le> real_of_float ?DIV"
+ using monotone by (rule order_trans)
+
+ have "arctan x = 2 * arctan (x / ?R)"
+ using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
+ also have "\<dots> \<le> 2 * arctan (?DIV)"
+ using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
+ also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>]
+ by (auto intro!: float_round_up_le)
+ finally show ?thesis
+ unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_not_P[OF False] .
+ qed
+ next
+ case False
+ hence "2 < real_of_float x" by auto
+ hence "1 \<le> real_of_float x" by auto
+ hence "0 < real_of_float x" by auto
+ hence "0 < x" by auto
+
+ let "?invx" = "float_divl prec 1 x"
+ have "0 \<le> arctan x"
+ using arctan_monotone'[OF \<open>0 \<le> real_of_float x\<close>] and arctan_tan[of 0, unfolded tan_zero] by auto
+
+ have "real_of_float ?invx \<le> 1"
+ unfolding less_float_def
+ by (rule order_trans[OF float_divl])
+ (auto simp add: \<open>1 \<le> real_of_float x\<close> divide_le_eq_1_pos[OF \<open>0 < real_of_float x\<close>])
+ have "0 \<le> real_of_float ?invx"
+ using \<open>0 < x\<close> by (intro float_divl_lower_bound) auto
+
+ have "1 / x \<noteq> 0" and "0 < 1 / x"
+ using \<open>0 < real_of_float x\<close> by auto
+
+ have "(?lb_horner ?invx) \<le> arctan (?invx)"
+ using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?invx\<close> \<open>real_of_float ?invx \<le> 1\<close>]
+ by (auto intro!: float_round_down_le)
+ also have "\<dots> \<le> arctan (1 / x)"
+ unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone') (rule float_divl)
+ finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)"
+ using \<open>0 \<le> arctan x\<close> arctan_inverse[OF \<open>1 / x \<noteq> 0\<close>]
+ unfolding sgn_pos[OF \<open>0 < 1 / x\<close>] le_diff_eq by auto
+ moreover
+ have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)"
+ unfolding Float_num times_divide_eq_right mult_1_right
+ using pi_boundaries by auto
+ ultimately
+ show ?thesis
+ unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_not_P[OF False]
+ by (auto intro!: float_round_up_le float_plus_up_le)
+ qed
+ qed
+qed
+
+lemma arctan_boundaries: "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
+proof (cases "0 \<le> x")
+ case True
+ hence "0 \<le> real_of_float x" by auto
+ show ?thesis
+ using ub_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float x\<close>]
+ unfolding atLeastAtMost_iff by auto
+next
+ case False
+ let ?mx = "-x"
+ from False have "x < 0" and "0 \<le> real_of_float ?mx"
+ by auto
+ hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx"
+ using ub_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] lb_arctan_bound'[OF \<open>0 \<le> real_of_float ?mx\<close>] by auto
+ show ?thesis
+ unfolding minus_float.rep_eq arctan_minus lb_arctan.simps[where x=x]
+ ub_arctan.simps[where x=x] Let_def if_P[OF \<open>x < 0\<close>]
+ unfolding atLeastAtMost_iff using bounds[unfolded minus_float.rep_eq arctan_minus]
+ by (simp add: arctan_minus)
+qed
+
+lemma bnds_arctan: "\<forall> (x::real) lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> arctan x \<and> arctan x \<le> u"
+proof (rule allI, rule allI, rule allI, rule impI)
+ fix x :: real
+ fix lx ux
+ assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}"
+ hence l: "lb_arctan prec lx = l "
+ and u: "ub_arctan prec ux = u"
+ and x: "x \<in> {lx .. ux}"
+ by auto
+ show "l \<le> arctan x \<and> arctan x \<le> u"
+ proof
+ show "l \<le> arctan x"
+ proof -
+ from arctan_boundaries[of lx prec, unfolded l]
+ have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps)
+ also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
+ finally show ?thesis .
+ qed
+ show "arctan x \<le> u"
+ proof -
+ have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone')
+ also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+
+section "Sinus and Cosinus"
+
+subsection "Compute the cosinus and sinus series"
+
+fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
+ "ub_sin_cos_aux prec 0 i k x = 0"
+| "ub_sin_cos_aux prec (Suc n) i k x = float_plus_up prec
+ (rapprox_rat prec 1 k) (-
+ float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
+| "lb_sin_cos_aux prec 0 i k x = 0"
+| "lb_sin_cos_aux prec (Suc n) i k x = float_plus_down prec
+ (lapprox_rat prec 1 k) (-
+ float_round_up prec (x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
+
+lemma cos_aux:
+ shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb")
+ and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
+proof -
+ have "0 \<le> real_of_float (x * x)" by auto
+ let "?f n" = "fact (2 * n) :: nat"
+ have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)" for n
+ proof -
+ have "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
+ then show ?thesis by auto
+ qed
+ from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
+ OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+ show ?lb and ?ub
+ by (auto simp add: power_mult power2_eq_square[of "real_of_float x"])
+qed
+
+lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1"
+ by (cases j n rule: nat.exhaust[case_product nat.exhaust])
+ (auto intro!: float_plus_down_le order_trans[OF lapprox_rat])
+
+lemma one_le_ub_sin_cos_aux: "odd n \<Longrightarrow> 1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0"
+ by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat])
+
+lemma cos_boundaries:
+ assumes "0 \<le> real_of_float x" and "x \<le> pi / 2"
+ shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
+proof (cases "real_of_float x = 0")
+ case False
+ hence "real_of_float x \<noteq> 0" by auto
+ hence "0 < x" and "0 < real_of_float x"
+ using \<open>0 \<le> real_of_float x\<close> by auto
+ have "0 < x * x"
+ using \<open>0 < x\<close> by simp
+
+ have morph_to_if_power: "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i)) =
+ (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)"
+ (is "?sum = ?ifsum") for x n
+ proof -
+ have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
+ also have "\<dots> =
+ (\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
+ also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)"
+ unfolding sum_split_even_odd atLeast0LessThan ..
+ also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)"
+ by (rule sum.cong) auto
+ finally show ?thesis .
+ qed
+
+ { fix n :: nat assume "0 < n"
+ hence "0 < 2 * n" by auto
+ obtain t where "0 < t" and "t < real_of_float x" and
+ cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i)
+ + (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)"
+ (is "_ = ?SUM + ?rest / ?fact * ?pow")
+ using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>]
+ unfolding cos_coeff_def atLeast0LessThan by auto
+
+ have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
+ also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add)
+ also have "\<dots> = ?rest" by auto
+ finally have "cos t * (- 1) ^ n = ?rest" .
+ moreover
+ have "t \<le> pi / 2" using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
+ hence "0 \<le> cos t" using \<open>0 < t\<close> and cos_ge_zero by auto
+ ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
+
+ have "0 < ?fact" by auto
+ have "0 < ?pow" using \<open>0 < real_of_float x\<close> by auto
+
+ {
+ assume "even n"
+ have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
+ unfolding morph_to_if_power[symmetric] using cos_aux by auto
+ also have "\<dots> \<le> cos x"
+ proof -
+ from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
+ have "0 \<le> (?rest / ?fact) * ?pow" by simp
+ thus ?thesis unfolding cos_eq by auto
+ qed
+ finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" .
+ } note lb = this
+
+ {
+ assume "odd n"
+ have "cos x \<le> ?SUM"
+ proof -
+ from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
+ have "0 \<le> (- ?rest) / ?fact * ?pow"
+ by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
+ thus ?thesis unfolding cos_eq by auto
+ qed
+ also have "\<dots> \<le> (ub_sin_cos_aux prec n 1 1 (x * x))"
+ unfolding morph_to_if_power[symmetric] using cos_aux by auto
+ finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" .
+ } note ub = this and lb
+ } note ub = this(1) and lb = this(2)
+
+ have "cos x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))"
+ using ub[OF odd_pos[OF get_odd] get_odd] .
+ moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x"
+ proof (cases "0 < get_even n")
+ case True
+ show ?thesis using lb[OF True get_even] .
+ next
+ case False
+ hence "get_even n = 0" by auto
+ have "- (pi / 2) \<le> x"
+ by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto
+ with \<open>x \<le> pi / 2\<close> show ?thesis
+ unfolding \<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq
+ using cos_ge_zero by auto
+ qed
+ ultimately show ?thesis by auto
+next
+ case True
+ hence "x = 0"
+ by transfer
+ thus ?thesis
+ using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux
+ by simp
+qed
+
+lemma sin_aux:
+ assumes "0 \<le> real_of_float x"
+ shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
+ (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
+ and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le>
+ (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
+proof -
+ have "0 \<le> real_of_float (x * x)" by auto
+ let "?f n" = "fact (2 * n + 1) :: nat"
+ have f_eq: "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)" for n
+ proof -
+ have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
+ show ?thesis
+ unfolding F by auto
+ qed
+ from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
+ OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+ show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close>
+ apply (simp_all only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric])
+ apply (simp_all only: mult.commute[where 'a=real] of_nat_fact)
+ apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"])
+ done
+qed
+
+lemma sin_boundaries:
+ assumes "0 \<le> real_of_float x"
+ and "x \<le> pi / 2"
+ shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
+proof (cases "real_of_float x = 0")
+ case False
+ hence "real_of_float x \<noteq> 0" by auto
+ hence "0 < x" and "0 < real_of_float x"
+ using \<open>0 \<le> real_of_float x\<close> by auto
+ have "0 < x * x"
+ using \<open>0 < x\<close> by simp
+
+ have sum_morph: "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((fact (2 * j + 1))) * x ^(2 * j + 1)) =
+ (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * x ^ i)"
+ (is "?SUM = _") for x :: real and n
+ proof -
+ have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)"
+ by auto
+ have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM"
+ by auto
+ also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)"
+ unfolding sum_split_even_odd atLeast0LessThan ..
+ also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)"
+ by (rule sum.cong) auto
+ finally show ?thesis .
+ qed
+
+ { fix n :: nat assume "0 < n"
+ hence "0 < 2 * n + 1" by auto
+ obtain t where "0 < t" and "t < real_of_float x" and
+ sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)
+ + (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real_of_float x)^(2*n + 1)"
+ (is "_ = ?SUM + ?rest / ?fact * ?pow")
+ using Maclaurin_sin_expansion3[OF \<open>0 < 2 * n + 1\<close> \<open>0 < real_of_float x\<close>]
+ unfolding sin_coeff_def atLeast0LessThan by auto
+
+ have "?rest = cos t * (- 1) ^ n"
+ unfolding sin_add cos_add of_nat_add distrib_right distrib_left by auto
+ moreover
+ have "t \<le> pi / 2"
+ using \<open>t < real_of_float x\<close> and \<open>x \<le> pi / 2\<close> by auto
+ hence "0 \<le> cos t"
+ using \<open>0 < t\<close> and cos_ge_zero by auto
+ ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest"
+ by auto
+
+ have "0 < ?fact"
+ by (simp del: fact_Suc)
+ have "0 < ?pow"
+ using \<open>0 < real_of_float x\<close> by (rule zero_less_power)
+
+ {
+ assume "even n"
+ have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
+ (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
+ using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
+ also have "\<dots> \<le> ?SUM" by auto
+ also have "\<dots> \<le> sin x"
+ proof -
+ from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close>
+ have "0 \<le> (?rest / ?fact) * ?pow" by simp
+ thus ?thesis unfolding sin_eq by auto
+ qed
+ finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" .
+ } note lb = this
+
+ {
+ assume "odd n"
+ have "sin x \<le> ?SUM"
+ proof -
+ from \<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>]
+ have "0 \<le> (- ?rest) / ?fact * ?pow"
+ by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
+ thus ?thesis unfolding sin_eq by auto
+ qed
+ also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real_of_float x) ^ i)"
+ by auto
+ also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
+ using sin_aux[OF \<open>0 \<le> real_of_float x\<close>] unfolding sum_morph[symmetric] by auto
+ finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
+ } note ub = this and lb
+ } note ub = this(1) and lb = this(2)
+
+ have "sin x \<le> (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))"
+ using ub[OF odd_pos[OF get_odd] get_odd] .
+ moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x"
+ proof (cases "0 < get_even n")
+ case True
+ show ?thesis
+ using lb[OF True get_even] .
+ next
+ case False
+ hence "get_even n = 0" by auto
+ with \<open>x \<le> pi / 2\<close> \<open>0 \<le> real_of_float x\<close>
+ show ?thesis
+ unfolding \<open>get_even n = 0\<close> ub_sin_cos_aux.simps minus_float.rep_eq
+ using sin_ge_zero by auto
+ qed
+ ultimately show ?thesis by auto
+next
+ case True
+ show ?thesis
+ proof (cases "n = 0")
+ case True
+ thus ?thesis
+ unfolding \<open>n = 0\<close> get_even_def get_odd_def
+ using \<open>real_of_float x = 0\<close> lapprox_rat[where x="-1" and y=1] by auto
+ next
+ case False
+ with not0_implies_Suc obtain m where "n = Suc m" by blast
+ thus ?thesis
+ unfolding \<open>n = Suc m\<close> get_even_def get_odd_def
+ using \<open>real_of_float x = 0\<close> rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1]
+ by (cases "even (Suc m)") auto
+ qed
+qed
+
+
+subsection "Compute the cosinus in the entire domain"
+
+definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
+"lb_cos prec x = (let
+ horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
+ half = \<lambda> x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)
+ in if x < Float 1 (- 1) then horner x
+else if x < 1 then half (horner (x * Float 1 (- 1)))
+ else half (half (horner (x * Float 1 (- 2)))))"
+
+definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
+"ub_cos prec x = (let
+ horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
+ half = \<lambda> x. float_plus_up prec (Float 1 1 * x * x) (- 1)
+ in if x < Float 1 (- 1) then horner x
+else if x < 1 then half (horner (x * Float 1 (- 1)))
+ else half (half (horner (x * Float 1 (- 2)))))"
+
+lemma lb_cos:
+ assumes "0 \<le> real_of_float x" and "x \<le> pi"
+ shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
+proof -
+ have x_half[symmetric]: "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" for x :: real
+ proof -
+ have "cos x = cos (x / 2 + x / 2)"
+ by auto
+ also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
+ unfolding cos_add by auto
+ also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1"
+ by algebra
+ finally show ?thesis .
+ qed
+
+ have "\<not> x < 0" using \<open>0 \<le> real_of_float x\<close> by auto
+ let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
+ let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
+ let "?ub_half x" = "float_plus_up prec (Float 1 1 * x * x) (- 1)"
+ let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)"
+
+ show ?thesis
+ proof (cases "x < Float 1 (- 1)")
+ case True
+ hence "x \<le> pi / 2"
+ using pi_ge_two by auto
+ show ?thesis
+ unfolding lb_cos_def[where x=x] ub_cos_def[where x=x]
+ if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF \<open>x < Float 1 (- 1)\<close>] Let_def
+ using cos_boundaries[OF \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi / 2\<close>] .
+ next
+ case False
+ { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
+ assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi"
+ hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
+ using pi_ge_two unfolding Float_num by auto
+ hence "0 \<le> cos ?x2"
+ by (rule cos_ge_zero)
+
+ have "(?lb_half y) \<le> cos x"
+ proof (cases "y < 0")
+ case True
+ show ?thesis
+ using cos_ge_minus_one unfolding if_P[OF True] by auto
+ next
+ case False
+ hence "0 \<le> real_of_float y" by auto
+ from mult_mono[OF \<open>y \<le> cos ?x2\<close> \<open>y \<le> cos ?x2\<close> \<open>0 \<le> cos ?x2\<close> this]
+ have "real_of_float y * real_of_float y \<le> cos ?x2 * cos ?x2" .
+ hence "2 * real_of_float y * real_of_float y \<le> 2 * cos ?x2 * cos ?x2"
+ by auto
+ hence "2 * real_of_float y * real_of_float y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1"
+ unfolding Float_num by auto
+ thus ?thesis
+ unfolding if_not_P[OF False] x_half Float_num
+ by (auto intro!: float_plus_down_le)
+ qed
+ } note lb_half = this
+
+ { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
+ assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi"
+ hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2"
+ using pi_ge_two unfolding Float_num by auto
+ hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
+
+ have "cos x \<le> (?ub_half y)"
+ proof -
+ have "0 \<le> real_of_float y"
+ using \<open>0 \<le> cos ?x2\<close> ub by (rule order_trans)
+ from mult_mono[OF ub ub this \<open>0 \<le> cos ?x2\<close>]
+ have "cos ?x2 * cos ?x2 \<le> real_of_float y * real_of_float y" .
+ hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real_of_float y * real_of_float y"
+ by auto
+ hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real_of_float y * real_of_float y - 1"
+ unfolding Float_num by auto
+ thus ?thesis
+ unfolding x_half Float_num
+ by (auto intro!: float_plus_up_le)
+ qed
+ } note ub_half = this
+
+ let ?x2 = "x * Float 1 (- 1)"
+ let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"
+
+ have "-pi \<le> x"
+ using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] \<open>0 \<le> real_of_float x\<close>
+ by (rule order_trans)
+
+ show ?thesis
+ proof (cases "x < 1")
+ case True
+ hence "real_of_float x \<le> 1" by auto
+ have "0 \<le> real_of_float ?x2" and "?x2 \<le> pi / 2"
+ using pi_ge_two \<open>0 \<le> real_of_float x\<close> using assms by auto
+ from cos_boundaries[OF this]
+ have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)"
+ by auto
+
+ have "(?lb x) \<le> ?cos x"
+ proof -
+ from lb_half[OF lb \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
+ show ?thesis
+ unfolding lb_cos_def[where x=x] Let_def
+ using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
+ qed
+ moreover have "?cos x \<le> (?ub x)"
+ proof -
+ from ub_half[OF ub \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>]
+ show ?thesis
+ unfolding ub_cos_def[where x=x] Let_def
+ using \<open>\<not> x < 0\<close> \<open>\<not> x < Float 1 (- 1)\<close> \<open>x < 1\<close> by auto
+ qed
+ ultimately show ?thesis by auto
+ next
+ case False
+ have "0 \<le> real_of_float ?x4" and "?x4 \<le> pi / 2"
+ using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> unfolding Float_num by auto
+ from cos_boundaries[OF this]
+ have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)"
+ by auto
+
+ have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)"
+ by transfer simp
+
+ have "(?lb x) \<le> ?cos x"
+ proof -
+ have "-pi \<le> ?x2" and "?x2 \<le> pi"
+ using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open>x \<le> pi\<close> by auto
+ from lb_half[OF lb_half[OF lb this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
+ show ?thesis
+ unfolding lb_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
+ qed
+ moreover have "?cos x \<le> (?ub x)"
+ proof -
+ have "-pi \<le> ?x2" and "?x2 \<le> pi"
+ using pi_ge_two \<open>0 \<le> real_of_float x\<close> \<open> x \<le> pi\<close> by auto
+ from ub_half[OF ub_half[OF ub this] \<open>-pi \<le> x\<close> \<open>x \<le> pi\<close>, unfolded eq_4]
+ show ?thesis
+ unfolding ub_cos_def[where x=x] if_not_P[OF \<open>\<not> x < 0\<close>]
+ if_not_P[OF \<open>\<not> x < Float 1 (- 1)\<close>] if_not_P[OF \<open>\<not> x < 1\<close>] Let_def .
+ qed
+ ultimately show ?thesis by auto
+ qed
+ qed
+qed
+
+lemma lb_cos_minus:
+ assumes "-pi \<le> x"
+ and "real_of_float x \<le> 0"
+ shows "cos (real_of_float(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
+proof -
+ have "0 \<le> real_of_float (-x)" and "(-x) \<le> pi"
+ using \<open>-pi \<le> x\<close> \<open>real_of_float x \<le> 0\<close> by auto
+ from lb_cos[OF this] show ?thesis .
+qed
+
+definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
+"bnds_cos prec lx ux = (let
+ lpi = float_round_down prec (lb_pi prec) ;
+ upi = float_round_up prec (ub_pi prec) ;
+ k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
+ lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)) ;
+ ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi))
+ in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux))
+ else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx)
+ else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
+ else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
+ else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
+ else (Float (- 1) 0, Float 1 0))"
+
+lemma floor_int: obtains k :: int where "real_of_int k = (floor_fl f)"
+ by (simp add: floor_fl_def)
+
+lemma cos_periodic_nat[simp]:
+ fixes n :: nat
+ shows "cos (x + n * (2 * pi)) = cos x"
+proof (induct n arbitrary: x)
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
+ unfolding Suc_eq_plus1 of_nat_add of_int_1 distrib_right by auto
+ show ?case
+ unfolding split_pi_off using Suc by auto
+qed
+
+lemma cos_periodic_int[simp]:
+ fixes i :: int
+ shows "cos (x + i * (2 * pi)) = cos x"
+proof (cases "0 \<le> i")
+ case True
+ hence i_nat: "real_of_int i = nat i" by auto
+ show ?thesis
+ unfolding i_nat by auto
+next
+ case False
+ hence i_nat: "i = - real (nat (-i))" by auto
+ have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))"
+ by auto
+ also have "\<dots> = cos (x + i * (2 * pi))"
+ unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
+ finally show ?thesis by auto
+qed
+
+lemma bnds_cos: "\<forall>(x::real) lx ux. (l, u) =
+ bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u"
+proof (rule allI | rule impI | erule conjE)+
+ fix x :: real
+ fix lx ux
+ assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}"
+
+ let ?lpi = "float_round_down prec (lb_pi prec)"
+ let ?upi = "float_round_up prec (ub_pi prec)"
+ let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
+ let ?lx2 = "(- ?k * 2 * (if ?k < 0 then ?lpi else ?upi))"
+ let ?ux2 = "(- ?k * 2 * (if ?k < 0 then ?upi else ?lpi))"
+ let ?lx = "float_plus_down prec lx ?lx2"
+ let ?ux = "float_plus_up prec ux ?ux2"
+
+ obtain k :: int where k: "k = real_of_float ?k"
+ by (rule floor_int)
+
+ have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
+ using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
+ float_round_down[of prec "lb_pi prec"]
+ by auto
+ hence "lx + ?lx2 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ux + ?ux2"
+ using x
+ by (cases "k = 0")
+ (auto intro!: add_mono
+ simp add: k [symmetric] uminus_add_conv_diff [symmetric]
+ simp del: float_of_numeral uminus_add_conv_diff)
+ hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
+ by (auto intro!: float_plus_down_le float_plus_up_le)
+ note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
+ hence lx_less_ux: "?lx \<le> real_of_float ?ux" by (rule order_trans)
+
+ { assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
+ with lpi[THEN le_imp_neg_le] lx
+ have pi_lx: "- pi \<le> ?lx" and lx_0: "real_of_float ?lx \<le> 0"
+ by simp_all
+
+ have "(lb_cos prec (- ?lx)) \<le> cos (real_of_float (- ?lx))"
+ using lb_cos_minus[OF pi_lx lx_0] by simp
+ also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
+ using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
+ by (simp only: uminus_float.rep_eq of_int_minus
+ cos_minus mult_minus_left) simp
+ finally have "(lb_cos prec (- ?lx)) \<le> cos x"
+ unfolding cos_periodic_int . }
+ note negative_lx = this
+
+ { assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
+ with lx
+ have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real_of_float ?lx"
+ by auto
+
+ have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
+ using cos_monotone_0_pi_le[OF lx_0 lx pi_x]
+ by (simp only: of_int_minus
+ cos_minus mult_minus_left) simp
+ also have "\<dots> \<le> (ub_cos prec ?lx)"
+ using lb_cos[OF lx_0 pi_lx] by simp
+ finally have "cos x \<le> (ub_cos prec ?lx)"
+ unfolding cos_periodic_int . }
+ note positive_lx = this
+
+ { assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
+ with ux
+ have pi_ux: "- pi \<le> ?ux" and ux_0: "real_of_float ?ux \<le> 0"
+ by simp_all
+
+ have "cos (x + (-k) * (2 * pi)) \<le> cos (real_of_float (- ?ux))"
+ using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
+ by (simp only: uminus_float.rep_eq of_int_minus
+ cos_minus mult_minus_left) simp
+ also have "\<dots> \<le> (ub_cos prec (- ?ux))"
+ using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
+ finally have "cos x \<le> (ub_cos prec (- ?ux))"
+ unfolding cos_periodic_int . }
+ note negative_ux = this
+
+ { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
+ with lpi ux
+ have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real_of_float ?ux"
+ by simp_all
+
+ have "(lb_cos prec ?ux) \<le> cos ?ux"
+ using lb_cos[OF ux_0 pi_ux] by simp
+ also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
+ using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux]
+ by (simp only: of_int_minus
+ cos_minus mult_minus_left) simp
+ finally have "(lb_cos prec ?ux) \<le> cos x"
+ unfolding cos_periodic_int . }
+ note positive_ux = this
+
+ show "l \<le> cos x \<and> cos x \<le> u"
+ proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
+ case True
+ with bnds have l: "l = lb_cos prec (-?lx)" and u: "u = ub_cos prec (-?ux)"
+ by (auto simp add: bnds_cos_def Let_def)
+ from True lpi[THEN le_imp_neg_le] lx ux
+ have "- pi \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> 0"
+ by auto
+ with True negative_ux negative_lx show ?thesis
+ unfolding l u by simp
+ next
+ case 1: False
+ show ?thesis
+ proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
+ case True with bnds 1
+ have l: "l = lb_cos prec ?ux"
+ and u: "u = ub_cos prec ?lx"
+ by (auto simp add: bnds_cos_def Let_def)
+ from True lpi lx ux
+ have "0 \<le> x - k * (2 * pi)" and "x - k * (2 * pi) \<le> pi"
+ by auto
+ with True positive_ux positive_lx show ?thesis
+ unfolding l u by simp
+ next
+ case 2: False
+ show ?thesis
+ proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
+ case Cond: True
+ with bnds 1 2 have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
+ and u: "u = Float 1 0"
+ by (auto simp add: bnds_cos_def Let_def)
+ show ?thesis
+ unfolding u l using negative_lx positive_ux Cond
+ by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min)
+ next
+ case 3: False
+ show ?thesis
+ proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
+ case Cond: True
+ with bnds 1 2 3
+ have l: "l = Float (- 1) 0"
+ and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
+ by (auto simp add: bnds_cos_def Let_def)
+
+ have "cos x \<le> real_of_float u"
+ proof (cases "x - k * (2 * pi) < pi")
+ case True
+ hence "x - k * (2 * pi) \<le> pi" by simp
+ from positive_lx[OF Cond[THEN conjunct1] this] show ?thesis
+ unfolding u by (simp add: real_of_float_max)
+ next
+ case False
+ hence "pi \<le> x - k * (2 * pi)" by simp
+ hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp
+
+ have "?ux \<le> 2 * pi"
+ using Cond lpi by auto
+ hence "x - k * (2 * pi) - 2 * pi \<le> 0"
+ using ux by simp
+
+ have ux_0: "real_of_float (?ux - 2 * ?lpi) \<le> 0"
+ using Cond by auto
+
+ from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
+ hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto
+ hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)"
+ using lpi[THEN le_imp_neg_le] by auto
+
+ have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)"
+ using ux lpi by auto
+ have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))"
+ unfolding cos_periodic_int ..
+ also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
+ using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
+ by (simp only: minus_float.rep_eq of_int_minus of_int_1
+ mult_minus_left mult_1_left) simp
+ also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
+ unfolding uminus_float.rep_eq cos_minus ..
+ also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
+ using lb_cos_minus[OF pi_ux ux_0] by simp
+ finally show ?thesis unfolding u by (simp add: real_of_float_max)
+ qed
+ thus ?thesis unfolding l by auto
+ next
+ case 4: False
+ show ?thesis
+ proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
+ case Cond: True
+ with bnds 1 2 3 4 have l: "l = Float (- 1) 0"
+ and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
+ by (auto simp add: bnds_cos_def Let_def)
+
+ have "cos x \<le> u"
+ proof (cases "-pi < x - k * (2 * pi)")
+ case True
+ hence "-pi \<le> x - k * (2 * pi)" by simp
+ from negative_ux[OF this Cond[THEN conjunct2]] show ?thesis
+ unfolding u by (simp add: real_of_float_max)
+ next
+ case False
+ hence "x - k * (2 * pi) \<le> -pi" by simp
+ hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp
+
+ have "-2 * pi \<le> ?lx" using Cond lpi by auto
+
+ hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp
+
+ have lx_0: "0 \<le> real_of_float (?lx + 2 * ?lpi)"
+ using Cond lpi by auto
+
+ from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
+ hence "?lx + 2 * ?lpi \<le> ?lpi" by auto
+ hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi"
+ using lpi[THEN le_imp_neg_le] by auto
+
+ have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi"
+ using lx lpi by auto
+
+ have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))"
+ unfolding cos_periodic_int ..
+ also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
+ using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x]
+ by (simp only: minus_float.rep_eq of_int_minus of_int_1
+ mult_minus_left mult_1_left) simp
+ also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
+ using lb_cos[OF lx_0 pi_lx] by simp
+ finally show ?thesis unfolding u by (simp add: real_of_float_max)
+ qed
+ thus ?thesis unfolding l by auto
+ next
+ case False
+ with bnds 1 2 3 4 show ?thesis
+ by (auto simp add: bnds_cos_def Let_def)
+ qed
+ qed
+ qed
+ qed
+ qed
+qed
+
+
+section "Exponential function"
+
+subsection "Compute the series of the exponential function"
+
+fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+ and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+where
+"ub_exp_horner prec 0 i k x = 0" |
+"ub_exp_horner prec (Suc n) i k x = float_plus_up prec
+ (rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" |
+"lb_exp_horner prec 0 i k x = 0" |
+"lb_exp_horner prec (Suc n) i k x = float_plus_down prec
+ (lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))"
+
+lemma bnds_exp_horner:
+ assumes "real_of_float x \<le> 0"
+ shows "exp x \<in> {lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x}"
+proof -
+ have f_eq: "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" for n
+ proof -
+ have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m"
+ by (induct n) auto
+ show ?thesis
+ unfolding F by auto
+ qed
+
+ note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
+ OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
+
+ have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x"
+ proof -
+ have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real_of_float x ^ j)"
+ using bounds(1) by auto
+ also have "\<dots> \<le> exp x"
+ proof -
+ obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real_of_float x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
+ using Maclaurin_exp_le unfolding atLeast0LessThan by blast
+ moreover have "0 \<le> exp t / (fact (get_even n)) * (real_of_float x) ^ (get_even n)"
+ by (auto simp: zero_le_even_power)
+ ultimately show ?thesis using get_odd exp_gt_zero by auto
+ qed
+ finally show ?thesis .
+ qed
+ moreover
+ have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x"
+ proof -
+ have x_less_zero: "real_of_float x ^ get_odd n \<le> 0"
+ proof (cases "real_of_float x = 0")
+ case True
+ have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
+ thus ?thesis unfolding True power_0_left by auto
+ next
+ case False hence "real_of_float x < 0" using \<open>real_of_float x \<le> 0\<close> by auto
+ show ?thesis by (rule less_imp_le, auto simp add: \<open>real_of_float x < 0\<close>)
+ qed
+ obtain t where "\<bar>t\<bar> \<le> \<bar>real_of_float x\<bar>"
+ and "exp x = (\<Sum>m = 0..<get_odd n. (real_of_float x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n)"
+ using Maclaurin_exp_le unfolding atLeast0LessThan by blast
+ moreover have "exp t / (fact (get_odd n)) * (real_of_float x) ^ (get_odd n) \<le> 0"
+ by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero)
+ ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real_of_float x ^ j)"
+ using get_odd exp_gt_zero by auto
+ also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
+ using bounds(2) by auto
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma ub_exp_horner_nonneg: "real_of_float x \<le> 0 \<Longrightarrow>
+ 0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
+ using bnds_exp_horner[of x prec n]
+ by (intro order_trans[OF exp_ge_zero]) auto
+
+
+subsection "Compute the exponential function on the entire domain"
+
+function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
+"lb_exp prec x =
+ (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
+ else
+ let
+ horner = (\<lambda> x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in
+ if y \<le> 0 then Float 1 (- 2) else y)
+ in
+ if x < - 1 then
+ power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x))
+ else horner x)" |
+"ub_exp prec x =
+ (if 0 < x then float_divr prec 1 (lb_exp prec (-x))
+ else if x < - 1 then
+ power_up_fl prec
+ (ub_exp_horner prec (get_odd (prec + 2)) 1 1
+ (float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x))
+ else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
+ by pat_completeness auto
+termination
+ by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))") auto
+
+lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
+proof -
+ have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
+ have "1 / 4 = (Float 1 (- 2))"
+ unfolding Float_num by auto
+ also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)"
+ by (subst less_eq_float.rep_eq [symmetric]) code_simp
+ also have "\<dots> \<le> exp (- 1 :: float)"
+ using bnds_exp_horner[where x="- 1"] by auto
+ finally show ?thesis
+ by simp
+qed
+
+lemma lb_exp_pos:
+ assumes "\<not> 0 < x"
+ shows "0 < lb_exp prec x"
+proof -
+ let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
+ let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 (- 2) else y"
+ have pos_horner: "0 < ?horner x" for x
+ unfolding Let_def by (cases "?lb_horner x \<le> 0") auto
+ moreover have "0 < real_of_float ((?horner x) ^ num)" for x :: float and num :: nat
+ proof -
+ have "0 < real_of_float (?horner x) ^ num" using \<open>0 < ?horner x\<close> by simp
+ also have "\<dots> = (?horner x) ^ num" by auto
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis
+ unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] Let_def
+ by (cases "floor_fl x", cases "x < - 1")
+ (auto simp: real_power_up_fl real_power_down_fl intro!: power_up_less power_down_pos)
+qed
+
+lemma exp_boundaries':
+ assumes "x \<le> 0"
+ shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}"
+proof -
+ let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
+ let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
+
+ have "real_of_float x \<le> 0" and "\<not> x > 0"
+ using \<open>x \<le> 0\<close> by auto
+ show ?thesis
+ proof (cases "x < - 1")
+ case False
+ hence "- 1 \<le> real_of_float x" by auto
+ show ?thesis
+ proof (cases "?lb_exp_horner x \<le> 0")
+ case True
+ from \<open>\<not> x < - 1\<close>
+ have "- 1 \<le> real_of_float x" by auto
+ hence "exp (- 1) \<le> exp x"
+ unfolding exp_le_cancel_iff .
+ from order_trans[OF exp_m1_ge_quarter this] have "Float 1 (- 2) \<le> exp x"
+ unfolding Float_num .
+ with True show ?thesis
+ using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by auto
+ next
+ case False
+ thus ?thesis
+ using bnds_exp_horner \<open>real_of_float x \<le> 0\<close> \<open>\<not> x > 0\<close> \<open>\<not> x < - 1\<close> by (auto simp add: Let_def)
+ qed
+ next
+ case True
+ let ?num = "nat (- int_floor_fl x)"
+
+ have "real_of_int (int_floor_fl x) < - 1"
+ using int_floor_fl[of x] \<open>x < - 1\<close> by simp
+ hence "real_of_int (int_floor_fl x) < 0" by simp
+ hence "int_floor_fl x < 0" by auto
+ hence "1 \<le> - int_floor_fl x" by auto
+ hence "0 < nat (- int_floor_fl x)" by auto
+ hence "0 < ?num" by auto
+ hence "real ?num \<noteq> 0" by auto
+ have num_eq: "real ?num = - int_floor_fl x"
+ using \<open>0 < nat (- int_floor_fl x)\<close> by auto
+ have "0 < - int_floor_fl x"
+ using \<open>0 < ?num\<close>[unfolded of_nat_less_iff[symmetric]] by simp
+ hence "real_of_int (int_floor_fl x) < 0"
+ unfolding less_float_def by auto
+ have fl_eq: "real_of_int (- int_floor_fl x) = real_of_float (- floor_fl x)"
+ by (simp add: floor_fl_def int_floor_fl_def)
+ from \<open>0 < - int_floor_fl x\<close> have "0 \<le> real_of_float (- floor_fl x)"
+ by (simp add: floor_fl_def int_floor_fl_def)
+ from \<open>real_of_int (int_floor_fl x) < 0\<close> have "real_of_float (floor_fl x) < 0"
+ by (simp add: floor_fl_def int_floor_fl_def)
+ have "exp x \<le> ub_exp prec x"
+ proof -
+ have div_less_zero: "real_of_float (float_divr prec x (- floor_fl x)) \<le> 0"
+ using float_divr_nonpos_pos_upper_bound[OF \<open>real_of_float x \<le> 0\<close> \<open>0 \<le> real_of_float (- floor_fl x)\<close>]
+ unfolding less_eq_float_def zero_float.rep_eq .
+
+ have "exp x = exp (?num * (x / ?num))"
+ using \<open>real ?num \<noteq> 0\<close> by auto
+ also have "\<dots> = exp (x / ?num) ^ ?num"
+ unfolding exp_of_nat_mult ..
+ also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num"
+ unfolding num_eq fl_eq
+ by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
+ also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
+ unfolding real_of_float_power
+ by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
+ also have "\<dots> \<le> real_of_float (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
+ by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero)
+ finally show ?thesis
+ unfolding ub_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>] floor_fl_def Let_def .
+ qed
+ moreover
+ have "lb_exp prec x \<le> exp x"
+ proof -
+ let ?divl = "float_divl prec x (- floor_fl x)"
+ let ?horner = "?lb_exp_horner ?divl"
+
+ show ?thesis
+ proof (cases "?horner \<le> 0")
+ case False
+ hence "0 \<le> real_of_float ?horner" by auto
+
+ have div_less_zero: "real_of_float (float_divl prec x (- floor_fl x)) \<le> 0"
+ using \<open>real_of_float (floor_fl x) < 0\<close> \<open>real_of_float x \<le> 0\<close>
+ by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
+
+ have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
+ exp (float_divl prec x (- floor_fl x)) ^ ?num"
+ using \<open>0 \<le> real_of_float ?horner\<close>[unfolded floor_fl_def[symmetric]]
+ bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1]
+ by (auto intro!: power_mono)
+ also have "\<dots> \<le> exp (x / ?num) ^ ?num"
+ unfolding num_eq fl_eq
+ using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq)
+ also have "\<dots> = exp (?num * (x / ?num))"
+ unfolding exp_of_nat_mult ..
+ also have "\<dots> = exp x"
+ using \<open>real ?num \<noteq> 0\<close> by auto
+ finally show ?thesis
+ using False
+ unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
+ int_floor_fl_def Let_def if_not_P[OF False]
+ by (auto simp: real_power_down_fl intro!: power_down_le)
+ next
+ case True
+ have "power_down_fl prec (Float 1 (- 2)) ?num \<le> (Float 1 (- 2)) ^ ?num"
+ by (metis Float_le_zero_iff less_imp_le linorder_not_less
+ not_numeral_le_zero numeral_One power_down_fl)
+ then have "power_down_fl prec (Float 1 (- 2)) ?num \<le> real_of_float (Float 1 (- 2)) ^ ?num"
+ by simp
+ also
+ have "real_of_float (floor_fl x) \<noteq> 0" and "real_of_float (floor_fl x) \<le> 0"
+ using \<open>real_of_float (floor_fl x) < 0\<close> by auto
+ from divide_right_mono_neg[OF floor_fl[of x] \<open>real_of_float (floor_fl x) \<le> 0\<close>, unfolded divide_self[OF \<open>real_of_float (floor_fl x) \<noteq> 0\<close>]]
+ have "- 1 \<le> x / (- floor_fl x)"
+ unfolding minus_float.rep_eq by auto
+ from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
+ have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))"
+ unfolding Float_num .
+ hence "real_of_float (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
+ by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral)
+ also have "\<dots> = exp x"
+ unfolding num_eq fl_eq exp_of_nat_mult[symmetric]
+ using \<open>real_of_float (floor_fl x) \<noteq> 0\<close> by auto
+ finally show ?thesis
+ unfolding lb_exp.simps if_not_P[OF \<open>\<not> 0 < x\<close>] if_P[OF \<open>x < - 1\<close>]
+ int_floor_fl_def Let_def if_P[OF True] real_of_float_power .
+ qed
+ qed
+ ultimately show ?thesis by auto
+ qed
+qed
+
+lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }"
+proof -
+ show ?thesis
+ proof (cases "0 < x")
+ case False
+ hence "x \<le> 0" by auto
+ from exp_boundaries'[OF this] show ?thesis .
+ next
+ case True
+ hence "-x \<le> 0" by auto
+
+ have "lb_exp prec x \<le> exp x"
+ proof -
+ from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
+ have ub_exp: "exp (- real_of_float x) \<le> ub_exp prec (-x)"
+ unfolding atLeastAtMost_iff minus_float.rep_eq by auto
+
+ have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)"
+ using float_divl[where x=1] by auto
+ also have "\<dots> \<le> exp x"
+ using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp]
+ exp_gt_zero, symmetric]]
+ unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide
+ by auto
+ finally show ?thesis
+ unfolding lb_exp.simps if_P[OF True] .
+ qed
+ moreover
+ have "exp x \<le> ub_exp prec x"
+ proof -
+ have "\<not> 0 < -x" using \<open>0 < x\<close> by auto
+
+ from exp_boundaries'[OF \<open>-x \<le> 0\<close>]
+ have lb_exp: "lb_exp prec (-x) \<le> exp (- real_of_float x)"
+ unfolding atLeastAtMost_iff minus_float.rep_eq by auto
+
+ have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
+ using lb_exp lb_exp_pos[OF \<open>\<not> 0 < -x\<close>, of prec]
+ by (simp del: lb_exp.simps add: exp_minus field_simps)
+ also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))"
+ using float_divr .
+ finally show ?thesis
+ unfolding ub_exp.simps if_P[OF True] .
+ qed
+ ultimately show ?thesis
+ by auto
+ qed
+qed
+
+lemma bnds_exp: "\<forall>(x::real) lx ux. (l, u) =
+ (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u"
+proof (rule allI, rule allI, rule allI, rule impI)
+ fix x :: real and lx ux
+ assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}"
+ hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}"
+ by auto
+ show "l \<le> exp x \<and> exp x \<le> u"
+ proof
+ show "l \<le> exp x"
+ proof -
+ from exp_boundaries[of lx prec, unfolded l]
+ have "l \<le> exp lx" by (auto simp del: lb_exp.simps)
+ also have "\<dots> \<le> exp x" using x by auto
+ finally show ?thesis .
+ qed
+ show "exp x \<le> u"
+ proof -
+ have "exp x \<le> exp ux" using x by auto
+ also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
+ finally show ?thesis .
+ qed
+ qed
+qed
+
+
+section "Logarithm"
+
+subsection "Compute the logarithm series"
+
+fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
+and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
+"ub_ln_horner prec 0 i x = 0" |
+"ub_ln_horner prec (Suc n) i x = float_plus_up prec
+ (rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" |
+"lb_ln_horner prec 0 i x = 0" |
+"lb_ln_horner prec (Suc n) i x = float_plus_down prec
+ (lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))"
+
+lemma ln_bounds:
+ assumes "0 \<le> x"
+ and "x < 1"
+ shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
+ and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
+proof -
+ let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
+
+ have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)"
+ using ln_series[of "x + 1"] \<open>0 \<le> x\<close> \<open>x < 1\<close> by auto
+
+ have "norm x < 1" using assms by auto
+ have "?a \<longlonglongrightarrow> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
+ using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF \<open>norm x < 1\<close>]]] by auto
+ have "0 \<le> ?a n" for n
+ by (rule mult_nonneg_nonneg) (auto simp: \<open>0 \<le> x\<close>)
+ have "?a (Suc n) \<le> ?a n" for n
+ unfolding inverse_eq_divide[symmetric]
+ proof (rule mult_mono)
+ show "0 \<le> x ^ Suc (Suc n)"
+ by (auto simp add: \<open>0 \<le> x\<close>)
+ have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1"
+ unfolding power_Suc2 mult.assoc[symmetric]
+ by (rule mult_left_mono, fact less_imp_le[OF \<open>x < 1\<close>]) (auto simp: \<open>0 \<le> x\<close>)
+ thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
+ qed auto
+ from summable_Leibniz'(2,4)[OF \<open>?a \<longlonglongrightarrow> 0\<close> \<open>\<And>n. 0 \<le> ?a n\<close>, OF \<open>\<And>n. ?a (Suc n) \<le> ?a n\<close>, unfolded ln_eq]
+ show ?lb and ?ub
+ unfolding atLeast0LessThan by auto
+qed
+
+lemma ln_float_bounds:
+ assumes "0 \<le> real_of_float x"
+ and "real_of_float x < 1"
+ shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
+ and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub")
+proof -
+ obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
+ obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
+
+ let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)"
+
+ have "?lb \<le> sum ?s {0 ..< 2 * ev}"
+ unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
+ unfolding mult.commute[of "real_of_float x"] ev
+ using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x"
+ and lb="\<lambda>n i k x. lb_ln_horner prec n k x"
+ and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
+ OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
+ unfolding real_of_float_power
+ by (rule mult_right_mono)
+ also have "\<dots> \<le> ?ln"
+ using ln_bounds(1)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
+ finally show "?lb \<le> ?ln" .
+
+ have "?ln \<le> sum ?s {0 ..< 2 * od + 1}"
+ using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
+ also have "\<dots> \<le> ?ub"
+ unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq sum_distrib_right[symmetric]
+ unfolding mult.commute[of "real_of_float x"] od
+ using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
+ OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
+ unfolding real_of_float_power
+ by (rule mult_right_mono)
+ finally show "?ln \<le> ?ub" .
+qed
+
+lemma ln_add:
+ fixes x :: real
+ assumes "0 < x" and "0 < y"
+ shows "ln (x + y) = ln x + ln (1 + y / x)"
+proof -
+ have "x \<noteq> 0" using assms by auto
+ have "x + y = x * (1 + y / x)"
+ unfolding distrib_left times_divide_eq_right nonzero_mult_div_cancel_left[OF \<open>x \<noteq> 0\<close>]
+ by auto
+ moreover
+ have "0 < y / x" using assms by auto
+ hence "0 < 1 + y / x" by auto
+ ultimately show ?thesis
+ using ln_mult assms by auto
+qed
+
+
+subsection "Compute the logarithm of 2"
+
+definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
+ in float_plus_up prec
+ ((Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))))
+ (float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))"
+definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
+ in float_plus_down prec
+ ((Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))))
+ (float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))"
+
+lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2")
+ and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2")
+proof -
+ let ?uthird = "rapprox_rat (max prec 1) 1 3"
+ let ?lthird = "lapprox_rat prec 1 3"
+
+ have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1::real)"
+ using ln_add[of "3 / 2" "1 / 2"] by auto
+ have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
+ hence lb3_ub: "real_of_float ?lthird < 1" by auto
+ have lb3_lb: "0 \<le> real_of_float ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
+ have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
+ hence ub3_lb: "0 \<le> real_of_float ?uthird" by auto
+
+ have lb2: "0 \<le> real_of_float (Float 1 (- 1))" and ub2: "real_of_float (Float 1 (- 1)) < 1"
+ unfolding Float_num by auto
+
+ have "0 \<le> (1::int)" and "0 < (3::int)" by auto
+ have ub3_ub: "real_of_float ?uthird < 1"
+ by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1)
+
+ have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
+ have uthird_gt0: "0 < real_of_float ?uthird + 1" using ub3_lb by auto
+ have lthird_gt0: "0 < real_of_float ?lthird + 1" using lb3_lb by auto
+
+ show ?ub_ln2
+ unfolding ub_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
+ proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
+ have "ln (1 / 3 + 1) \<le> ln (real_of_float ?uthird + 1)"
+ unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
+ also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
+ using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
+ also note float_round_up
+ finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
+ qed
+ show ?lb_ln2
+ unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
+ proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
+ have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real_of_float ?lthird + 1)"
+ using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
+ note float_round_down_le[OF this]
+ also have "\<dots> \<le> ln (1 / 3 + 1)"
+ unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0]
+ using lb3 by auto
+ finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le>
+ ln (1 / 3 + 1)" .
+ qed
+qed
+
+
+subsection "Compute the logarithm in the entire domain"
+
+function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
+"ub_ln prec x = (if x \<le> 0 then None
+ else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
+ else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
+ if x \<le> Float 3 (- 1) then Some (horner (x - 1))
+ else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
+ else let l = bitlen (mantissa x) - 1 in
+ Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
+"lb_ln prec x = (if x \<le> 0 then None
+ else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
+ else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
+ if x \<le> Float 3 (- 1) then Some (horner (x - 1))
+ else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
+ horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
+ else let l = bitlen (mantissa x) - 1 in
+ Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))"
+ by pat_completeness auto
+
+termination
+proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
+ fix prec and x :: float
+ assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1"
+ hence "0 < real_of_float x" "1 \<le> max prec (Suc 0)" "real_of_float x < 1"
+ by auto
+ from float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x < 1\<close>[THEN less_imp_le] \<open>1 \<le> max prec (Suc 0)\<close>]
+ show False
+ using \<open>real_of_float (float_divl (max prec (Suc 0)) 1 x) < 1\<close> by auto
+next
+ fix prec x
+ assume "\<not> real_of_float x \<le> 0" and "real_of_float x < 1" and "real_of_float (float_divr prec 1 x) < 1"
+ hence "0 < x" by auto
+ from float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close>, of prec] \<open>real_of_float x < 1\<close> show False
+ using \<open>real_of_float (float_divr prec 1 x) < 1\<close> by auto
+qed
+
+lemma float_pos_eq_mantissa_pos: "x > 0 \<longleftrightarrow> mantissa x > 0"
+ apply (subst Float_mantissa_exponent[of x, symmetric])
+ apply (auto simp add: zero_less_mult_iff zero_float_def dest: less_zeroE)
+ apply (metis not_le powr_ge_pzero)
+ done
+
+lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0"
+ using powr_gt_zero[of 2 "e"]
+ by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)
+
+lemma Float_representation_aux:
+ fixes m e
+ defines "x \<equiv> Float m e"
+ assumes "x > 0"
+ shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1)
+ and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))" (is ?th2)
+proof -
+ from assms have mantissa_pos: "m > 0" "mantissa x > 0"
+ using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all
+ thus ?th1
+ using bitlen_Float[of m e] assms
+ by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float])
+ have "x \<noteq> float_of 0"
+ unfolding zero_float_def[symmetric] using \<open>0 < x\<close> by auto
+ from denormalize_shift[OF assms(1) this] guess i . note i = this
+
+ have "2 powr (1 - (real_of_int (bitlen (mantissa x)) + real_of_int i)) =
+ 2 powr (1 - (real_of_int (bitlen (mantissa x)))) * inverse (2 powr (real i))"
+ by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
+ hence "real_of_int (mantissa x) * 2 powr (1 - real_of_int (bitlen (mantissa x))) =
+ (real_of_int (mantissa x) * 2 ^ i) * 2 powr (1 - real_of_int (bitlen (mantissa x * 2 ^ i)))"
+ using \<open>mantissa x > 0\<close> by (simp add: powr_realpow)
+ then show ?th2
+ unfolding i by transfer auto
+qed
+
+lemma compute_ln[code]:
+ fixes m e
+ defines "x \<equiv> Float m e"
+ shows "ub_ln prec x = (if x \<le> 0 then None
+ else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
+ else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
+ if x \<le> Float 3 (- 1) then Some (horner (x - 1))
+ else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
+ else let l = bitlen m - 1 in
+ Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
+ (is ?th1)
+ and "lb_ln prec x = (if x \<le> 0 then None
+ else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
+ else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
+ if x \<le> Float 3 (- 1) then Some (horner (x - 1))
+ else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
+ horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
+ else let l = bitlen m - 1 in
+ Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
+ (is ?th2)
+proof -
+ from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0"
+ by simp
+ thus ?th1 ?th2
+ using Float_representation_aux[of m e]
+ unfolding x_def[symmetric]
+ by (auto dest: not_le_imp_less)
+qed
+
+lemma ln_shifted_float:
+ assumes "0 < m"
+ shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))"
+proof -
+ let ?B = "2^nat (bitlen m - 1)"
+ define bl where "bl = bitlen m - 1"
+ have "0 < real_of_int m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0"
+ using assms by auto
+ hence "0 \<le> bl" by (simp add: bitlen_alt_def bl_def)
+ show ?thesis
+ proof (cases "0 \<le> e")
+ case True
+ thus ?thesis
+ unfolding bl_def[symmetric] using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
+ apply (simp add: ln_mult)
+ apply (cases "e=0")
+ apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
+ apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps)
+ done
+ next
+ case False
+ hence "0 < -e" by auto
+ have lne: "ln (2 powr real_of_int e) = ln (inverse (2 powr - e))"
+ by (simp add: powr_minus)
+ hence pow_gt0: "(0::real) < 2^nat (-e)"
+ by auto
+ hence inv_gt0: "(0::real) < inverse (2^nat (-e))"
+ by auto
+ show ?thesis
+ using False unfolding bl_def[symmetric]
+ using \<open>0 < real_of_int m\<close> \<open>0 \<le> bl\<close>
+ by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
+ qed
+qed
+
+lemma ub_ln_lb_ln_bounds':
+ assumes "1 \<le> x"
+ shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
+ (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
+proof (cases "x < Float 1 1")
+ case True
+ hence "real_of_float (x - 1) < 1" and "real_of_float x < 2" by auto
+ have "\<not> x \<le> 0" and "\<not> x < 1" using \<open>1 \<le> x\<close> by auto
+ hence "0 \<le> real_of_float (x - 1)" using \<open>1 \<le> x\<close> by auto
+
+ have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp
+
+ show ?thesis
+ proof (cases "x \<le> Float 3 (- 1)")
+ case True
+ show ?thesis
+ unfolding lb_ln.simps
+ unfolding ub_ln.simps Let_def
+ using ln_float_bounds[OF \<open>0 \<le> real_of_float (x - 1)\<close> \<open>real_of_float (x - 1) < 1\<close>, of prec]
+ \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True
+ by (auto intro!: float_round_down_le float_round_up_le)
+ next
+ case False
+ hence *: "3 / 2 < x" by auto
+
+ with ln_add[of "3 / 2" "x - 3 / 2"]
+ have add: "ln x = ln (3 / 2) + ln (real_of_float x * 2 / 3)"
+ by (auto simp add: algebra_simps diff_divide_distrib)
+
+ let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)"
+ let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)"
+
+ { have up: "real_of_float (rapprox_rat prec 2 3) \<le> 1"
+ by (rule rapprox_rat_le1) simp_all
+ have low: "2 / 3 \<le> rapprox_rat prec 2 3"
+ by (rule order_trans[OF _ rapprox_rat]) simp
+ from mult_less_le_imp_less[OF * low] *
+ have pos: "0 < real_of_float (x * rapprox_rat prec 2 3 - 1)" by auto
+
+ have "ln (real_of_float x * 2/3)
+ \<le> ln (real_of_float (x * rapprox_rat prec 2 3 - 1) + 1)"
+ proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
+ show "real_of_float x * 2 / 3 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1) + 1"
+ using * low by auto
+ show "0 < real_of_float x * 2 / 3" using * by simp
+ show "0 < real_of_float (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
+ qed
+ also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
+ proof (rule float_round_up_le, rule ln_float_bounds(2))
+ from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] low *
+ show "real_of_float (x * rapprox_rat prec 2 3 - 1) < 1" by auto
+ show "0 \<le> real_of_float (x * rapprox_rat prec 2 3 - 1)" using pos by auto
+ qed
+ finally have "ln x \<le> ?ub_horner (Float 1 (-1))
+ + ?ub_horner ((x * rapprox_rat prec 2 3 - 1))"
+ using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add
+ by (auto intro!: add_mono float_round_up_le)
+ note float_round_up_le[OF this, of prec]
+ }
+ moreover
+ { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
+
+ have up: "lapprox_rat prec 2 3 \<le> 2/3"
+ by (rule order_trans[OF lapprox_rat], simp)
+
+ have low: "0 \<le> real_of_float (lapprox_rat prec 2 3)"
+ using lapprox_rat_nonneg[of 2 3 prec] by simp
+
+ have "?lb_horner ?max
+ \<le> ln (real_of_float ?max + 1)"
+ proof (rule float_round_down_le, rule ln_float_bounds(1))
+ from mult_less_le_imp_less[OF \<open>real_of_float x < 2\<close> up] * low
+ show "real_of_float ?max < 1" by (cases "real_of_float (lapprox_rat prec 2 3) = 0",
+ auto simp add: real_of_float_max)
+ show "0 \<le> real_of_float ?max" by (auto simp add: real_of_float_max)
+ qed
+ also have "\<dots> \<le> ln (real_of_float x * 2/3)"
+ proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
+ show "0 < real_of_float ?max + 1" by (auto simp add: real_of_float_max)
+ show "0 < real_of_float x * 2/3" using * by auto
+ show "real_of_float ?max + 1 \<le> real_of_float x * 2/3" using * up
+ by (cases "0 < real_of_float x * real_of_float (lapprox_posrat prec 2 3) - 1",
+ auto simp add: max_def)
+ qed
+ finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x"
+ using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add
+ by (auto intro!: add_mono float_round_down_le)
+ note float_round_down_le[OF this, of prec]
+ }
+ ultimately
+ show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
+ using \<open>\<not> x \<le> 0\<close> \<open>\<not> x < 1\<close> True False by auto
+ qed
+next
+ case False
+ hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)"
+ using \<open>1 \<le> x\<close> by auto
+ show ?thesis
+ proof -
+ define m where "m = mantissa x"
+ define e where "e = exponent x"
+ from Float_mantissa_exponent[of x] have Float: "x = Float m e"
+ by (simp add: m_def e_def)
+ let ?s = "Float (e + (bitlen m - 1)) 0"
+ let ?x = "Float m (- (bitlen m - 1))"
+
+ have "0 < m" and "m \<noteq> 0" using \<open>0 < x\<close> Float powr_gt_zero[of 2 e]
+ apply (auto simp add: zero_less_mult_iff)
+ using not_le powr_ge_pzero apply blast
+ done
+ define bl where "bl = bitlen m - 1"
+ hence "bl \<ge> 0"
+ using \<open>m > 0\<close> by (simp add: bitlen_alt_def)
+ have "1 \<le> Float m e"
+ using \<open>1 \<le> x\<close> Float unfolding less_eq_float_def by auto
+ from bitlen_div[OF \<open>0 < m\<close>] float_gt1_scale[OF \<open>1 \<le> Float m e\<close>] \<open>bl \<ge> 0\<close>
+ have x_bnds: "0 \<le> real_of_float (?x - 1)" "real_of_float (?x - 1) < 1"
+ unfolding bl_def[symmetric]
+ by (auto simp: powr_realpow[symmetric] field_simps)
+ (auto simp : powr_minus field_simps)
+
+ {
+ have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))"
+ (is "real_of_float ?lb2 \<le> _")
+ apply (rule float_round_down_le)
+ unfolding nat_0 power_0 mult_1_right times_float.rep_eq
+ using lb_ln2[of prec]
+ proof (rule mult_mono)
+ from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
+ show "0 \<le> real_of_float (Float (e + (bitlen m - 1)) 0)" by simp
+ qed auto
+ moreover
+ from ln_float_bounds(1)[OF x_bnds]
+ have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real_of_float ?lb_horner \<le> _")
+ by (auto intro!: float_round_down_le)
+ ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x"
+ unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e] by (auto intro!: float_plus_down_le)
+ }
+ moreover
+ {
+ from ln_float_bounds(2)[OF x_bnds]
+ have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))"
+ (is "_ \<le> real_of_float ?ub_horner")
+ by (auto intro!: float_round_up_le)
+ moreover
+ have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)"
+ (is "_ \<le> real_of_float ?ub2")
+ apply (rule float_round_up_le)
+ unfolding nat_0 power_0 mult_1_right times_float.rep_eq
+ using ub_ln2[of prec]
+ proof (rule mult_mono)
+ from float_gt1_scale[OF \<open>1 \<le> Float m e\<close>]
+ show "0 \<le> real_of_int (e + (bitlen m - 1))" by auto
+ have "0 \<le> ln (2 :: real)" by simp
+ thus "0 \<le> real_of_float (ub_ln2 prec)" using ub_ln2[of prec] by arith
+ qed auto
+ ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner"
+ unfolding Float ln_shifted_float[OF \<open>0 < m\<close>, of e]
+ by (auto intro!: float_plus_up_le)
+ }
+ ultimately show ?thesis
+ unfolding lb_ln.simps
+ unfolding ub_ln.simps
+ unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_not_P[OF \<open>\<not> x < 1\<close>]
+ if_not_P[OF False] if_not_P[OF \<open>\<not> x \<le> Float 3 (- 1)\<close>] Let_def
+ unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric]
+ by simp
+ qed
+qed
+
+lemma ub_ln_lb_ln_bounds:
+ assumes "0 < x"
+ shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
+ (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
+proof (cases "x < 1")
+ case False
+ hence "1 \<le> x"
+ unfolding less_float_def less_eq_float_def by auto
+ show ?thesis
+ using ub_ln_lb_ln_bounds'[OF \<open>1 \<le> x\<close>] .
+next
+ case True
+ have "\<not> x \<le> 0" using \<open>0 < x\<close> by auto
+ from True have "real_of_float x \<le> 1" "x \<le> 1"
+ by simp_all
+ have "0 < real_of_float x" and "real_of_float x \<noteq> 0"
+ using \<open>0 < x\<close> by auto
+ hence A: "0 < 1 / real_of_float x" by auto
+
+ {
+ let ?divl = "float_divl (max prec 1) 1 x"
+ have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF \<open>0 < real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by auto
+ hence B: "0 < real_of_float ?divl" by auto
+
+ have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
+ hence "ln x \<le> - ln ?divl" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
+ from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
+ have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans)
+ } moreover
+ {
+ let ?divr = "float_divr prec 1 x"
+ have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF \<open>0 < x\<close> \<open>x \<le> 1\<close>] unfolding less_eq_float_def less_float_def by auto
+ hence B: "0 < real_of_float ?divr" by auto
+
+ have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
+ hence "- ln ?divr \<le> ln x" unfolding nonzero_inverse_eq_divide[OF \<open>real_of_float x \<noteq> 0\<close>, symmetric] ln_inverse[OF \<open>0 < real_of_float x\<close>] by auto
+ from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
+ have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans)
+ }
+ ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x]
+ unfolding if_not_P[OF \<open>\<not> x \<le> 0\<close>] if_P[OF True] by auto
+qed
+
+lemma lb_ln:
+ assumes "Some y = lb_ln prec x"
+ shows "y \<le> ln x" and "0 < real_of_float x"
+proof -
+ have "0 < x"
+ proof (rule ccontr)
+ assume "\<not> 0 < x"
+ hence "x \<le> 0"
+ unfolding less_eq_float_def less_float_def by auto
+ thus False
+ using assms by auto
+ qed
+ thus "0 < real_of_float x" by auto
+ have "the (lb_ln prec x) \<le> ln x"
+ using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
+ thus "y \<le> ln x"
+ unfolding assms[symmetric] by auto
+qed
+
+lemma ub_ln:
+ assumes "Some y = ub_ln prec x"
+ shows "ln x \<le> y" and "0 < real_of_float x"
+proof -
+ have "0 < x"
+ proof (rule ccontr)
+ assume "\<not> 0 < x"
+ hence "x \<le> 0" by auto
+ thus False
+ using assms by auto
+ qed
+ thus "0 < real_of_float x" by auto
+ have "ln x \<le> the (ub_ln prec x)"
+ using ub_ln_lb_ln_bounds[OF \<open>0 < x\<close>] ..
+ thus "ln x \<le> y"
+ unfolding assms[symmetric] by auto
+qed
+
+lemma bnds_ln: "\<forall>(x::real) lx ux. (Some l, Some u) =
+ (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u"
+proof (rule allI, rule allI, rule allI, rule impI)
+ fix x :: real
+ fix lx ux
+ assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}"
+ hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}"
+ by auto
+
+ have "ln ux \<le> u" and "0 < real_of_float ux"
+ using ub_ln u by auto
+ have "l \<le> ln lx" and "0 < real_of_float lx" and "0 < x"
+ using lb_ln[OF l] x by auto
+
+ from ln_le_cancel_iff[OF \<open>0 < real_of_float lx\<close> \<open>0 < x\<close>] \<open>l \<le> ln lx\<close>
+ have "l \<le> ln x"
+ using x unfolding atLeastAtMost_iff by auto
+ moreover
+ from ln_le_cancel_iff[OF \<open>0 < x\<close> \<open>0 < real_of_float ux\<close>] \<open>ln ux \<le> real_of_float u\<close>
+ have "ln x \<le> u"
+ using x unfolding atLeastAtMost_iff by auto
+ ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
+qed
+
+
+section \<open>Real power function\<close>
+
+definition bnds_powr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float \<times> float) option" where
+ "bnds_powr prec l1 u1 l2 u2 = (
+ if l1 = 0 \<and> u1 = 0 then
+ Some (0, 0)
+ else if l1 = 0 \<and> l2 \<ge> 1 then
+ let uln = the (ub_ln prec u1)
+ in Some (0, ub_exp prec (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2))))
+ else if l1 \<le> 0 then
+ None
+ else
+ Some (map_bnds lb_exp ub_exp prec
+ (bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2)))"
+
+lemmas [simp del] = lb_exp.simps ub_exp.simps
+
+lemma mono_exp_real: "mono (exp :: real \<Rightarrow> real)"
+ by (auto simp: mono_def)
+
+lemma ub_exp_nonneg: "real_of_float (ub_exp prec x) \<ge> 0"
+proof -
+ have "0 \<le> exp (real_of_float x)" by simp
+ also from exp_boundaries[of x prec]
+ have "\<dots> \<le> real_of_float (ub_exp prec x)" by simp
+ finally show ?thesis .
+qed
+
+lemma bnds_powr:
+ assumes lu: "Some (l, u) = bnds_powr prec l1 u1 l2 u2"
+ assumes x: "x \<in> {real_of_float l1..real_of_float u1}"
+ assumes y: "y \<in> {real_of_float l2..real_of_float u2}"
+ shows "x powr y \<in> {real_of_float l..real_of_float u}"
+proof -
+ consider "l1 = 0" "u1 = 0" | "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1" |
+ "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))" | "l1 > 0" by force
+ thus ?thesis
+ proof cases
+ assume "l1 = 0" "u1 = 0"
+ with x lu show ?thesis by (auto simp: bnds_powr_def)
+ next
+ assume A: "l1 = 0" "u1 \<noteq> 0" "l2 \<ge> 1"
+ define uln where "uln = the (ub_ln prec u1)"
+ show ?thesis
+ proof (cases "x = 0")
+ case False
+ with A x y have "x powr y = exp (ln x * y)" by (simp add: powr_def)
+ also {
+ from A x False have "ln x \<le> ln (real_of_float u1)" by simp
+ also from ub_ln_lb_ln_bounds[of u1 prec] A y x False
+ have "ln (real_of_float u1) \<le> real_of_float uln" by (simp add: uln_def del: lb_ln.simps)
+ also from A x y have "\<dots> * y \<le> real_of_float uln * (if uln \<ge> 0 then u2 else l2)"
+ by (auto intro: mult_left_mono mult_left_mono_neg)
+ also have "\<dots> \<le> real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))"
+ by (simp add: float_round_up_le)
+ finally have "ln x * y \<le> \<dots>" using A y by - simp
+ }
+ also have "exp (real_of_float (float_round_up prec (uln * (if uln \<ge> 0 then u2 else l2)))) \<le>
+ real_of_float (ub_exp prec (float_round_up prec
+ (uln * (if uln \<ge> 0 then u2 else l2))))"
+ using exp_boundaries by simp
+ finally show ?thesis using A x y lu
+ by (simp add: bnds_powr_def uln_def Let_def del: lb_ln.simps ub_ln.simps)
+ qed (insert x y lu A, simp_all add: bnds_powr_def Let_def ub_exp_nonneg
+ del: lb_ln.simps ub_ln.simps)
+ next
+ assume "l1 \<le> 0" "\<not>(l1 = 0 \<and> (u1 = 0 \<or> l2 \<ge> 1))"
+ with lu show ?thesis by (simp add: bnds_powr_def split: if_split_asm)
+ next
+ assume l1: "l1 > 0"
+ obtain lm um where lmum:
+ "(lm, um) = bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2"
+ by (cases "bnds_mult prec (the (lb_ln prec l1)) (the (ub_ln prec u1)) l2 u2") simp
+ with l1 have "(l, u) = map_bnds lb_exp ub_exp prec (lm, um)"
+ using lu by (simp add: bnds_powr_def del: lb_ln.simps ub_ln.simps split: if_split_asm)
+ hence "exp (ln x * y) \<in> {real_of_float l..real_of_float u}"
+ proof (rule map_bnds[OF _ mono_exp_real], goal_cases)
+ case 1
+ let ?lln = "the (lb_ln prec l1)" and ?uln = "the (ub_ln prec u1)"
+ from ub_ln_lb_ln_bounds[of l1 prec] ub_ln_lb_ln_bounds[of u1 prec] x l1
+ have "real_of_float ?lln \<le> ln (real_of_float l1) \<and>
+ ln (real_of_float u1) \<le> real_of_float ?uln"
+ by (auto simp del: lb_ln.simps ub_ln.simps)
+ moreover from l1 x have "ln (real_of_float l1) \<le> ln x \<and> ln x \<le> ln (real_of_float u1)"
+ by auto
+ ultimately have ln: "real_of_float ?lln \<le> ln x \<and> ln x \<le> real_of_float ?uln" by simp
+ from lmum show ?case
+ by (rule bnds_mult) (insert y ln, simp_all)
+ qed (insert exp_boundaries[of lm prec] exp_boundaries[of um prec], simp_all)
+ with x l1 show ?thesis
+ by (simp add: powr_def mult_ac)
+ qed
+qed
+
+end