src/HOL/Decision_Procs/Approximation.thy
author hoelzl
Thu Jun 25 18:12:40 2009 +0200 (2009-06-25)
changeset 31811 64dea9a15031
parent 31810 a6b800855cdd
child 31863 e391eee8bf14
permissions -rw-r--r--
Improved computation of bounds and implemented interval splitting for 'approximation'.
hoelzl@30443
     1
(* Author:     Johannes Hoelzl <hoelzl@in.tum.de> 2008 / 2009 *)
wenzelm@30122
     2
wenzelm@30886
     3
header {* Prove Real Valued Inequalities by Computation *}
wenzelm@30122
     4
hoelzl@29805
     5
theory Approximation
haftmann@29823
     6
imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat
hoelzl@29805
     7
begin
hoelzl@29805
     8
hoelzl@29805
     9
section "Horner Scheme"
hoelzl@29805
    10
hoelzl@29805
    11
subsection {* Define auxiliary helper @{text horner} function *}
hoelzl@29805
    12
hoelzl@31098
    13
primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
hoelzl@29805
    14
"horner F G 0 i k x       = 0" |
hoelzl@29805
    15
"horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x"
hoelzl@29805
    16
hoelzl@29805
    17
lemma horner_schema': fixes x :: real  and a :: "nat \<Rightarrow> real"
hoelzl@29805
    18
  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)"
hoelzl@29805
    19
proof -
hoelzl@29805
    20
  have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto
hoelzl@29805
    21
  show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc]
hoelzl@29805
    22
    setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
hoelzl@29805
    23
qed
hoelzl@29805
    24
hoelzl@29805
    25
lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
haftmann@30971
    26
  assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
haftmann@30971
    27
  shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. -1 ^ j * (1 / real (f (j' + j))) * x ^ j)"
hoelzl@29805
    28
proof (induct n arbitrary: i k j')
hoelzl@29805
    29
  case (Suc n)
hoelzl@29805
    30
hoelzl@29805
    31
  show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
hoelzl@29805
    32
    using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto
hoelzl@29805
    33
qed auto
hoelzl@29805
    34
hoelzl@29805
    35
lemma horner_bounds':
hoelzl@31098
    36
  assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
hoelzl@29805
    37
  and lb_0: "\<And> i k x. lb 0 i k x = 0"
hoelzl@29805
    38
  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
hoelzl@29805
    39
  and ub_0: "\<And> i k x. ub 0 i k x = 0"
hoelzl@29805
    40
  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
hoelzl@31809
    41
  shows "real (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real x) \<and>
hoelzl@31098
    42
         horner F G n ((F ^^ j') s) (f j') (real x) \<le> real (ub n ((F ^^ j') s) (f j') x)"
hoelzl@29805
    43
  (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
hoelzl@29805
    44
proof (induct n arbitrary: j')
hoelzl@29805
    45
  case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
hoelzl@29805
    46
next
hoelzl@29805
    47
  case (Suc n)
hoelzl@31098
    48
  have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps real_of_float_sub diff_def
hoelzl@29805
    49
  proof (rule add_mono)
hoelzl@31098
    50
    show "real (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1  "int (f j')"] by auto
hoelzl@31098
    51
    from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> real x`
hoelzl@31098
    52
    show "- real (x * ub n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x) \<le> - (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x))"
hoelzl@31098
    53
      unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
hoelzl@29805
    54
  qed
hoelzl@31098
    55
  moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps real_of_float_sub diff_def
hoelzl@29805
    56
  proof (rule add_mono)
hoelzl@31098
    57
    show "1 / real (f j') \<le> real (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
hoelzl@31098
    58
    from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> real x`
hoelzl@31809
    59
    show "- (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x)) \<le>
hoelzl@31098
    60
          - real (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
hoelzl@31098
    61
      unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
hoelzl@29805
    62
  qed
hoelzl@29805
    63
  ultimately show ?case by blast
hoelzl@29805
    64
qed
hoelzl@29805
    65
hoelzl@29805
    66
subsection "Theorems for floating point functions implementing the horner scheme"
hoelzl@29805
    67
hoelzl@29805
    68
text {*
hoelzl@29805
    69
hoelzl@29805
    70
Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
hoelzl@29805
    71
all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
hoelzl@29805
    72
hoelzl@29805
    73
*}
hoelzl@29805
    74
hoelzl@29805
    75
lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
hoelzl@31098
    76
  assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
hoelzl@29805
    77
  and lb_0: "\<And> i k x. lb 0 i k x = 0"
hoelzl@29805
    78
  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
hoelzl@29805
    79
  and ub_0: "\<And> i k x. ub 0 i k x = 0"
hoelzl@29805
    80
  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
hoelzl@31809
    81
  shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
hoelzl@31098
    82
    "(\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
hoelzl@29805
    83
proof -
hoelzl@31809
    84
  have "?lb  \<and> ?ub"
hoelzl@31098
    85
    using horner_bounds'[where lb=lb, OF `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
hoelzl@29805
    86
    unfolding horner_schema[where f=f, OF f_Suc] .
hoelzl@29805
    87
  thus "?lb" and "?ub" by auto
hoelzl@29805
    88
qed
hoelzl@29805
    89
hoelzl@29805
    90
lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
hoelzl@31098
    91
  assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
hoelzl@29805
    92
  and lb_0: "\<And> i k x. lb 0 i k x = 0"
hoelzl@29805
    93
  and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)"
hoelzl@29805
    94
  and ub_0: "\<And> i k x. ub 0 i k x = 0"
hoelzl@29805
    95
  and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)"
hoelzl@31809
    96
  shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
hoelzl@31098
    97
    "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
hoelzl@29805
    98
proof -
hoelzl@29805
    99
  { fix x y z :: float have "x - y * z = x + - y * z"
haftmann@30968
   100
      by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
hoelzl@29805
   101
  } note diff_mult_minus = this
hoelzl@29805
   102
hoelzl@29805
   103
  { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
hoelzl@29805
   104
hoelzl@31098
   105
  have move_minus: "real (-x) = -1 * real x" by auto
hoelzl@29805
   106
hoelzl@31809
   107
  have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j) =
hoelzl@31098
   108
    (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j)"
hoelzl@29805
   109
  proof (rule setsum_cong, simp)
hoelzl@29805
   110
    fix j assume "j \<in> {0 ..< n}"
hoelzl@31098
   111
    show "1 / real (f (j' + j)) * real x ^ j = -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j"
hoelzl@29805
   112
      unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
haftmann@30952
   113
      unfolding real_mult_commute unfolding real_mult_assoc[of "-1 ^ j", symmetric] power_mult_distrib[symmetric]
hoelzl@29805
   114
      by auto
hoelzl@29805
   115
  qed
hoelzl@29805
   116
hoelzl@31098
   117
  have "0 \<le> real (-x)" using assms by auto
hoelzl@29805
   118
  from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
hoelzl@29805
   119
    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,
hoelzl@29805
   120
    OF this f_Suc lb_0 refl ub_0 refl]
hoelzl@29805
   121
  show "?lb" and "?ub" unfolding minus_minus sum_eq
hoelzl@29805
   122
    by auto
hoelzl@29805
   123
qed
hoelzl@29805
   124
hoelzl@29805
   125
subsection {* Selectors for next even or odd number *}
hoelzl@29805
   126
hoelzl@29805
   127
text {*
hoelzl@29805
   128
hoelzl@29805
   129
The horner scheme computes alternating series. To get the upper and lower bounds we need to
hoelzl@29805
   130
guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
hoelzl@29805
   131
hoelzl@29805
   132
*}
hoelzl@29805
   133
hoelzl@29805
   134
definition get_odd :: "nat \<Rightarrow> nat" where
hoelzl@29805
   135
  "get_odd n = (if odd n then n else (Suc n))"
hoelzl@29805
   136
hoelzl@29805
   137
definition get_even :: "nat \<Rightarrow> nat" where
hoelzl@29805
   138
  "get_even n = (if even n then n else (Suc n))"
hoelzl@29805
   139
hoelzl@29805
   140
lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
hoelzl@29805
   141
lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
hoelzl@29805
   142
lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
hoelzl@29805
   143
proof (cases "odd n")
hoelzl@29805
   144
  case True hence "0 < n" by (rule odd_pos)
hoelzl@31467
   145
  from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto
hoelzl@29805
   146
  thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast
hoelzl@29805
   147
next
hoelzl@29805
   148
  case False hence "odd (Suc n)" by auto
hoelzl@29805
   149
  thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
hoelzl@29805
   150
qed
hoelzl@29805
   151
hoelzl@29805
   152
lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] .
hoelzl@29805
   153
lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto
hoelzl@29805
   154
hoelzl@29805
   155
section "Power function"
hoelzl@29805
   156
hoelzl@29805
   157
definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@29805
   158
"float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n)
hoelzl@29805
   159
                      else if u < 0         then (u ^ n, l ^ n)
hoelzl@29805
   160
                                            else (0, (max (-l) u) ^ n))"
hoelzl@29805
   161
hoelzl@31098
   162
lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {real l .. real u}"
hoelzl@31098
   163
  shows "x ^ n \<in> {real l1..real u1}"
hoelzl@29805
   164
proof (cases "even n")
hoelzl@31467
   165
  case True
hoelzl@29805
   166
  show ?thesis
hoelzl@29805
   167
  proof (cases "0 < l")
hoelzl@31098
   168
    case True hence "odd n \<or> 0 < l" and "0 \<le> real l" unfolding less_float_def by auto
hoelzl@29805
   169
    have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
hoelzl@31098
   170
    have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using `0 \<le> real l` and assms unfolding atLeastAtMost_iff using power_mono[of "real l" x] power_mono[of x "real u"] by auto
hoelzl@29805
   171
    thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
hoelzl@29805
   172
  next
hoelzl@29805
   173
    case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
hoelzl@29805
   174
    show ?thesis
hoelzl@29805
   175
    proof (cases "u < 0")
hoelzl@31098
   176
      case True hence "0 \<le> - real u" and "- real u \<le> - x" and "0 \<le> - x" and "-x \<le> - real l" using assms unfolding less_float_def by auto
hoelzl@31809
   177
      hence "real u ^ n \<le> x ^ n" and "x ^ n \<le> real l ^ n" using power_mono[of  "-x" "-real l" n] power_mono[of "-real u" "-x" n]
hoelzl@29805
   178
	unfolding power_minus_even[OF `even n`] by auto
hoelzl@29805
   179
      moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto
hoelzl@29805
   180
      ultimately show ?thesis using float_power by auto
hoelzl@29805
   181
    next
hoelzl@31467
   182
      case False
hoelzl@31098
   183
      have "\<bar>x\<bar> \<le> real (max (-l) u)"
hoelzl@29805
   184
      proof (cases "-l \<le> u")
hoelzl@29805
   185
	case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto
hoelzl@29805
   186
      next
hoelzl@29805
   187
	case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto
hoelzl@29805
   188
      qed
hoelzl@31098
   189
      hence x_abs: "\<bar>x\<bar> \<le> \<bar>real (max (-l) u)\<bar>" by auto
hoelzl@29805
   190
      have u1: "u1 = (max (-l) u) ^ n" and l1: "l1 = 0" using assms unfolding float_power_bnds_def if_not_P[OF P] if_not_P[OF False] by auto
hoelzl@29805
   191
      show ?thesis unfolding atLeastAtMost_iff l1 u1 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto
hoelzl@29805
   192
    qed
hoelzl@29805
   193
  qed
hoelzl@29805
   194
next
hoelzl@29805
   195
  case False hence "odd n \<or> 0 < l" by auto
hoelzl@29805
   196
  have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
hoelzl@31098
   197
  have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
hoelzl@29805
   198
  thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
hoelzl@29805
   199
qed
hoelzl@29805
   200
hoelzl@31098
   201
lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {real l .. real u} \<longrightarrow> real l1 \<le> x ^ n \<and> x ^ n \<le> real u1"
hoelzl@29805
   202
  using float_power_bnds by auto
hoelzl@29805
   203
hoelzl@29805
   204
section "Square root"
hoelzl@29805
   205
hoelzl@29805
   206
text {*
hoelzl@29805
   207
hoelzl@29805
   208
The square root computation is implemented as newton iteration. As first first step we use the
hoelzl@29805
   209
nearest power of two greater than the square root.
hoelzl@29805
   210
hoelzl@29805
   211
*}
hoelzl@29805
   212
hoelzl@29805
   213
fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   214
"sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" |
hoelzl@31467
   215
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
hoelzl@29805
   216
                                  in Float 1 -1 * (y + float_divr prec x y))"
hoelzl@29805
   217
hoelzl@31467
   218
function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@31467
   219
"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
hoelzl@31467
   220
              else if x < 0 then - lb_sqrt prec (- x)
hoelzl@31467
   221
                            else 0)" |
hoelzl@31467
   222
"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
hoelzl@31467
   223
              else if x < 0 then - ub_sqrt prec (- x)
hoelzl@31467
   224
                            else 0)"
hoelzl@31467
   225
by pat_completeness auto
hoelzl@31467
   226
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
hoelzl@29805
   227
hoelzl@31467
   228
declare lb_sqrt.simps[simp del]
hoelzl@31467
   229
declare ub_sqrt.simps[simp del]
hoelzl@29805
   230
hoelzl@29805
   231
lemma sqrt_ub_pos_pos_1:
hoelzl@29805
   232
  assumes "sqrt x < b" and "0 < b" and "0 < x"
hoelzl@29805
   233
  shows "sqrt x < (b + x / b)/2"
hoelzl@29805
   234
proof -
hoelzl@29805
   235
  from assms have "0 < (b - sqrt x) ^ 2 " by simp
hoelzl@29805
   236
  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
hoelzl@29805
   237
  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
hoelzl@29805
   238
  finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
hoelzl@29805
   239
  hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
hoelzl@29805
   240
    by (simp add: field_simps power2_eq_square)
hoelzl@29805
   241
  thus ?thesis by (simp add: field_simps)
hoelzl@29805
   242
qed
hoelzl@29805
   243
hoelzl@31098
   244
lemma sqrt_iteration_bound: assumes "0 < real x"
hoelzl@31098
   245
  shows "sqrt (real x) < real (sqrt_iteration prec n x)"
hoelzl@29805
   246
proof (induct n)
hoelzl@29805
   247
  case 0
hoelzl@29805
   248
  show ?case
hoelzl@29805
   249
  proof (cases x)
hoelzl@29805
   250
    case (Float m e)
hoelzl@29805
   251
    hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
hoelzl@29805
   252
    hence "0 < sqrt (real m)" by auto
hoelzl@29805
   253
hoelzl@29805
   254
    have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
hoelzl@29805
   255
hoelzl@31098
   256
    have "real x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
hoelzl@31098
   257
      unfolding pow2_add pow2_int Float real_of_float_simp by auto
hoelzl@29805
   258
    also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
hoelzl@29805
   259
    proof (rule mult_strict_right_mono, auto)
hoelzl@31467
   260
      show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
hoelzl@29805
   261
	unfolding real_of_int_less_iff[of m, symmetric] by auto
hoelzl@29805
   262
    qed
hoelzl@31098
   263
    finally have "sqrt (real x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
hoelzl@29805
   264
    also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
hoelzl@29805
   265
    proof -
hoelzl@29805
   266
      let ?E = "e + bitlen m"
hoelzl@29805
   267
      have E_mod_pow: "pow2 (?E mod 2) < 4"
hoelzl@29805
   268
      proof (cases "?E mod 2 = 1")
hoelzl@29805
   269
	case True thus ?thesis by auto
hoelzl@29805
   270
      next
hoelzl@31467
   271
	case False
hoelzl@31467
   272
	have "0 \<le> ?E mod 2" by auto
hoelzl@29805
   273
	have "?E mod 2 < 2" by auto
hoelzl@29805
   274
	from this[THEN zless_imp_add1_zle]
hoelzl@29805
   275
	have "?E mod 2 \<le> 0" using False by auto
hoelzl@29805
   276
	from xt1(5)[OF `0 \<le> ?E mod 2` this]
hoelzl@29805
   277
	show ?thesis by auto
hoelzl@29805
   278
      qed
hoelzl@29805
   279
      hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
hoelzl@29805
   280
      hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
hoelzl@29805
   281
hoelzl@29805
   282
      have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
hoelzl@29805
   283
      have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
hoelzl@29805
   284
	unfolding E_eq unfolding pow2_add ..
hoelzl@29805
   285
      also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
hoelzl@29805
   286
	unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
hoelzl@31467
   287
      also have "\<dots> < pow2 (?E div 2) * 2"
hoelzl@29805
   288
	by (rule mult_strict_left_mono, auto intro: E_mod_pow)
hoelzl@29805
   289
      also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
hoelzl@29805
   290
      finally show ?thesis by auto
hoelzl@29805
   291
    qed
hoelzl@31467
   292
    finally show ?thesis
hoelzl@31098
   293
      unfolding Float sqrt_iteration.simps real_of_float_simp by auto
hoelzl@29805
   294
  qed
hoelzl@29805
   295
next
hoelzl@29805
   296
  case (Suc n)
hoelzl@29805
   297
  let ?b = "sqrt_iteration prec n x"
hoelzl@31098
   298
  have "0 < sqrt (real x)" using `0 < real x` by auto
hoelzl@31098
   299
  also have "\<dots> < real ?b" using Suc .
hoelzl@31098
   300
  finally have "sqrt (real x) < (real ?b + real x / real ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] by auto
hoelzl@31098
   301
  also have "\<dots> \<le> (real ?b + real (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
hoelzl@31098
   302
  also have "\<dots> = real (Float 1 -1) * (real ?b + real (float_divr prec x ?b))" by auto
hoelzl@31098
   303
  finally show ?case unfolding sqrt_iteration.simps Let_def real_of_float_mult real_of_float_add right_distrib .
hoelzl@29805
   304
qed
hoelzl@29805
   305
hoelzl@31098
   306
lemma sqrt_iteration_lower_bound: assumes "0 < real x"
hoelzl@31098
   307
  shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
hoelzl@29805
   308
proof -
hoelzl@31098
   309
  have "0 < sqrt (real x)" using assms by auto
hoelzl@29805
   310
  also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
hoelzl@29805
   311
  finally show ?thesis .
hoelzl@29805
   312
qed
hoelzl@29805
   313
hoelzl@31098
   314
lemma lb_sqrt_lower_bound: assumes "0 \<le> real x"
hoelzl@31467
   315
  shows "0 \<le> real (lb_sqrt prec x)"
hoelzl@29805
   316
proof (cases "0 < x")
hoelzl@31098
   317
  case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` unfolding less_float_def le_float_def by auto
hoelzl@31809
   318
  hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto
hoelzl@31098
   319
  hence "0 \<le> real (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto
hoelzl@31467
   320
  thus ?thesis unfolding lb_sqrt.simps using True by auto
hoelzl@29805
   321
next
hoelzl@31098
   322
  case False with `0 \<le> real x` have "real x = 0" unfolding less_float_def by auto
hoelzl@31467
   323
  thus ?thesis unfolding lb_sqrt.simps less_float_def by auto
hoelzl@29805
   324
qed
hoelzl@29805
   325
hoelzl@31467
   326
lemma bnds_sqrt':
hoelzl@31467
   327
  shows "sqrt (real x) \<in> { real (lb_sqrt prec x) .. real (ub_sqrt prec x) }"
hoelzl@31467
   328
proof -
hoelzl@31467
   329
  { fix x :: float assume "0 < x"
hoelzl@31467
   330
    hence "0 < real x" and "0 \<le> real x" unfolding less_float_def by auto
hoelzl@31467
   331
    hence sqrt_gt0: "0 < sqrt (real x)" by auto
hoelzl@31467
   332
    hence sqrt_ub: "sqrt (real x) < real (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
hoelzl@31467
   333
hoelzl@31467
   334
    have "real (float_divl prec x (sqrt_iteration prec prec x)) \<le>
hoelzl@31467
   335
          real x / real (sqrt_iteration prec prec x)" by (rule float_divl)
hoelzl@31467
   336
    also have "\<dots> < real x / sqrt (real x)"
hoelzl@31467
   337
      by (rule divide_strict_left_mono[OF sqrt_ub `0 < real x`
hoelzl@31467
   338
               mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
hoelzl@31809
   339
    also have "\<dots> = sqrt (real x)"
hoelzl@31467
   340
      unfolding inverse_eq_iff_eq[of _ "sqrt (real x)", symmetric]
hoelzl@31467
   341
	        sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
hoelzl@31467
   342
    finally have "real (lb_sqrt prec x) \<le> sqrt (real x)"
hoelzl@31467
   343
      unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
hoelzl@31467
   344
  note lb = this
hoelzl@31467
   345
hoelzl@31467
   346
  { fix x :: float assume "0 < x"
hoelzl@31467
   347
    hence "0 < real x" unfolding less_float_def by auto
hoelzl@31467
   348
    hence "0 < sqrt (real x)" by auto
hoelzl@31467
   349
    hence "sqrt (real x) < real (sqrt_iteration prec prec x)"
hoelzl@31467
   350
      using sqrt_iteration_bound by auto
hoelzl@31467
   351
    hence "sqrt (real x) \<le> real (ub_sqrt prec x)"
hoelzl@31467
   352
      unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
hoelzl@31467
   353
  note ub = this
hoelzl@31467
   354
hoelzl@31467
   355
  show ?thesis
hoelzl@31467
   356
  proof (cases "0 < x")
hoelzl@31467
   357
    case True with lb ub show ?thesis by auto
hoelzl@31467
   358
  next case False show ?thesis
hoelzl@31467
   359
  proof (cases "real x = 0")
hoelzl@31809
   360
    case True thus ?thesis
hoelzl@31467
   361
      by (auto simp add: less_float_def lb_sqrt.simps ub_sqrt.simps)
hoelzl@31467
   362
  next
hoelzl@31467
   363
    case False with `\<not> 0 < x` have "x < 0" and "0 < -x"
hoelzl@31467
   364
      by (auto simp add: less_float_def)
hoelzl@31467
   365
hoelzl@31467
   366
    with `\<not> 0 < x`
hoelzl@31467
   367
    show ?thesis using lb[OF `0 < -x`] ub[OF `0 < -x`]
hoelzl@31467
   368
      by (auto simp add: real_sqrt_minus lb_sqrt.simps ub_sqrt.simps)
hoelzl@31467
   369
  qed qed
hoelzl@29805
   370
qed
hoelzl@29805
   371
hoelzl@31467
   372
lemma bnds_sqrt: "\<forall> x lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> sqrt x \<and> sqrt x \<le> real u"
hoelzl@31467
   373
proof ((rule allI) +, rule impI, erule conjE, rule conjI)
hoelzl@31467
   374
  fix x lx ux
hoelzl@31467
   375
  assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
hoelzl@31467
   376
    and x: "x \<in> {real lx .. real ux}"
hoelzl@31467
   377
  hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
hoelzl@29805
   378
hoelzl@31467
   379
  have "sqrt (real lx) \<le> sqrt x" using x by auto
hoelzl@31467
   380
  from order_trans[OF _ this]
hoelzl@31467
   381
  show "real l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
hoelzl@29805
   382
hoelzl@31467
   383
  have "sqrt x \<le> sqrt (real ux)" using x by auto
hoelzl@31467
   384
  from order_trans[OF this]
hoelzl@31467
   385
  show "sqrt x \<le> real u" unfolding u using bnds_sqrt'[of ux prec] by auto
hoelzl@29805
   386
qed
hoelzl@29805
   387
hoelzl@29805
   388
section "Arcus tangens and \<pi>"
hoelzl@29805
   389
hoelzl@29805
   390
subsection "Compute arcus tangens series"
hoelzl@29805
   391
hoelzl@29805
   392
text {*
hoelzl@29805
   393
hoelzl@29805
   394
As first step we implement the computation of the arcus tangens series. This is only valid in the range
hoelzl@29805
   395
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
hoelzl@29805
   396
hoelzl@29805
   397
*}
hoelzl@29805
   398
hoelzl@29805
   399
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805
   400
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   401
  "ub_arctan_horner prec 0 k x = 0"
hoelzl@31809
   402
| "ub_arctan_horner prec (Suc n) k x =
hoelzl@29805
   403
    (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
hoelzl@29805
   404
| "lb_arctan_horner prec 0 k x = 0"
hoelzl@31809
   405
| "lb_arctan_horner prec (Suc n) k x =
hoelzl@29805
   406
    (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
hoelzl@29805
   407
hoelzl@31098
   408
lemma arctan_0_1_bounds': assumes "0 \<le> real x" "real x \<le> 1" and "even n"
hoelzl@31098
   409
  shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec n 1 (x * x)) .. real (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
hoelzl@29805
   410
proof -
hoelzl@31098
   411
  let "?c i" = "-1^i * (1 / real (i * 2 + 1) * real x ^ (i * 2 + 1))"
hoelzl@29805
   412
  let "?S n" = "\<Sum> i=0..<n. ?c i"
hoelzl@29805
   413
hoelzl@31098
   414
  have "0 \<le> real (x * x)" by auto
hoelzl@29805
   415
  from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
hoelzl@31809
   416
hoelzl@31098
   417
  have "arctan (real x) \<in> { ?S n .. ?S (Suc n) }"
hoelzl@31098
   418
  proof (cases "real x = 0")
hoelzl@29805
   419
    case False
hoelzl@31098
   420
    hence "0 < real x" using `0 \<le> real x` by auto
hoelzl@31809
   421
    hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
hoelzl@29805
   422
hoelzl@31098
   423
    have "\<bar> real x \<bar> \<le> 1"  using `0 \<le> real x` `real x \<le> 1` by auto
hoelzl@29805
   424
    from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded `2 * m = n`]
nipkow@31790
   425
    show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1  .
hoelzl@29805
   426
  qed auto
hoelzl@29805
   427
  note arctan_bounds = this[unfolded atLeastAtMost_iff]
hoelzl@29805
   428
hoelzl@29805
   429
  have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
hoelzl@29805
   430
hoelzl@31809
   431
  note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
hoelzl@29805
   432
    and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
hoelzl@31809
   433
    and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
hoelzl@31098
   434
    OF `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
hoelzl@29805
   435
hoelzl@31098
   436
  { have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
hoelzl@31098
   437
      using bounds(1) `0 \<le> real x`
hoelzl@31098
   438
      unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@31098
   439
      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
hoelzl@29805
   440
      by (auto intro!: mult_left_mono)
hoelzl@31098
   441
    also have "\<dots> \<le> arctan (real x)" using arctan_bounds ..
hoelzl@31098
   442
    finally have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (real x)" . }
hoelzl@29805
   443
  moreover
hoelzl@31098
   444
  { have "arctan (real x) \<le> ?S (Suc n)" using arctan_bounds ..
hoelzl@31098
   445
    also have "\<dots> \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
hoelzl@31098
   446
      using bounds(2)[of "Suc n"] `0 \<le> real x`
hoelzl@31098
   447
      unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@31098
   448
      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
hoelzl@29805
   449
      by (auto intro!: mult_left_mono)
hoelzl@31098
   450
    finally have "arctan (real x) \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
hoelzl@29805
   451
  ultimately show ?thesis by auto
hoelzl@29805
   452
qed
hoelzl@29805
   453
hoelzl@31098
   454
lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
hoelzl@31098
   455
  shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
hoelzl@29805
   456
proof (cases "even n")
hoelzl@29805
   457
  case True
hoelzl@29805
   458
  obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
nipkow@31148
   459
  hence "even n'" unfolding even_Suc by auto
hoelzl@31098
   460
  have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
hoelzl@31098
   461
    unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
hoelzl@29805
   462
  moreover
hoelzl@31098
   463
  have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
hoelzl@31098
   464
    unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n`] by auto
hoelzl@29805
   465
  ultimately show ?thesis by auto
hoelzl@29805
   466
next
hoelzl@29805
   467
  case False hence "0 < n" by (rule odd_pos)
hoelzl@29805
   468
  from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
nipkow@31148
   469
  from False[unfolded this even_Suc]
hoelzl@29805
   470
  have "even n'" and "even (Suc (Suc n'))" by auto
hoelzl@29805
   471
  have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
hoelzl@29805
   472
hoelzl@31098
   473
  have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
hoelzl@31098
   474
    unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
hoelzl@29805
   475
  moreover
hoelzl@31098
   476
  have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
hoelzl@31098
   477
    unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even (Suc (Suc n'))`] by auto
hoelzl@29805
   478
  ultimately show ?thesis by auto
hoelzl@29805
   479
qed
hoelzl@29805
   480
hoelzl@29805
   481
subsection "Compute \<pi>"
hoelzl@29805
   482
hoelzl@29805
   483
definition ub_pi :: "nat \<Rightarrow> float" where
hoelzl@31809
   484
  "ub_pi prec = (let A = rapprox_rat prec 1 5 ;
hoelzl@29805
   485
                     B = lapprox_rat prec 1 239
hoelzl@31809
   486
                 in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) -
hoelzl@29805
   487
                                                  B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805
   488
hoelzl@29805
   489
definition lb_pi :: "nat \<Rightarrow> float" where
hoelzl@31809
   490
  "lb_pi prec = (let A = lapprox_rat prec 1 5 ;
hoelzl@29805
   491
                     B = rapprox_rat prec 1 239
hoelzl@31809
   492
                 in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
hoelzl@29805
   493
                                                  B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805
   494
hoelzl@31098
   495
lemma pi_boundaries: "pi \<in> {real (lb_pi n) .. real (ub_pi n)}"
hoelzl@29805
   496
proof -
hoelzl@29805
   497
  have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
hoelzl@29805
   498
hoelzl@29805
   499
  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
hoelzl@29805
   500
    let ?k = "rapprox_rat prec 1 k"
hoelzl@29805
   501
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@31809
   502
hoelzl@31098
   503
    have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
hoelzl@31098
   504
    have "real ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
hoelzl@29805
   505
      by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
hoelzl@29805
   506
hoelzl@31098
   507
    have "1 / real k \<le> real ?k" using rapprox_rat[where x=1 and y=k] by auto
hoelzl@31098
   508
    hence "arctan (1 / real k) \<le> arctan (real ?k)" by (rule arctan_monotone')
hoelzl@31098
   509
    also have "\<dots> \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
hoelzl@31098
   510
      using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
hoelzl@31098
   511
    finally have "arctan (1 / (real k)) \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
hoelzl@29805
   512
  } note ub_arctan = this
hoelzl@29805
   513
hoelzl@29805
   514
  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
hoelzl@29805
   515
    let ?k = "lapprox_rat prec 1 k"
hoelzl@29805
   516
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@29805
   517
    have "1 / real k \<le> 1" using `1 < k` by auto
hoelzl@29805
   518
hoelzl@31098
   519
    have "\<And>n. 0 \<le> real ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
hoelzl@31098
   520
    have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
hoelzl@29805
   521
hoelzl@31098
   522
    have "real ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
hoelzl@29805
   523
hoelzl@31098
   524
    have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (real ?k)"
hoelzl@31098
   525
      using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
hoelzl@31098
   526
    also have "\<dots> \<le> arctan (1 / real k)" using `real ?k \<le> 1 / real k` by (rule arctan_monotone')
hoelzl@31098
   527
    finally have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
hoelzl@29805
   528
  } note lb_arctan = this
hoelzl@29805
   529
hoelzl@31098
   530
  have "pi \<le> real (ub_pi n)"
hoelzl@31098
   531
    unfolding ub_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub unfolding Float_num
hoelzl@29805
   532
    using lb_arctan[of 239] ub_arctan[of 5]
hoelzl@29805
   533
    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
hoelzl@29805
   534
  moreover
hoelzl@31098
   535
  have "real (lb_pi n) \<le> pi"
hoelzl@31098
   536
    unfolding lb_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub Float_num
hoelzl@29805
   537
    using lb_arctan[of 5] ub_arctan[of 239]
hoelzl@29805
   538
    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
hoelzl@29805
   539
  ultimately show ?thesis by auto
hoelzl@29805
   540
qed
hoelzl@29805
   541
hoelzl@29805
   542
subsection "Compute arcus tangens in the entire domain"
hoelzl@29805
   543
hoelzl@31467
   544
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   545
  "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805
   546
                           lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
hoelzl@29805
   547
    in (if x < 0          then - ub_arctan prec (-x) else
hoelzl@29805
   548
        if x \<le> Float 1 -1 then lb_horner x else
hoelzl@31467
   549
        if x \<le> Float 1 1  then Float 1 1 * lb_horner (float_divl prec x (1 + ub_sqrt prec (1 + x * x)))
hoelzl@31467
   550
                          else (let inv = float_divr prec 1 x
hoelzl@31467
   551
                                in if inv > 1 then 0
hoelzl@29805
   552
                                              else lb_pi prec * Float 1 -1 - ub_horner inv)))"
hoelzl@29805
   553
hoelzl@29805
   554
| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805
   555
                           ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
hoelzl@29805
   556
    in (if x < 0          then - lb_arctan prec (-x) else
hoelzl@29805
   557
        if x \<le> Float 1 -1 then ub_horner x else
hoelzl@31467
   558
        if x \<le> Float 1 1  then let y = float_divr prec x (1 + lb_sqrt prec (1 + x * x))
hoelzl@31467
   559
                               in if y > 1 then ub_pi prec * Float 1 -1
hoelzl@31467
   560
                                           else Float 1 1 * ub_horner y
hoelzl@29805
   561
                          else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
hoelzl@29805
   562
by pat_completeness auto
hoelzl@29805
   563
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
hoelzl@29805
   564
hoelzl@29805
   565
declare ub_arctan_horner.simps[simp del]
hoelzl@29805
   566
declare lb_arctan_horner.simps[simp del]
hoelzl@29805
   567
hoelzl@31098
   568
lemma lb_arctan_bound': assumes "0 \<le> real x"
hoelzl@31098
   569
  shows "real (lb_arctan prec x) \<le> arctan (real x)"
hoelzl@29805
   570
proof -
hoelzl@31098
   571
  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
hoelzl@29805
   572
  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   573
    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   574
hoelzl@29805
   575
  show ?thesis
hoelzl@29805
   576
  proof (cases "x \<le> Float 1 -1")
hoelzl@31098
   577
    case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
hoelzl@29805
   578
    show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@31098
   579
      using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
hoelzl@29805
   580
  next
hoelzl@31098
   581
    case False hence "0 < real x" unfolding le_float_def Float_num by auto
hoelzl@31098
   582
    let ?R = "1 + sqrt (1 + real x * real x)"
hoelzl@31467
   583
    let ?fR = "1 + ub_sqrt prec (1 + x * x)"
hoelzl@29805
   584
    let ?DIV = "float_divl prec x ?fR"
hoelzl@31467
   585
hoelzl@31098
   586
    have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
hoelzl@29805
   587
    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805
   588
hoelzl@31467
   589
    have "sqrt (real (1 + x * x)) \<le> real (ub_sqrt prec (1 + x * x))"
hoelzl@31467
   590
      using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@31467
   591
hoelzl@31098
   592
    hence "?R \<le> real ?fR" by auto
hoelzl@31098
   593
    hence "0 < ?fR" and "0 < real ?fR" unfolding less_float_def using `0 < ?R` by auto
hoelzl@29805
   594
hoelzl@31098
   595
    have monotone: "real (float_divl prec x ?fR) \<le> real x / ?R"
hoelzl@29805
   596
    proof -
hoelzl@31098
   597
      have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
hoelzl@31098
   598
      also have "\<dots> \<le> real x / ?R" by (rule divide_left_mono[OF `?R \<le> real ?fR` `0 \<le> real x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> real ?fR`] divisor_gt0]])
hoelzl@29805
   599
      finally show ?thesis .
hoelzl@29805
   600
    qed
hoelzl@29805
   601
hoelzl@29805
   602
    show ?thesis
hoelzl@29805
   603
    proof (cases "x \<le> Float 1 1")
hoelzl@29805
   604
      case True
hoelzl@31467
   605
hoelzl@31098
   606
      have "real x \<le> sqrt (real (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
hoelzl@31467
   607
      also have "\<dots> \<le> real (ub_sqrt prec (1 + x * x))"
hoelzl@31467
   608
	using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@31098
   609
      finally have "real x \<le> real ?fR" by auto
hoelzl@31098
   610
      moreover have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
hoelzl@31098
   611
      ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto
hoelzl@29805
   612
hoelzl@31098
   613
      have "0 \<le> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
hoelzl@29805
   614
hoelzl@31098
   615
      have "real (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (real (float_divl prec x ?fR))" unfolding real_of_float_mult[of "Float 1 1"] Float_num
hoelzl@31098
   616
	using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
hoelzl@31098
   617
      also have "\<dots> \<le> 2 * arctan (real x / ?R)"
hoelzl@29805
   618
	using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@31809
   619
      also have "2 * arctan (real x / ?R) = arctan (real x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
hoelzl@29805
   620
      finally show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] .
hoelzl@29805
   621
    next
hoelzl@29805
   622
      case False
hoelzl@31098
   623
      hence "2 < real x" unfolding le_float_def Float_num by auto
hoelzl@31098
   624
      hence "1 \<le> real x" by auto
hoelzl@29805
   625
hoelzl@29805
   626
      let "?invx" = "float_divr prec 1 x"
hoelzl@31098
   627
      have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
hoelzl@29805
   628
hoelzl@29805
   629
      show ?thesis
hoelzl@29805
   630
      proof (cases "1 < ?invx")
hoelzl@29805
   631
	case True
hoelzl@31809
   632
	show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] if_P[OF True]
hoelzl@31098
   633
	  using `0 \<le> arctan (real x)` by auto
hoelzl@29805
   634
      next
hoelzl@29805
   635
	case False
hoelzl@31098
   636
	hence "real ?invx \<le> 1" unfolding less_float_def by auto
hoelzl@31098
   637
	have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
hoelzl@29805
   638
hoelzl@31098
   639
	have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
hoelzl@31467
   640
hoelzl@31098
   641
	have "arctan (1 / real x) \<le> arctan (real ?invx)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divr)
hoelzl@31098
   642
	also have "\<dots> \<le> real (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
hoelzl@31467
   643
	finally have "pi / 2 - real (?ub_horner ?invx) \<le> arctan (real x)"
hoelzl@31467
   644
	  using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
hoelzl@31098
   645
	  unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
hoelzl@29805
   646
	moreover
hoelzl@31098
   647
	have "real (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
hoelzl@29805
   648
	ultimately
hoelzl@29805
   649
	show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
hoelzl@29805
   650
	  by auto
hoelzl@29805
   651
      qed
hoelzl@29805
   652
    qed
hoelzl@29805
   653
  qed
hoelzl@29805
   654
qed
hoelzl@29805
   655
hoelzl@31098
   656
lemma ub_arctan_bound': assumes "0 \<le> real x"
hoelzl@31098
   657
  shows "arctan (real x) \<le> real (ub_arctan prec x)"
hoelzl@29805
   658
proof -
hoelzl@31098
   659
  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
hoelzl@29805
   660
hoelzl@29805
   661
  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   662
    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   663
hoelzl@29805
   664
  show ?thesis
hoelzl@29805
   665
  proof (cases "x \<le> Float 1 -1")
hoelzl@31098
   666
    case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
hoelzl@29805
   667
    show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@31098
   668
      using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
hoelzl@29805
   669
  next
hoelzl@31098
   670
    case False hence "0 < real x" unfolding le_float_def Float_num by auto
hoelzl@31098
   671
    let ?R = "1 + sqrt (1 + real x * real x)"
hoelzl@31467
   672
    let ?fR = "1 + lb_sqrt prec (1 + x * x)"
hoelzl@29805
   673
    let ?DIV = "float_divr prec x ?fR"
hoelzl@31467
   674
hoelzl@31098
   675
    have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
hoelzl@31098
   676
    hence "0 \<le> real (1 + x*x)" by auto
hoelzl@31467
   677
hoelzl@29805
   678
    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805
   679
hoelzl@31467
   680
    have "real (lb_sqrt prec (1 + x * x)) \<le> sqrt (real (1 + x * x))"
hoelzl@31467
   681
      using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@31098
   682
    hence "real ?fR \<le> ?R" by auto
hoelzl@31098
   683
    have "0 < real ?fR" unfolding real_of_float_add real_of_float_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> real (1 + x*x)`])
hoelzl@29805
   684
hoelzl@31098
   685
    have monotone: "real x / ?R \<le> real (float_divr prec x ?fR)"
hoelzl@29805
   686
    proof -
hoelzl@31098
   687
      from divide_left_mono[OF `real ?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
hoelzl@31098
   688
      have "real x / ?R \<le> real x / real ?fR" .
hoelzl@31098
   689
      also have "\<dots> \<le> real ?DIV" by (rule float_divr)
hoelzl@29805
   690
      finally show ?thesis .
hoelzl@29805
   691
    qed
hoelzl@29805
   692
hoelzl@29805
   693
    show ?thesis
hoelzl@29805
   694
    proof (cases "x \<le> Float 1 1")
hoelzl@29805
   695
      case True
hoelzl@29805
   696
      show ?thesis
hoelzl@29805
   697
      proof (cases "?DIV > 1")
hoelzl@29805
   698
	case True
hoelzl@31098
   699
	have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
hoelzl@29805
   700
	from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
hoelzl@29805
   701
	show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_P[OF True] .
hoelzl@29805
   702
      next
hoelzl@29805
   703
	case False
hoelzl@31098
   704
	hence "real ?DIV \<le> 1" unfolding less_float_def by auto
hoelzl@31467
   705
hoelzl@31098
   706
	have "0 \<le> real x / ?R" using `0 \<le> real x` `0 < ?R` unfolding real_0_le_divide_iff by auto
hoelzl@31098
   707
	hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
hoelzl@29805
   708
hoelzl@31098
   709
	have "arctan (real x) = 2 * arctan (real x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
hoelzl@31098
   710
	also have "\<dots> \<le> 2 * arctan (real ?DIV)"
hoelzl@29805
   711
	  using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@31098
   712
	also have "\<dots> \<le> real (Float 1 1 * ?ub_horner ?DIV)" unfolding real_of_float_mult[of "Float 1 1"] Float_num
hoelzl@31098
   713
	  using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
hoelzl@29805
   714
	finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_not_P[OF False] .
hoelzl@29805
   715
      qed
hoelzl@29805
   716
    next
hoelzl@29805
   717
      case False
hoelzl@31098
   718
      hence "2 < real x" unfolding le_float_def Float_num by auto
hoelzl@31098
   719
      hence "1 \<le> real x" by auto
hoelzl@31098
   720
      hence "0 < real x" by auto
hoelzl@29805
   721
      hence "0 < x" unfolding less_float_def by auto
hoelzl@29805
   722
hoelzl@29805
   723
      let "?invx" = "float_divl prec 1 x"
hoelzl@31098
   724
      have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
hoelzl@29805
   725
hoelzl@31098
   726
      have "real ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> real x` divide_le_eq_1_pos[OF `0 < real x`])
hoelzl@31098
   727
      have "0 \<le> real ?invx" unfolding real_of_float_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
hoelzl@31467
   728
hoelzl@31098
   729
      have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
hoelzl@31467
   730
hoelzl@31098
   731
      have "real (?lb_horner ?invx) \<le> arctan (real ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
hoelzl@31098
   732
      also have "\<dots> \<le> arctan (1 / real x)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divl)
hoelzl@31098
   733
      finally have "arctan (real x) \<le> pi / 2 - real (?lb_horner ?invx)"
hoelzl@31809
   734
	using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
hoelzl@31098
   735
	unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
hoelzl@29805
   736
      moreover
hoelzl@31098
   737
      have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
hoelzl@29805
   738
      ultimately
hoelzl@29805
   739
      show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
hoelzl@29805
   740
	by auto
hoelzl@29805
   741
    qed
hoelzl@29805
   742
  qed
hoelzl@29805
   743
qed
hoelzl@29805
   744
hoelzl@29805
   745
lemma arctan_boundaries:
hoelzl@31098
   746
  "arctan (real x) \<in> {real (lb_arctan prec x) .. real (ub_arctan prec x)}"
hoelzl@29805
   747
proof (cases "0 \<le> x")
hoelzl@31098
   748
  case True hence "0 \<le> real x" unfolding le_float_def by auto
hoelzl@31098
   749
  show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
hoelzl@29805
   750
next
hoelzl@29805
   751
  let ?mx = "-x"
hoelzl@31098
   752
  case False hence "x < 0" and "0 \<le> real ?mx" unfolding le_float_def less_float_def by auto
hoelzl@31098
   753
  hence bounds: "real (lb_arctan prec ?mx) \<le> arctan (real ?mx) \<and> arctan (real ?mx) \<le> real (ub_arctan prec ?mx)"
hoelzl@31098
   754
    using ub_arctan_bound'[OF `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] by auto
hoelzl@31098
   755
  show ?thesis unfolding real_of_float_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
hoelzl@31098
   756
    unfolding atLeastAtMost_iff using bounds[unfolded real_of_float_minus arctan_minus] by auto
hoelzl@29805
   757
qed
hoelzl@29805
   758
hoelzl@31098
   759
lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> arctan x \<and> arctan x \<le> real u"
hoelzl@29805
   760
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
   761
  fix x lx ux
hoelzl@31098
   762
  assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux}"
hoelzl@31098
   763
  hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
hoelzl@29805
   764
hoelzl@29805
   765
  { from arctan_boundaries[of lx prec, unfolded l]
hoelzl@31098
   766
    have "real l \<le> arctan (real lx)" by (auto simp del: lb_arctan.simps)
hoelzl@29805
   767
    also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
hoelzl@31098
   768
    finally have "real l \<le> arctan x" .
hoelzl@29805
   769
  } moreover
hoelzl@31098
   770
  { have "arctan x \<le> arctan (real ux)" using x by (auto intro: arctan_monotone')
hoelzl@31098
   771
    also have "\<dots> \<le> real u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
hoelzl@31098
   772
    finally have "arctan x \<le> real u" .
hoelzl@31098
   773
  } ultimately show "real l \<le> arctan x \<and> arctan x \<le> real u" ..
hoelzl@29805
   774
qed
hoelzl@29805
   775
hoelzl@29805
   776
section "Sinus and Cosinus"
hoelzl@29805
   777
hoelzl@29805
   778
subsection "Compute the cosinus and sinus series"
hoelzl@29805
   779
hoelzl@29805
   780
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805
   781
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   782
  "ub_sin_cos_aux prec 0 i k x = 0"
hoelzl@31809
   783
| "ub_sin_cos_aux prec (Suc n) i k x =
hoelzl@29805
   784
    (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@29805
   785
| "lb_sin_cos_aux prec 0 i k x = 0"
hoelzl@31809
   786
| "lb_sin_cos_aux prec (Suc n) i k x =
hoelzl@29805
   787
    (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@29805
   788
hoelzl@29805
   789
lemma cos_aux:
hoelzl@31098
   790
  shows "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i))" (is "?lb")
hoelzl@31098
   791
  and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i)) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
hoelzl@29805
   792
proof -
hoelzl@31098
   793
  have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
hoelzl@29805
   794
  let "?f n" = "fact (2 * n)"
hoelzl@29805
   795
hoelzl@31809
   796
  { fix n
haftmann@30971
   797
    have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
haftmann@30971
   798
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
hoelzl@29805
   799
      unfolding F by auto } note f_eq = this
hoelzl@31809
   800
hoelzl@31809
   801
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
hoelzl@31098
   802
    OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@31098
   803
  show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "real x"])
hoelzl@29805
   804
qed
hoelzl@29805
   805
hoelzl@31098
   806
lemma cos_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
hoelzl@31098
   807
  shows "cos (real x) \<in> {real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
hoelzl@31098
   808
proof (cases "real x = 0")
hoelzl@31098
   809
  case False hence "real x \<noteq> 0" by auto
hoelzl@31098
   810
  hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
hoelzl@31098
   811
  have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
hoelzl@31098
   812
    using mult_pos_pos[where a="real x" and b="real x"] by auto
hoelzl@29805
   813
haftmann@30952
   814
  { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
hoelzl@29805
   815
    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
hoelzl@29805
   816
  proof -
hoelzl@29805
   817
    have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
hoelzl@31809
   818
    also have "\<dots> =
hoelzl@29805
   819
      (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
hoelzl@29805
   820
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
hoelzl@29805
   821
      unfolding sum_split_even_odd ..
hoelzl@29805
   822
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
hoelzl@29805
   823
      by (rule setsum_cong2) auto
hoelzl@29805
   824
    finally show ?thesis by assumption
hoelzl@29805
   825
  qed } note morph_to_if_power = this
hoelzl@29805
   826
hoelzl@29805
   827
hoelzl@29805
   828
  { fix n :: nat assume "0 < n"
hoelzl@29805
   829
    hence "0 < 2 * n" by auto
hoelzl@31098
   830
    obtain t where "0 < t" and "t < real x" and
hoelzl@31809
   831
      cos_eq: "cos (real x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
hoelzl@31809
   832
      + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
hoelzl@29805
   833
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
hoelzl@31098
   834
      using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`] by auto
hoelzl@29805
   835
hoelzl@29805
   836
    have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
hoelzl@29805
   837
    also have "\<dots> = cos (t + real n * pi)"  using cos_add by auto
hoelzl@29805
   838
    also have "\<dots> = ?rest" by auto
hoelzl@29805
   839
    finally have "cos t * -1^n = ?rest" .
hoelzl@29805
   840
    moreover
hoelzl@31098
   841
    have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
hoelzl@29805
   842
    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805
   843
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805
   844
hoelzl@29805
   845
    have "0 < ?fact" by auto
hoelzl@31098
   846
    have "0 < ?pow" using `0 < real x` by auto
hoelzl@29805
   847
hoelzl@29805
   848
    {
hoelzl@29805
   849
      assume "even n"
hoelzl@31098
   850
      have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
hoelzl@31809
   851
	unfolding morph_to_if_power[symmetric] using cos_aux by auto
hoelzl@31098
   852
      also have "\<dots> \<le> cos (real x)"
hoelzl@29805
   853
      proof -
hoelzl@29805
   854
	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@29805
   855
	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   856
	thus ?thesis unfolding cos_eq by auto
hoelzl@29805
   857
      qed
hoelzl@31098
   858
      finally have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (real x)" .
hoelzl@29805
   859
    } note lb = this
hoelzl@29805
   860
hoelzl@29805
   861
    {
hoelzl@29805
   862
      assume "odd n"
hoelzl@31098
   863
      have "cos (real x) \<le> ?SUM"
hoelzl@29805
   864
      proof -
hoelzl@29805
   865
	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
hoelzl@29805
   866
	have "0 \<le> (- ?rest) / ?fact * ?pow"
hoelzl@29805
   867
	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   868
	thus ?thesis unfolding cos_eq by auto
hoelzl@29805
   869
      qed
hoelzl@31098
   870
      also have "\<dots> \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))"
hoelzl@29805
   871
	unfolding morph_to_if_power[symmetric] using cos_aux by auto
hoelzl@31098
   872
      finally have "cos (real x) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" .
hoelzl@29805
   873
    } note ub = this and lb
hoelzl@29805
   874
  } note ub = this(1) and lb = this(2)
hoelzl@29805
   875
hoelzl@31098
   876
  have "cos (real x) \<le> real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
hoelzl@31809
   877
  moreover have "real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (real x)"
hoelzl@29805
   878
  proof (cases "0 < get_even n")
hoelzl@29805
   879
    case True show ?thesis using lb[OF True get_even] .
hoelzl@29805
   880
  next
hoelzl@29805
   881
    case False
hoelzl@29805
   882
    hence "get_even n = 0" by auto
hoelzl@31098
   883
    have "- (pi / 2) \<le> real x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
hoelzl@31098
   884
    with `real x \<le> pi / 2`
hoelzl@31098
   885
    show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps real_of_float_minus real_of_float_0 using cos_ge_zero by auto
hoelzl@29805
   886
  qed
hoelzl@29805
   887
  ultimately show ?thesis by auto
hoelzl@29805
   888
next
hoelzl@29805
   889
  case True
hoelzl@29805
   890
  show ?thesis
hoelzl@29805
   891
  proof (cases "n = 0")
hoelzl@31809
   892
    case True
hoelzl@31098
   893
    thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
hoelzl@29805
   894
  next
hoelzl@29805
   895
    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@31098
   896
    thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
hoelzl@29805
   897
  qed
hoelzl@29805
   898
qed
hoelzl@29805
   899
hoelzl@31098
   900
lemma sin_aux: assumes "0 \<le> real x"
hoelzl@31098
   901
  shows "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1))" (is "?lb")
hoelzl@31098
   902
  and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1)) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
hoelzl@29805
   903
proof -
hoelzl@31098
   904
  have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
hoelzl@29805
   905
  let "?f n" = "fact (2 * n + 1)"
hoelzl@29805
   906
hoelzl@31809
   907
  { fix n
haftmann@30971
   908
    have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
haftmann@30971
   909
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
hoelzl@29805
   910
      unfolding F by auto } note f_eq = this
hoelzl@31809
   911
hoelzl@29805
   912
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
hoelzl@31098
   913
    OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@31098
   914
  show "?lb" and "?ub" using `0 \<le> real x` unfolding real_of_float_mult
hoelzl@29805
   915
    unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@29805
   916
    unfolding real_mult_commute
hoelzl@31098
   917
    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
hoelzl@29805
   918
qed
hoelzl@29805
   919
hoelzl@31098
   920
lemma sin_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
hoelzl@31098
   921
  shows "sin (real x) \<in> {real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
hoelzl@31098
   922
proof (cases "real x = 0")
hoelzl@31098
   923
  case False hence "real x \<noteq> 0" by auto
hoelzl@31098
   924
  hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
hoelzl@31098
   925
  have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
hoelzl@31098
   926
    using mult_pos_pos[where a="real x" and b="real x"] by auto
hoelzl@29805
   927
hoelzl@29805
   928
  { fix x n have "(\<Sum> j = 0 ..< n. -1 ^ (((2 * j + 1) - Suc 0) div 2) / (real (fact (2 * j + 1))) * x ^(2 * j + 1))
hoelzl@29805
   929
    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
hoelzl@29805
   930
    proof -
hoelzl@29805
   931
      have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
hoelzl@29805
   932
      have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
hoelzl@29805
   933
      also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)"
hoelzl@29805
   934
	unfolding sum_split_even_odd ..
hoelzl@29805
   935
      also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)"
hoelzl@29805
   936
	by (rule setsum_cong2) auto
hoelzl@29805
   937
      finally show ?thesis by assumption
hoelzl@29805
   938
    qed } note setsum_morph = this
hoelzl@29805
   939
hoelzl@29805
   940
  { fix n :: nat assume "0 < n"
hoelzl@29805
   941
    hence "0 < 2 * n + 1" by auto
hoelzl@31098
   942
    obtain t where "0 < t" and "t < real x" and
hoelzl@31809
   943
      sin_eq: "sin (real x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
hoelzl@31809
   944
      + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
hoelzl@29805
   945
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
hoelzl@31098
   946
      using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`] by auto
hoelzl@29805
   947
hoelzl@29805
   948
    have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
hoelzl@29805
   949
    moreover
hoelzl@31098
   950
    have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
hoelzl@29805
   951
    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805
   952
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805
   953
hoelzl@29805
   954
    have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
hoelzl@31098
   955
    have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
hoelzl@29805
   956
hoelzl@29805
   957
    {
hoelzl@29805
   958
      assume "even n"
hoelzl@31809
   959
      have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
hoelzl@31098
   960
            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
hoelzl@31098
   961
	using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
hoelzl@29805
   962
      also have "\<dots> \<le> ?SUM" by auto
hoelzl@31098
   963
      also have "\<dots> \<le> sin (real x)"
hoelzl@29805
   964
      proof -
hoelzl@29805
   965
	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@29805
   966
	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   967
	thus ?thesis unfolding sin_eq by auto
hoelzl@29805
   968
      qed
hoelzl@31098
   969
      finally have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (real x)" .
hoelzl@29805
   970
    } note lb = this
hoelzl@29805
   971
hoelzl@29805
   972
    {
hoelzl@29805
   973
      assume "odd n"
hoelzl@31098
   974
      have "sin (real x) \<le> ?SUM"
hoelzl@29805
   975
      proof -
hoelzl@29805
   976
	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
hoelzl@29805
   977
	have "0 \<le> (- ?rest) / ?fact * ?pow"
hoelzl@29805
   978
	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   979
	thus ?thesis unfolding sin_eq by auto
hoelzl@29805
   980
      qed
hoelzl@31098
   981
      also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
hoelzl@29805
   982
	 by auto
hoelzl@31809
   983
      also have "\<dots> \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))"
hoelzl@31098
   984
	using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
hoelzl@31098
   985
      finally have "sin (real x) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
hoelzl@29805
   986
    } note ub = this and lb
hoelzl@29805
   987
  } note ub = this(1) and lb = this(2)
hoelzl@29805
   988
hoelzl@31098
   989
  have "sin (real x) \<le> real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
hoelzl@31809
   990
  moreover have "real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (real x)"
hoelzl@29805
   991
  proof (cases "0 < get_even n")
hoelzl@29805
   992
    case True show ?thesis using lb[OF True get_even] .
hoelzl@29805
   993
  next
hoelzl@29805
   994
    case False
hoelzl@29805
   995
    hence "get_even n = 0" by auto
hoelzl@31098
   996
    with `real x \<le> pi / 2` `0 \<le> real x`
hoelzl@31098
   997
    show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps real_of_float_minus real_of_float_0 using sin_ge_zero by auto
hoelzl@29805
   998
  qed
hoelzl@29805
   999
  ultimately show ?thesis by auto
hoelzl@29805
  1000
next
hoelzl@29805
  1001
  case True
hoelzl@29805
  1002
  show ?thesis
hoelzl@29805
  1003
  proof (cases "n = 0")
hoelzl@31809
  1004
    case True
hoelzl@31098
  1005
    thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
hoelzl@29805
  1006
  next
hoelzl@29805
  1007
    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@31098
  1008
    thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
hoelzl@29805
  1009
  qed
hoelzl@29805
  1010
qed
hoelzl@29805
  1011
hoelzl@29805
  1012
subsection "Compute the cosinus in the entire domain"
hoelzl@29805
  1013
hoelzl@29805
  1014
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1015
"lb_cos prec x = (let
hoelzl@29805
  1016
    horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805
  1017
    half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
hoelzl@29805
  1018
  in if x < Float 1 -1 then horner x
hoelzl@29805
  1019
else if x < 1          then half (horner (x * Float 1 -1))
hoelzl@29805
  1020
                       else half (half (horner (x * Float 1 -2))))"
hoelzl@29805
  1021
hoelzl@29805
  1022
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1023
"ub_cos prec x = (let
hoelzl@29805
  1024
    horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805
  1025
    half = \<lambda> x. Float 1 1 * x * x - 1
hoelzl@29805
  1026
  in if x < Float 1 -1 then horner x
hoelzl@29805
  1027
else if x < 1          then half (horner (x * Float 1 -1))
hoelzl@29805
  1028
                       else half (half (horner (x * Float 1 -2))))"
hoelzl@29805
  1029
hoelzl@31467
  1030
lemma lb_cos: assumes "0 \<le> real x" and "real x \<le> pi"
hoelzl@31098
  1031
  shows "cos (real x) \<in> {real (lb_cos prec x) .. real (ub_cos prec x)}" (is "?cos x \<in> { real (?lb x) .. real (?ub x) }")
hoelzl@29805
  1032
proof -
hoelzl@29805
  1033
  { fix x :: real
hoelzl@29805
  1034
    have "cos x = cos (x / 2 + x / 2)" by auto
hoelzl@29805
  1035
    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"
hoelzl@29805
  1036
      unfolding cos_add by auto
hoelzl@29805
  1037
    also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
hoelzl@29805
  1038
    finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
hoelzl@29805
  1039
  } note x_half = this[symmetric]
hoelzl@29805
  1040
hoelzl@31098
  1041
  have "\<not> x < 0" using `0 \<le> real x` unfolding less_float_def by auto
hoelzl@29805
  1042
  let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805
  1043
  let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805
  1044
  let "?ub_half x" = "Float 1 1 * x * x - 1"
hoelzl@29805
  1045
  let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
hoelzl@29805
  1046
hoelzl@29805
  1047
  show ?thesis
hoelzl@29805
  1048
  proof (cases "x < Float 1 -1")
hoelzl@31098
  1049
    case True hence "real x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
hoelzl@29805
  1050
    show ?thesis unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_P[OF `x < Float 1 -1`] Let_def
hoelzl@31098
  1051
      using cos_boundaries[OF `0 \<le> real x` `real x \<le> pi / 2`] .
hoelzl@29805
  1052
  next
hoelzl@29805
  1053
    case False
hoelzl@31098
  1054
    { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
hoelzl@31098
  1055
      assume "real y \<le> cos ?x2" and "-pi \<le> real x" and "real x \<le> pi"
hoelzl@31098
  1056
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
hoelzl@29805
  1057
      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@31467
  1058
hoelzl@31098
  1059
      have "real (?lb_half y) \<le> cos (real x)"
hoelzl@29805
  1060
      proof (cases "y < 0")
hoelzl@29805
  1061
	case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
hoelzl@29805
  1062
      next
hoelzl@29805
  1063
	case False
hoelzl@31098
  1064
	hence "0 \<le> real y" unfolding less_float_def by auto
hoelzl@31098
  1065
	from mult_mono[OF `real y \<le> cos ?x2` `real y \<le> cos ?x2` `0 \<le> cos ?x2` this]
hoelzl@31098
  1066
	have "real y * real y \<le> cos ?x2 * cos ?x2" .
hoelzl@31098
  1067
	hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
hoelzl@31098
  1068
	hence "2 * real y * real y - 1 \<le> 2 * cos (real x / 2) * cos (real x / 2) - 1" unfolding Float_num real_of_float_mult by auto
hoelzl@31098
  1069
	thus ?thesis unfolding if_not_P[OF False] x_half Float_num real_of_float_mult real_of_float_sub by auto
hoelzl@29805
  1070
      qed
hoelzl@29805
  1071
    } note lb_half = this
hoelzl@31467
  1072
hoelzl@31098
  1073
    { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
hoelzl@31098
  1074
      assume ub: "cos ?x2 \<le> real y" and "- pi \<le> real x" and "real x \<le> pi"
hoelzl@31098
  1075
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
hoelzl@29805
  1076
      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@31467
  1077
hoelzl@31098
  1078
      have "cos (real x) \<le> real (?ub_half y)"
hoelzl@29805
  1079
      proof -
hoelzl@31098
  1080
	have "0 \<le> real y" using `0 \<le> cos ?x2` ub by (rule order_trans)
hoelzl@29805
  1081
	from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
hoelzl@31098
  1082
	have "cos ?x2 * cos ?x2 \<le> real y * real y" .
hoelzl@31098
  1083
	hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
hoelzl@31098
  1084
	hence "2 * cos (real x / 2) * cos (real x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num real_of_float_mult by auto
hoelzl@31098
  1085
	thus ?thesis unfolding x_half real_of_float_mult Float_num real_of_float_sub by auto
hoelzl@29805
  1086
      qed
hoelzl@29805
  1087
    } note ub_half = this
hoelzl@31467
  1088
hoelzl@29805
  1089
    let ?x2 = "x * Float 1 -1"
hoelzl@29805
  1090
    let ?x4 = "x * Float 1 -1 * Float 1 -1"
hoelzl@31467
  1091
hoelzl@31098
  1092
    have "-pi \<le> real x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> real x` by (rule order_trans)
hoelzl@31467
  1093
hoelzl@29805
  1094
    show ?thesis
hoelzl@29805
  1095
    proof (cases "x < 1")
hoelzl@31098
  1096
      case True hence "real x \<le> 1" unfolding less_float_def by auto
hoelzl@31098
  1097
      have "0 \<le> real ?x2" and "real ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` unfolding real_of_float_mult Float_num using assms by auto
hoelzl@29805
  1098
      from cos_boundaries[OF this]
hoelzl@31098
  1099
      have lb: "real (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> real (?ub_horner ?x2)" by auto
hoelzl@31467
  1100
hoelzl@31098
  1101
      have "real (?lb x) \<le> ?cos x"
hoelzl@29805
  1102
      proof -
hoelzl@31098
  1103
	from lb_half[OF lb `-pi \<le> real x` `real x \<le> pi`]
hoelzl@29805
  1104
	show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
hoelzl@29805
  1105
      qed
hoelzl@31098
  1106
      moreover have "?cos x \<le> real (?ub x)"
hoelzl@29805
  1107
      proof -
hoelzl@31098
  1108
	from ub_half[OF ub `-pi \<le> real x` `real x \<le> pi`]
hoelzl@31809
  1109
	show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
hoelzl@29805
  1110
      qed
hoelzl@29805
  1111
      ultimately show ?thesis by auto
hoelzl@29805
  1112
    next
hoelzl@29805
  1113
      case False
hoelzl@31098
  1114
      have "0 \<le> real ?x4" and "real ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> real x` `real x \<le> pi` unfolding real_of_float_mult Float_num by auto
hoelzl@29805
  1115
      from cos_boundaries[OF this]
hoelzl@31098
  1116
      have lb: "real (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> real (?ub_horner ?x4)" by auto
hoelzl@31467
  1117
hoelzl@29805
  1118
      have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
hoelzl@31467
  1119
hoelzl@31098
  1120
      have "real (?lb x) \<le> ?cos x"
hoelzl@29805
  1121
      proof -
hoelzl@31098
  1122
	have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
hoelzl@31098
  1123
	from lb_half[OF lb_half[OF lb this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
hoelzl@29805
  1124
	show ?thesis unfolding lb_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def .
hoelzl@29805
  1125
      qed
hoelzl@31098
  1126
      moreover have "?cos x \<le> real (?ub x)"
hoelzl@29805
  1127
      proof -
hoelzl@31098
  1128
	have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
hoelzl@31098
  1129
	from ub_half[OF ub_half[OF ub this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
hoelzl@29805
  1130
	show ?thesis unfolding ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def .
hoelzl@29805
  1131
      qed
hoelzl@29805
  1132
      ultimately show ?thesis by auto
hoelzl@29805
  1133
    qed
hoelzl@29805
  1134
  qed
hoelzl@29805
  1135
qed
hoelzl@29805
  1136
hoelzl@31467
  1137
lemma lb_cos_minus: assumes "-pi \<le> real x" and "real x \<le> 0"
hoelzl@31098
  1138
  shows "cos (real (-x)) \<in> {real (lb_cos prec (-x)) .. real (ub_cos prec (-x))}"
hoelzl@29805
  1139
proof -
hoelzl@31098
  1140
  have "0 \<le> real (-x)" and "real (-x) \<le> pi" using `-pi \<le> real x` `real x \<le> 0` by auto
hoelzl@29805
  1141
  from lb_cos[OF this] show ?thesis .
hoelzl@29805
  1142
qed
hoelzl@29805
  1143
hoelzl@31467
  1144
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@31467
  1145
"bnds_cos prec lx ux = (let
hoelzl@31467
  1146
    lpi = round_down prec (lb_pi prec) ;
hoelzl@31467
  1147
    upi = round_up prec (ub_pi prec) ;
hoelzl@31467
  1148
    k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
hoelzl@31467
  1149
    lx = lx - k * 2 * (if k < 0 then lpi else upi) ;
hoelzl@31467
  1150
    ux = ux - k * 2 * (if k < 0 then upi else lpi)
hoelzl@31467
  1151
  in   if - lpi \<le> lx \<and> ux \<le> 0    then (lb_cos prec (-lx), ub_cos prec (-ux))
hoelzl@31467
  1152
  else if 0 \<le> lx \<and> ux \<le> lpi      then (lb_cos prec ux, ub_cos prec lx)
hoelzl@31467
  1153
  else if - lpi \<le> lx \<and> ux \<le> lpi  then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
hoelzl@31467
  1154
  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))))
hoelzl@31508
  1155
  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)))
hoelzl@31467
  1156
                                 else (Float -1 0, Float 1 0))"
hoelzl@29805
  1157
hoelzl@31467
  1158
lemma floor_int:
hoelzl@31467
  1159
  obtains k :: int where "real k = real (floor_fl f)"
hoelzl@31467
  1160
proof -
hoelzl@31467
  1161
  assume *: "\<And> k :: int. real k = real (floor_fl f) \<Longrightarrow> thesis"
hoelzl@31467
  1162
  obtain m e where fl: "Float m e = floor_fl f" by (cases "floor_fl f", auto)
hoelzl@31467
  1163
  from floor_pos_exp[OF this]
hoelzl@31467
  1164
  have "real (m* 2^(nat e)) = real (floor_fl f)"
hoelzl@31467
  1165
    by (auto simp add: fl[symmetric] real_of_float_def pow2_def)
hoelzl@31467
  1166
  from *[OF this] show thesis by blast
hoelzl@31467
  1167
qed
hoelzl@29805
  1168
hoelzl@31467
  1169
lemma float_remove_real_numeral[simp]: "real (number_of k :: float) = number_of k"
hoelzl@31467
  1170
proof -
hoelzl@31467
  1171
  have "real (number_of k :: float) = real k"
hoelzl@31467
  1172
    unfolding number_of_float_def real_of_float_def pow2_def by auto
hoelzl@31467
  1173
  also have "\<dots> = real (number_of k :: int)"
hoelzl@31467
  1174
    by (simp add: number_of_is_id)
hoelzl@31467
  1175
  finally show ?thesis by auto
hoelzl@31467
  1176
qed
hoelzl@29805
  1177
hoelzl@31467
  1178
lemma cos_periodic_nat[simp]: fixes n :: nat shows "cos (x + real n * 2 * pi) = cos x"
hoelzl@31467
  1179
proof (induct n arbitrary: x)
hoelzl@31467
  1180
  case (Suc n)
hoelzl@31467
  1181
  have split_pi_off: "x + real (Suc n) * 2 * pi = (x + real n * 2 * pi) + 2 * pi"
nipkow@31790
  1182
    unfolding Suc_eq_plus1 real_of_nat_add real_of_one real_add_mult_distrib by auto
hoelzl@31467
  1183
  show ?case unfolding split_pi_off using Suc by auto
hoelzl@31467
  1184
qed auto
hoelzl@31467
  1185
hoelzl@31467
  1186
lemma cos_periodic_int[simp]: fixes i :: int shows "cos (x + real i * 2 * pi) = cos x"
hoelzl@31467
  1187
proof (cases "0 \<le> i")
hoelzl@31467
  1188
  case True hence i_nat: "real i = real (nat i)" by auto
hoelzl@31467
  1189
  show ?thesis unfolding i_nat by auto
hoelzl@31467
  1190
next
hoelzl@31467
  1191
  case False hence i_nat: "real i = - real (nat (-i))" by auto
hoelzl@31467
  1192
  have "cos x = cos (x + real i * 2 * pi - real i * 2 * pi)" by auto
hoelzl@31467
  1193
  also have "\<dots> = cos (x + real i * 2 * pi)"
hoelzl@31467
  1194
    unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
hoelzl@31467
  1195
  finally show ?thesis by auto
hoelzl@29805
  1196
qed
hoelzl@29805
  1197
hoelzl@31467
  1198
lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> cos x \<and> cos x \<le> real u"
hoelzl@31467
  1199
proof ((rule allI | rule impI | erule conjE) +)
hoelzl@31467
  1200
  fix x lx ux
hoelzl@31467
  1201
  assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {real lx .. real ux}"
hoelzl@31467
  1202
hoelzl@31467
  1203
  let ?lpi = "round_down prec (lb_pi prec)"
hoelzl@31467
  1204
  let ?upi = "round_up prec (ub_pi prec)"
hoelzl@31467
  1205
  let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
hoelzl@31467
  1206
  let ?lx = "lx - ?k * 2 * (if ?k < 0 then ?lpi else ?upi)"
hoelzl@31467
  1207
  let ?ux = "ux - ?k * 2 * (if ?k < 0 then ?upi else ?lpi)"
hoelzl@31467
  1208
hoelzl@31467
  1209
  obtain k :: int where k: "real k = real ?k" using floor_int .
hoelzl@31467
  1210
hoelzl@31467
  1211
  have upi: "pi \<le> real ?upi" and lpi: "real ?lpi \<le> pi"
hoelzl@31467
  1212
    using round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
hoelzl@31467
  1213
          round_down[of prec "lb_pi prec"] by auto
hoelzl@31467
  1214
  hence "real ?lx \<le> x - real k * 2 * pi \<and> x - real k * 2 * pi \<le> real ?ux"
hoelzl@31467
  1215
    using x by (cases "k = 0") (auto intro!: add_mono
hoelzl@31467
  1216
                simp add: real_diff_def k[symmetric] less_float_def)
hoelzl@31467
  1217
  note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
hoelzl@31467
  1218
  hence lx_less_ux: "real ?lx \<le> real ?ux" by (rule order_trans)
hoelzl@31467
  1219
hoelzl@31467
  1220
  { assume "- ?lpi \<le> ?lx" and x_le_0: "x - real k * 2 * pi \<le> 0"
hoelzl@31467
  1221
    with lpi[THEN le_imp_neg_le] lx
hoelzl@31467
  1222
    have pi_lx: "- pi \<le> real ?lx" and lx_0: "real ?lx \<le> 0"
hoelzl@31467
  1223
      by (simp_all add: le_float_def)
hoelzl@29805
  1224
hoelzl@31467
  1225
    have "real (lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
hoelzl@31467
  1226
      using lb_cos_minus[OF pi_lx lx_0] by simp
hoelzl@31467
  1227
    also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
hoelzl@31467
  1228
      using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
hoelzl@31467
  1229
      by (simp only: real_of_float_minus real_of_int_minus
hoelzl@31467
  1230
	cos_minus real_diff_def mult_minus_left)
hoelzl@31467
  1231
    finally have "real (lb_cos prec (- ?lx)) \<le> cos x"
hoelzl@31467
  1232
      unfolding cos_periodic_int . }
hoelzl@31467
  1233
  note negative_lx = this
hoelzl@31467
  1234
hoelzl@31467
  1235
  { assume "0 \<le> ?lx" and pi_x: "x - real k * 2 * pi \<le> pi"
hoelzl@31467
  1236
    with lx
hoelzl@31467
  1237
    have pi_lx: "real ?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
hoelzl@31467
  1238
      by (auto simp add: le_float_def)
hoelzl@29805
  1239
hoelzl@31467
  1240
    have "cos (x + real (-k) * 2 * pi) \<le> cos (real ?lx)"
hoelzl@31467
  1241
      using cos_monotone_0_pi'[OF lx_0 lx pi_x]
hoelzl@31467
  1242
      by (simp only: real_of_float_minus real_of_int_minus
hoelzl@31467
  1243
	cos_minus real_diff_def mult_minus_left)
hoelzl@31467
  1244
    also have "\<dots> \<le> real (ub_cos prec ?lx)"
hoelzl@31467
  1245
      using lb_cos[OF lx_0 pi_lx] by simp
hoelzl@31467
  1246
    finally have "cos x \<le> real (ub_cos prec ?lx)"
hoelzl@31467
  1247
      unfolding cos_periodic_int . }
hoelzl@31467
  1248
  note positive_lx = this
hoelzl@31467
  1249
hoelzl@31467
  1250
  { assume pi_x: "- pi \<le> x - real k * 2 * pi" and "?ux \<le> 0"
hoelzl@31467
  1251
    with ux
hoelzl@31467
  1252
    have pi_ux: "- pi \<le> real ?ux" and ux_0: "real ?ux \<le> 0"
hoelzl@31467
  1253
      by (simp_all add: le_float_def)
hoelzl@29805
  1254
hoelzl@31467
  1255
    have "cos (x + real (-k) * 2 * pi) \<le> cos (real (- ?ux))"
hoelzl@31467
  1256
      using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
hoelzl@31467
  1257
      by (simp only: real_of_float_minus real_of_int_minus
hoelzl@31467
  1258
	  cos_minus real_diff_def mult_minus_left)
hoelzl@31467
  1259
    also have "\<dots> \<le> real (ub_cos prec (- ?ux))"
hoelzl@31467
  1260
      using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
hoelzl@31467
  1261
    finally have "cos x \<le> real (ub_cos prec (- ?ux))"
hoelzl@31467
  1262
      unfolding cos_periodic_int . }
hoelzl@31467
  1263
  note negative_ux = this
hoelzl@31467
  1264
hoelzl@31467
  1265
  { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - real k * 2 * pi"
hoelzl@31467
  1266
    with lpi ux
hoelzl@31467
  1267
    have pi_ux: "real ?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
hoelzl@31467
  1268
      by (simp_all add: le_float_def)
hoelzl@31467
  1269
hoelzl@31467
  1270
    have "real (lb_cos prec ?ux) \<le> cos (real ?ux)"
hoelzl@31467
  1271
      using lb_cos[OF ux_0 pi_ux] by simp
hoelzl@31467
  1272
    also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
hoelzl@31467
  1273
      using cos_monotone_0_pi'[OF x_ge_0 ux pi_ux]
hoelzl@31467
  1274
      by (simp only: real_of_float_minus real_of_int_minus
hoelzl@31467
  1275
	cos_minus real_diff_def mult_minus_left)
hoelzl@31467
  1276
    finally have "real (lb_cos prec ?ux) \<le> cos x"
hoelzl@31467
  1277
      unfolding cos_periodic_int . }
hoelzl@31467
  1278
  note positive_ux = this
hoelzl@31467
  1279
hoelzl@31467
  1280
  show "real l \<le> cos x \<and> cos x \<le> real u"
hoelzl@31467
  1281
  proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
hoelzl@31467
  1282
    case True with bnds
hoelzl@31467
  1283
    have l: "l = lb_cos prec (-?lx)"
hoelzl@31467
  1284
      and u: "u = ub_cos prec (-?ux)"
hoelzl@31467
  1285
      by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805
  1286
hoelzl@31467
  1287
    from True lpi[THEN le_imp_neg_le] lx ux
hoelzl@31467
  1288
    have "- pi \<le> x - real k * 2 * pi"
hoelzl@31467
  1289
      and "x - real k * 2 * pi \<le> 0"
hoelzl@31467
  1290
      by (auto simp add: le_float_def)
hoelzl@31467
  1291
    with True negative_ux negative_lx
hoelzl@31467
  1292
    show ?thesis unfolding l u by simp
hoelzl@31467
  1293
  next case False note 1 = this show ?thesis
hoelzl@31467
  1294
  proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
hoelzl@31467
  1295
    case True with bnds 1
hoelzl@31467
  1296
    have l: "l = lb_cos prec ?ux"
hoelzl@31467
  1297
      and u: "u = ub_cos prec ?lx"
hoelzl@31467
  1298
      by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805
  1299
hoelzl@31467
  1300
    from True lpi lx ux
hoelzl@31467
  1301
    have "0 \<le> x - real k * 2 * pi"
hoelzl@31467
  1302
      and "x - real k * 2 * pi \<le> pi"
hoelzl@31467
  1303
      by (auto simp add: le_float_def)
hoelzl@31467
  1304
    with True positive_ux positive_lx
hoelzl@31467
  1305
    show ?thesis unfolding l u by simp
hoelzl@31467
  1306
  next case False note 2 = this show ?thesis
hoelzl@31467
  1307
  proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
hoelzl@31467
  1308
    case True note Cond = this with bnds 1 2
hoelzl@31467
  1309
    have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
hoelzl@31467
  1310
      and u: "u = Float 1 0"
hoelzl@31467
  1311
      by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805
  1312
hoelzl@31467
  1313
    show ?thesis unfolding u l using negative_lx positive_ux Cond
hoelzl@31467
  1314
      by (cases "x - real k * 2 * pi < 0", simp_all add: real_of_float_min)
hoelzl@31467
  1315
  next case False note 3 = this show ?thesis
hoelzl@31467
  1316
  proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
hoelzl@31467
  1317
    case True note Cond = this with bnds 1 2 3
hoelzl@31467
  1318
    have l: "l = Float -1 0"
hoelzl@31467
  1319
      and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
hoelzl@31467
  1320
      by (auto simp add: bnds_cos_def Let_def)
hoelzl@31467
  1321
hoelzl@31467
  1322
    have "cos x \<le> real u"
hoelzl@31467
  1323
    proof (cases "x - real k * 2 * pi < pi")
hoelzl@31467
  1324
      case True hence "x - real k * 2 * pi \<le> pi" by simp
hoelzl@31467
  1325
      from positive_lx[OF Cond[THEN conjunct1] this]
hoelzl@31467
  1326
      show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@29805
  1327
    next
hoelzl@31467
  1328
      case False hence "pi \<le> x - real k * 2 * pi" by simp
hoelzl@31467
  1329
      hence pi_x: "- pi \<le> x - real k * 2 * pi - 2 * pi" by simp
hoelzl@31467
  1330
hoelzl@31467
  1331
      have "real ?ux \<le> 2 * pi" using Cond lpi by (auto simp add: le_float_def)
hoelzl@31467
  1332
      hence "x - real k * 2 * pi - 2 * pi \<le> 0" using ux by simp
hoelzl@31467
  1333
hoelzl@31467
  1334
      have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
hoelzl@31467
  1335
	using Cond by (auto simp add: le_float_def)
hoelzl@31467
  1336
hoelzl@31467
  1337
      from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
hoelzl@31467
  1338
      hence "- ?lpi \<le> ?ux - 2 * ?lpi" by (auto simp add: le_float_def)
hoelzl@31467
  1339
      hence pi_ux: "- pi \<le> real (?ux - 2 * ?lpi)"
hoelzl@31467
  1340
	using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
hoelzl@31467
  1341
hoelzl@31467
  1342
      have x_le_ux: "x - real k * 2 * pi - 2 * pi \<le> real (?ux - 2 * ?lpi)"
hoelzl@31467
  1343
	using ux lpi by auto
hoelzl@31467
  1344
hoelzl@31467
  1345
      have "cos x = cos (x + real (-k) * 2 * pi + real (-1 :: int) * 2 * pi)"
hoelzl@31467
  1346
	unfolding cos_periodic_int ..
hoelzl@31467
  1347
      also have "\<dots> \<le> cos (real (?ux - 2 * ?lpi))"
hoelzl@31467
  1348
	using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
hoelzl@31467
  1349
	by (simp only: real_of_float_minus real_of_int_minus real_of_one
hoelzl@31467
  1350
	    number_of_Min real_diff_def mult_minus_left real_mult_1)
hoelzl@31467
  1351
      also have "\<dots> = cos (real (- (?ux - 2 * ?lpi)))"
hoelzl@31467
  1352
	unfolding real_of_float_minus cos_minus ..
hoelzl@31467
  1353
      also have "\<dots> \<le> real (ub_cos prec (- (?ux - 2 * ?lpi)))"
hoelzl@31467
  1354
	using lb_cos_minus[OF pi_ux ux_0] by simp
hoelzl@31467
  1355
      finally show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@29805
  1356
    qed
hoelzl@31467
  1357
    thus ?thesis unfolding l by auto
hoelzl@31508
  1358
  next case False note 4 = this show ?thesis
hoelzl@31508
  1359
  proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
hoelzl@31508
  1360
    case True note Cond = this with bnds 1 2 3 4
hoelzl@31508
  1361
    have l: "l = Float -1 0"
hoelzl@31508
  1362
      and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
hoelzl@31508
  1363
      by (auto simp add: bnds_cos_def Let_def)
hoelzl@31508
  1364
hoelzl@31508
  1365
    have "cos x \<le> real u"
hoelzl@31508
  1366
    proof (cases "-pi < x - real k * 2 * pi")
hoelzl@31508
  1367
      case True hence "-pi \<le> x - real k * 2 * pi" by simp
hoelzl@31508
  1368
      from negative_ux[OF this Cond[THEN conjunct2]]
hoelzl@31508
  1369
      show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@31508
  1370
    next
hoelzl@31508
  1371
      case False hence "x - real k * 2 * pi \<le> -pi" by simp
hoelzl@31508
  1372
      hence pi_x: "x - real k * 2 * pi + 2 * pi \<le> pi" by simp
hoelzl@31508
  1373
hoelzl@31508
  1374
      have "-2 * pi \<le> real ?lx" using Cond lpi by (auto simp add: le_float_def)
hoelzl@31508
  1375
hoelzl@31508
  1376
      hence "0 \<le> x - real k * 2 * pi + 2 * pi" using lx by simp
hoelzl@31508
  1377
hoelzl@31508
  1378
      have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
hoelzl@31508
  1379
	using Cond lpi by (auto simp add: le_float_def)
hoelzl@31508
  1380
hoelzl@31508
  1381
      from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
hoelzl@31508
  1382
      hence "?lx + 2 * ?lpi \<le> ?lpi" by (auto simp add: le_float_def)
hoelzl@31508
  1383
      hence pi_lx: "real (?lx + 2 * ?lpi) \<le> pi"
hoelzl@31508
  1384
	using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
hoelzl@31508
  1385
hoelzl@31508
  1386
      have lx_le_x: "real (?lx + 2 * ?lpi) \<le> x - real k * 2 * pi + 2 * pi"
hoelzl@31508
  1387
	using lx lpi by auto
hoelzl@31508
  1388
hoelzl@31508
  1389
      have "cos x = cos (x + real (-k) * 2 * pi + real (1 :: int) * 2 * pi)"
hoelzl@31508
  1390
	unfolding cos_periodic_int ..
hoelzl@31508
  1391
      also have "\<dots> \<le> cos (real (?lx + 2 * ?lpi))"
hoelzl@31508
  1392
	using cos_monotone_0_pi'[OF lx_0 lx_le_x pi_x]
hoelzl@31508
  1393
	by (simp only: real_of_float_minus real_of_int_minus real_of_one
hoelzl@31508
  1394
	  number_of_Min real_diff_def mult_minus_left real_mult_1)
hoelzl@31508
  1395
      also have "\<dots> \<le> real (ub_cos prec (?lx + 2 * ?lpi))"
hoelzl@31508
  1396
	using lb_cos[OF lx_0 pi_lx] by simp
hoelzl@31508
  1397
      finally show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@31508
  1398
    qed
hoelzl@31508
  1399
    thus ?thesis unfolding l by auto
hoelzl@29805
  1400
  next
hoelzl@31508
  1401
    case False with bnds 1 2 3 4 show ?thesis by (auto simp add: bnds_cos_def Let_def)
hoelzl@31508
  1402
  qed qed qed qed qed
hoelzl@29805
  1403
qed
hoelzl@29805
  1404
hoelzl@29805
  1405
section "Exponential function"
hoelzl@29805
  1406
hoelzl@29805
  1407
subsection "Compute the series of the exponential function"
hoelzl@29805
  1408
hoelzl@29805
  1409
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
hoelzl@29805
  1410
"ub_exp_horner prec 0 i k x       = 0" |
hoelzl@29805
  1411
"ub_exp_horner prec (Suc n) i k x = rapprox_rat prec 1 (int k) + x * lb_exp_horner prec n (i + 1) (k * i) x" |
hoelzl@29805
  1412
"lb_exp_horner prec 0 i k x       = 0" |
hoelzl@29805
  1413
"lb_exp_horner prec (Suc n) i k x = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x"
hoelzl@29805
  1414
hoelzl@31098
  1415
lemma bnds_exp_horner: assumes "real x \<le> 0"
hoelzl@31098
  1416
  shows "exp (real x) \<in> { real (lb_exp_horner prec (get_even n) 1 1 x) .. real (ub_exp_horner prec (get_odd n) 1 1 x) }"
hoelzl@29805
  1417
proof -
hoelzl@29805
  1418
  { fix n
haftmann@30971
  1419
    have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
haftmann@30971
  1420
    have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this
hoelzl@31467
  1421
hoelzl@29805
  1422
  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,
hoelzl@29805
  1423
    OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
hoelzl@29805
  1424
hoelzl@31098
  1425
  { have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
hoelzl@29805
  1426
      using bounds(1) by auto
hoelzl@31098
  1427
    also have "\<dots> \<le> exp (real x)"
hoelzl@29805
  1428
    proof -
hoelzl@31098
  1429
      obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_even n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
hoelzl@29805
  1430
	using Maclaurin_exp_le by blast
hoelzl@31098
  1431
      moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
hoelzl@29805
  1432
	by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
hoelzl@29805
  1433
      ultimately show ?thesis
hoelzl@29805
  1434
	using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
hoelzl@29805
  1435
    qed
hoelzl@31098
  1436
    finally have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (real x)" .
hoelzl@29805
  1437
  } moreover
hoelzl@31809
  1438
  {
hoelzl@31098
  1439
    have x_less_zero: "real x ^ get_odd n \<le> 0"
hoelzl@31098
  1440
    proof (cases "real x = 0")
hoelzl@29805
  1441
      case True
hoelzl@29805
  1442
      have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
hoelzl@29805
  1443
      thus ?thesis unfolding True power_0_left by auto
hoelzl@29805
  1444
    next
hoelzl@31098
  1445
      case False hence "real x < 0" using `real x \<le> 0` by auto
hoelzl@31098
  1446
      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `real x < 0`)
hoelzl@29805
  1447
    qed
hoelzl@29805
  1448
hoelzl@31098
  1449
    obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
hoelzl@29805
  1450
      using Maclaurin_exp_le by blast
hoelzl@31098
  1451
    moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
hoelzl@29805
  1452
      by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
hoelzl@31098
  1453
    ultimately have "exp (real x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
hoelzl@29805
  1454
      using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
hoelzl@31098
  1455
    also have "\<dots> \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)"
hoelzl@29805
  1456
      using bounds(2) by auto
hoelzl@31098
  1457
    finally have "exp (real x) \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)" .
hoelzl@29805
  1458
  } ultimately show ?thesis by auto
hoelzl@29805
  1459
qed
hoelzl@29805
  1460
hoelzl@29805
  1461
subsection "Compute the exponential function on the entire domain"
hoelzl@29805
  1462
hoelzl@29805
  1463
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1464
"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
hoelzl@31809
  1465
             else let
hoelzl@29805
  1466
                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)
hoelzl@29805
  1467
             in if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow> (horner (float_divl prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
hoelzl@29805
  1468
                           else horner x)" |
hoelzl@29805
  1469
"ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
hoelzl@31809
  1470
             else if x < - 1  then (case floor_fl x of (Float m e) \<Rightarrow>
hoelzl@29805
  1471
                                    (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
hoelzl@29805
  1472
                              else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
hoelzl@29805
  1473
by pat_completeness auto
hoelzl@29805
  1474
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if 0 < x then 1 else 0))", auto simp add: less_float_def)
hoelzl@29805
  1475
hoelzl@29805
  1476
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
hoelzl@29805
  1477
proof -
hoelzl@29805
  1478
  have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
hoelzl@29805
  1479
hoelzl@31098
  1480
  have "1 / 4 = real (Float 1 -2)" unfolding Float_num by auto
hoelzl@31098
  1481
  also have "\<dots> \<le> real (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
hoelzl@31809
  1482
    unfolding get_even_def eq4
hoelzl@29805
  1483
    by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
hoelzl@31098
  1484
  also have "\<dots> \<le> exp (real (- 1 :: float))" using bnds_exp_horner[where x="- 1"] by auto
hoelzl@31809
  1485
  finally show ?thesis unfolding real_of_float_minus real_of_float_1 .
hoelzl@29805
  1486
qed
hoelzl@29805
  1487
hoelzl@29805
  1488
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
hoelzl@29805
  1489
proof -
hoelzl@29805
  1490
  let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
hoelzl@29805
  1491
  let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 -2 else y"
hoelzl@29805
  1492
  have pos_horner: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto simp add: le_float_def less_float_def)
hoelzl@29805
  1493
  moreover { fix x :: float fix num :: nat
hoelzl@31098
  1494
    have "0 < real (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def real_of_float_0] by (rule zero_less_power)
hoelzl@31098
  1495
    also have "\<dots> = real ((?horner x) ^ num)" using float_power by auto
hoelzl@31098
  1496
    finally have "0 < real ((?horner x) ^ num)" .
hoelzl@29805
  1497
  }
hoelzl@29805
  1498
  ultimately show ?thesis
haftmann@30968
  1499
    unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
haftmann@30968
  1500
    by (cases "floor_fl x", cases "x < - 1", auto simp add: float_power le_float_def less_float_def)
hoelzl@29805
  1501
qed
hoelzl@29805
  1502
hoelzl@29805
  1503
lemma exp_boundaries': assumes "x \<le> 0"
hoelzl@31098
  1504
  shows "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
hoelzl@29805
  1505
proof -
hoelzl@29805
  1506
  let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
hoelzl@29805
  1507
  let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
hoelzl@29805
  1508
hoelzl@31098
  1509
  have "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
hoelzl@29805
  1510
  show ?thesis
hoelzl@29805
  1511
  proof (cases "x < - 1")
hoelzl@31098
  1512
    case False hence "- 1 \<le> real x" unfolding less_float_def by auto
hoelzl@29805
  1513
    show ?thesis
hoelzl@29805
  1514
    proof (cases "?lb_exp_horner x \<le> 0")
hoelzl@31098
  1515
      from `\<not> x < - 1` have "- 1 \<le> real x" unfolding less_float_def by auto
hoelzl@31098
  1516
      hence "exp (- 1) \<le> exp (real x)" unfolding exp_le_cancel_iff .
hoelzl@29805
  1517
      from order_trans[OF exp_m1_ge_quarter this]
hoelzl@31098
  1518
      have "real (Float 1 -2) \<le> exp (real x)" unfolding Float_num .
hoelzl@29805
  1519
      moreover case True
hoelzl@31098
  1520
      ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
hoelzl@29805
  1521
    next
hoelzl@31098
  1522
      case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
hoelzl@29805
  1523
    qed
hoelzl@29805
  1524
  next
hoelzl@29805
  1525
    case True
hoelzl@31809
  1526
hoelzl@29805
  1527
    obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
hoelzl@29805
  1528
    let ?num = "nat (- m) * 2 ^ nat e"
hoelzl@31809
  1529
hoelzl@31098
  1530
    have "real (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def real_of_float_minus real_of_float_1 by (rule order_le_less_trans)
hoelzl@31098
  1531
    hence "real (floor_fl x) < 0" unfolding Float_floor real_of_float_simp using zero_less_pow2[of xe] by auto
hoelzl@29805
  1532
    hence "m < 0"
hoelzl@31098
  1533
      unfolding less_float_def real_of_float_0 Float_floor real_of_float_simp
hoelzl@29805
  1534
      unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
hoelzl@29805
  1535
    hence "1 \<le> - m" by auto
hoelzl@29805
  1536
    hence "0 < nat (- m)" by auto
hoelzl@29805
  1537
    moreover
hoelzl@29805
  1538
    have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
hoelzl@29805
  1539
    hence "(0::nat) < 2 ^ nat e" by auto
hoelzl@29805
  1540
    ultimately have "0 < ?num"  by auto
hoelzl@29805
  1541
    hence "real ?num \<noteq> 0" by auto
hoelzl@29805
  1542
    have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
hoelzl@31098
  1543
    have num_eq: "real ?num = real (- floor_fl x)" using `0 < nat (- m)`
hoelzl@31098
  1544
      unfolding Float_floor real_of_float_minus real_of_float_simp real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
hoelzl@31098
  1545
    have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] real_of_float_0 real_of_nat_zero .
hoelzl@31098
  1546
    hence "real (floor_fl x) < 0" unfolding less_float_def by auto
hoelzl@31809
  1547
hoelzl@31098
  1548
    have "exp (real x) \<le> real (ub_exp prec x)"
hoelzl@29805
  1549
    proof -
hoelzl@31809
  1550
      have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
hoelzl@31098
  1551
	using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def real_of_float_0 .
hoelzl@31809
  1552
hoelzl@31098
  1553
      have "exp (real x) = exp (real ?num * (real x / real ?num))" using `real ?num \<noteq> 0` by auto
hoelzl@31098
  1554
      also have "\<dots> = exp (real x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
hoelzl@31098
  1555
      also have "\<dots> \<le> exp (real (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
hoelzl@29805
  1556
	by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
hoelzl@31098
  1557
      also have "\<dots> \<le> real ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
hoelzl@29805
  1558
	by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
hoelzl@29805
  1559
      finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def .
hoelzl@29805
  1560
    qed
hoelzl@31809
  1561
    moreover
hoelzl@31098
  1562
    have "real (lb_exp prec x) \<le> exp (real x)"
hoelzl@29805
  1563
    proof -
hoelzl@29805
  1564
      let ?divl = "float_divl prec x (- Float m e)"
hoelzl@29805
  1565
      let ?horner = "?lb_exp_horner ?divl"
hoelzl@31809
  1566
hoelzl@29805
  1567
      show ?thesis
hoelzl@29805
  1568
      proof (cases "?horner \<le> 0")
hoelzl@31098
  1569
	case False hence "0 \<le> real ?horner" unfolding le_float_def by auto
hoelzl@31809
  1570
hoelzl@31098
  1571
	have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
hoelzl@31098
  1572
	  using `real (floor_fl x) < 0` `real x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
hoelzl@31809
  1573
hoelzl@31809
  1574
	have "real ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>
hoelzl@31809
  1575
          exp (real (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power
hoelzl@31098
  1576
	  using `0 \<le> real ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
hoelzl@31098
  1577
	also have "\<dots> \<le> exp (real x / real ?num) ^ ?num" unfolding num_eq
hoelzl@31098
  1578
	  using float_divl by (auto intro!: power_mono simp del: real_of_float_minus)
hoelzl@31098
  1579
	also have "\<dots> = exp (real ?num * (real x / real ?num))" unfolding exp_real_of_nat_mult ..
hoelzl@31098
  1580
	also have "\<dots> = exp (real x)" using `real ?num \<noteq> 0` by auto
hoelzl@29805
  1581
	finally show ?thesis
hoelzl@29805
  1582
	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_not_P[OF False] by auto
hoelzl@29805
  1583
      next
hoelzl@29805
  1584
	case True
hoelzl@31098
  1585
	have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
hoelzl@31098
  1586
	from divide_right_mono_neg[OF floor_fl[of x] `real (floor_fl x) \<le> 0`, unfolded divide_self[OF `real (floor_fl x) \<noteq> 0`]]
hoelzl@31098
  1587
	have "- 1 \<le> real x / real (- floor_fl x)" unfolding real_of_float_minus by auto
hoelzl@29805
  1588
	from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
hoelzl@31098
  1589
	have "real (Float 1 -2) \<le> exp (real x / real (- floor_fl x))" unfolding Float_num .
hoelzl@31098
  1590
	hence "real (Float 1 -2) ^ ?num \<le> exp (real x / real (- floor_fl x)) ^ ?num"
hoelzl@29805
  1591
	  by (auto intro!: power_mono simp add: Float_num)
hoelzl@31098
  1592
	also have "\<dots> = exp (real x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `real (floor_fl x) \<noteq> 0` by auto
hoelzl@29805
  1593
	finally show ?thesis
hoelzl@29805
  1594
	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_P[OF True] float_power .
hoelzl@29805
  1595
      qed
hoelzl@29805
  1596
    qed
hoelzl@29805
  1597
    ultimately show ?thesis by auto
hoelzl@29805
  1598
  qed
hoelzl@29805
  1599
qed
hoelzl@29805
  1600
hoelzl@31098
  1601
lemma exp_boundaries: "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
hoelzl@29805
  1602
proof -
hoelzl@29805
  1603
  show ?thesis
hoelzl@29805
  1604
  proof (cases "0 < x")
hoelzl@31809
  1605
    case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1606
    from exp_boundaries'[OF this] show ?thesis .
hoelzl@29805
  1607
  next
hoelzl@29805
  1608
    case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
hoelzl@31809
  1609
hoelzl@31098
  1610
    have "real (lb_exp prec x) \<le> exp (real x)"
hoelzl@29805
  1611
    proof -
hoelzl@29805
  1612
      from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@31098
  1613
      have ub_exp: "exp (- real x) \<le> real (ub_exp prec (-x))" unfolding atLeastAtMost_iff real_of_float_minus by auto
hoelzl@31809
  1614
hoelzl@31098
  1615
      have "real (float_divl prec 1 (ub_exp prec (-x))) \<le> 1 / real (ub_exp prec (-x))" using float_divl[where x=1] by auto
hoelzl@31098
  1616
      also have "\<dots> \<le> exp (real x)"
hoelzl@29805
  1617
	using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
hoelzl@29805
  1618
	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
hoelzl@29805
  1619
      finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
hoelzl@29805
  1620
    qed
hoelzl@29805
  1621
    moreover
hoelzl@31098
  1622
    have "exp (real x) \<le> real (ub_exp prec x)"
hoelzl@29805
  1623
    proof -
hoelzl@29805
  1624
      have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
hoelzl@31809
  1625
hoelzl@29805
  1626
      from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@31098
  1627
      have lb_exp: "real (lb_exp prec (-x)) \<le> exp (- real x)" unfolding atLeastAtMost_iff real_of_float_minus by auto
hoelzl@31809
  1628
hoelzl@31098
  1629
      have "exp (real x) \<le> real (1 :: float) / real (lb_exp prec (-x))"
hoelzl@31809
  1630
	using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def real_of_float_0],
hoelzl@31098
  1631
	                                        symmetric]]
hoelzl@31098
  1632
	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide real_of_float_1 by auto
hoelzl@31098
  1633
      also have "\<dots> \<le> real (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
hoelzl@29805
  1634
      finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
hoelzl@29805
  1635
    qed
hoelzl@29805
  1636
    ultimately show ?thesis by auto
hoelzl@29805
  1637
  qed
hoelzl@29805
  1638
qed
hoelzl@29805
  1639
hoelzl@31098
  1640
lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> exp x \<and> exp x \<le> real u"
hoelzl@29805
  1641
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  1642
  fix x lx ux
hoelzl@31098
  1643
  assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux}"
hoelzl@31098
  1644
  hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
hoelzl@29805
  1645
hoelzl@29805
  1646
  { from exp_boundaries[of lx prec, unfolded l]
hoelzl@31098
  1647
    have "real l \<le> exp (real lx)" by (auto simp del: lb_exp.simps)
hoelzl@29805
  1648
    also have "\<dots> \<le> exp x" using x by auto
hoelzl@31098
  1649
    finally have "real l \<le> exp x" .
hoelzl@29805
  1650
  } moreover
hoelzl@31098
  1651
  { have "exp x \<le> exp (real ux)" using x by auto
hoelzl@31098
  1652
    also have "\<dots> \<le> real u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
hoelzl@31098
  1653
    finally have "exp x \<le> real u" .
hoelzl@31098
  1654
  } ultimately show "real l \<le> exp x \<and> exp x \<le> real u" ..
hoelzl@29805
  1655
qed
hoelzl@29805
  1656
hoelzl@29805
  1657
section "Logarithm"
hoelzl@29805
  1658
hoelzl@29805
  1659
subsection "Compute the logarithm series"
hoelzl@29805
  1660
hoelzl@31809
  1661
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805
  1662
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1663
"ub_ln_horner prec 0 i x       = 0" |
hoelzl@29805
  1664
"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
hoelzl@29805
  1665
"lb_ln_horner prec 0 i x       = 0" |
hoelzl@29805
  1666
"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
hoelzl@29805
  1667
hoelzl@29805
  1668
lemma ln_bounds:
hoelzl@29805
  1669
  assumes "0 \<le> x" and "x < 1"
haftmann@30952
  1670
  shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
haftmann@30952
  1671
  and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
hoelzl@29805
  1672
proof -
haftmann@30952
  1673
  let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
hoelzl@29805
  1674
hoelzl@29805
  1675
  have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
hoelzl@29805
  1676
    using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
hoelzl@29805
  1677
hoelzl@29805
  1678
  have "norm x < 1" using assms by auto
hoelzl@31809
  1679
  have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
hoelzl@29805
  1680
    using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
hoelzl@29805
  1681
  { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
hoelzl@29805
  1682
  { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
hoelzl@29805
  1683
    proof (rule mult_mono)
hoelzl@29805
  1684
      show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
hoelzl@31809
  1685
      have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric]
hoelzl@29805
  1686
	by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
hoelzl@29805
  1687
      thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
hoelzl@29805
  1688
    qed auto }
hoelzl@29805
  1689
  from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq]
hoelzl@29805
  1690
  show "?lb" and "?ub" by auto
hoelzl@29805
  1691
qed
hoelzl@29805
  1692
hoelzl@31809
  1693
lemma ln_float_bounds:
hoelzl@31098
  1694
  assumes "0 \<le> real x" and "real x < 1"
hoelzl@31098
  1695
  shows "real (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (real x + 1)" (is "?lb \<le> ?ln")
hoelzl@31098
  1696
  and "ln (real x + 1) \<le> real (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
hoelzl@29805
  1697
proof -
hoelzl@29805
  1698
  obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
hoelzl@29805
  1699
  obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
hoelzl@29805
  1700
hoelzl@31098
  1701
  let "?s n" = "-1^n * (1 / real (1 + n)) * (real x)^(Suc n)"
hoelzl@29805
  1702
hoelzl@31098
  1703
  have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] ev
hoelzl@29805
  1704
    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",
hoelzl@31098
  1705
      OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
hoelzl@29805
  1706
    by (rule mult_right_mono)
hoelzl@31098
  1707
  also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
hoelzl@31809
  1708
  finally show "?lb \<le> ?ln" .
hoelzl@29805
  1709
hoelzl@31098
  1710
  have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
hoelzl@31098
  1711
  also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] od
hoelzl@29805
  1712
    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",
hoelzl@31098
  1713
      OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
hoelzl@29805
  1714
    by (rule mult_right_mono)
hoelzl@31809
  1715
  finally show "?ln \<le> ?ub" .
hoelzl@29805
  1716
qed
hoelzl@29805
  1717
hoelzl@29805
  1718
lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
hoelzl@29805
  1719
proof -
hoelzl@29805
  1720
  have "x \<noteq> 0" using assms by auto
hoelzl@29805
  1721
  have "x + y = x * (1 + y / x)" unfolding right_distrib times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \<noteq> 0`] by auto
hoelzl@31809
  1722
  moreover
hoelzl@29805
  1723
  have "0 < y / x" using assms divide_pos_pos by auto
hoelzl@29805
  1724
  hence "0 < 1 + y / x" by auto
hoelzl@29805
  1725
  ultimately show ?thesis using ln_mult assms by auto
hoelzl@29805
  1726
qed
hoelzl@29805
  1727
hoelzl@29805
  1728
subsection "Compute the logarithm of 2"
hoelzl@29805
  1729
hoelzl@31809
  1730
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
hoelzl@31809
  1731
                                        in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) +
hoelzl@29805
  1732
                                           (third * ub_ln_horner prec (get_odd prec) 1 third))"
hoelzl@31809
  1733
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
hoelzl@31809
  1734
                                        in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) +
hoelzl@29805
  1735
                                           (third * lb_ln_horner prec (get_even prec) 1 third))"
hoelzl@29805
  1736
hoelzl@31098
  1737
lemma ub_ln2: "ln 2 \<le> real (ub_ln2 prec)" (is "?ub_ln2")
hoelzl@31098
  1738
  and lb_ln2: "real (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
hoelzl@29805
  1739
proof -
hoelzl@29805
  1740
  let ?uthird = "rapprox_rat (max prec 1) 1 3"
hoelzl@29805
  1741
  let ?lthird = "lapprox_rat prec 1 3"
hoelzl@29805
  1742
hoelzl@29805
  1743
  have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
hoelzl@29805
  1744
    using ln_add[of "3 / 2" "1 / 2"] by auto
hoelzl@31098
  1745
  have lb3: "real ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
hoelzl@31098
  1746
  hence lb3_ub: "real ?lthird < 1" by auto
hoelzl@31098
  1747
  have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_bottom[of 1 3] by auto
hoelzl@31098
  1748
  have ub3: "1 / 3 \<le> real ?uthird" using rapprox_rat[of 1 3] by auto
hoelzl@31098
  1749
  hence ub3_lb: "0 \<le> real ?uthird" by auto
hoelzl@29805
  1750
hoelzl@31098
  1751
  have lb2: "0 \<le> real (Float 1 -1)" and ub2: "real (Float 1 -1) < 1" unfolding Float_num by auto
hoelzl@29805
  1752
hoelzl@29805
  1753
  have "0 \<le> (1::int)" and "0 < (3::int)" by auto
hoelzl@31098
  1754
  have ub3_ub: "real ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
hoelzl@29805
  1755
    by (rule rapprox_posrat_less1, auto)
hoelzl@29805
  1756
hoelzl@29805
  1757
  have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
hoelzl@31098
  1758
  have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
hoelzl@31098
  1759
  have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
hoelzl@29805
  1760
hoelzl@31098
  1761
  show ?ub_ln2 unfolding ub_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
hoelzl@29805
  1762
  proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
hoelzl@31098
  1763
    have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
hoelzl@31098
  1764
    also have "\<dots> \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
hoelzl@29805
  1765
      using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
hoelzl@31098
  1766
    finally show "ln (1 / 3 + 1) \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
hoelzl@29805
  1767
  qed
hoelzl@31098
  1768
  show ?lb_ln2 unfolding lb_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
hoelzl@29805
  1769
  proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
hoelzl@31098
  1770
    have "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (real ?lthird + 1)"
hoelzl@29805
  1771
      using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
hoelzl@29805
  1772
    also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
hoelzl@31098
  1773
    finally show "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
hoelzl@29805
  1774
  qed
hoelzl@29805
  1775
qed
hoelzl@29805
  1776
hoelzl@29805
  1777
subsection "Compute the logarithm in the entire domain"
hoelzl@29805
  1778
hoelzl@29805
  1779
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
hoelzl@31468
  1780
"ub_ln prec x = (if x \<le> 0          then None
hoelzl@31468
  1781
            else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
hoelzl@31468
  1782
            else let horner = \<lambda>x. x * ub_ln_horner prec (get_odd prec) 1 x in
hoelzl@31468
  1783
                 if x \<le> Float 3 -1 then Some (horner (x - 1))
hoelzl@31468
  1784
            else if x < Float 1 1  then Some (horner (Float 1 -1) + horner (x * rapprox_rat prec 2 3 - 1))
hoelzl@31468
  1785
                                   else let l = bitlen (mantissa x) - 1 in
hoelzl@31468
  1786
                                        Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))" |
hoelzl@31468
  1787
"lb_ln prec x = (if x \<le> 0          then None
hoelzl@31468
  1788
            else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
hoelzl@31468
  1789
            else let horner = \<lambda>x. x * lb_ln_horner prec (get_even prec) 1 x in
hoelzl@31468
  1790
                 if x \<le> Float 3 -1 then Some (horner (x - 1))
hoelzl@31468
  1791
            else if x < Float 1 1  then Some (horner (Float 1 -1) +
hoelzl@31468
  1792
                                              horner (max (x * lapprox_rat prec 2 3 - 1) 0))
hoelzl@31468
  1793
                                   else let l = bitlen (mantissa x) - 1 in
hoelzl@31468
  1794
                                        Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))"
hoelzl@29805
  1795
by pat_completeness auto
hoelzl@29805
  1796
hoelzl@29805
  1797
termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
hoelzl@29805
  1798
  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
hoelzl@29805
  1799
  hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1800
  from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
hoelzl@29805
  1801
  show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1802
next
hoelzl@29805
  1803
  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
hoelzl@29805
  1804
  hence "0 < x" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1805
  from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
hoelzl@29805
  1806
  show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1807
qed
hoelzl@29805
  1808
hoelzl@31098
  1809
lemma ln_shifted_float: assumes "0 < m" shows "ln (real (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (real (Float m (- (bitlen m - 1))))"
hoelzl@29805
  1810
proof -
hoelzl@29805
  1811
  let ?B = "2^nat (bitlen m - 1)"
hoelzl@29805
  1812
  have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
hoelzl@29805
  1813
  hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
hoelzl@31468
  1814
  show ?thesis
hoelzl@29805
  1815
  proof (cases "0 \<le> e")
hoelzl@29805
  1816
    case True
hoelzl@29805
  1817
    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
hoelzl@31468
  1818
      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
hoelzl@31468
  1819
      unfolding real_of_float_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`]
hoelzl@29805
  1820
      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
hoelzl@29805
  1821
  next
hoelzl@29805
  1822
    case False hence "0 < -e" by auto
hoelzl@29805
  1823
    hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
hoelzl@29805
  1824
    hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
hoelzl@29805
  1825
    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
hoelzl@31468
  1826
      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
hoelzl@31098
  1827
      unfolding real_of_float_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
hoelzl@29805
  1828
      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
hoelzl@29805
  1829
  qed
hoelzl@29805
  1830
qed
hoelzl@29805
  1831
hoelzl@29805
  1832
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
hoelzl@31098
  1833
  shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
hoelzl@29805
  1834
  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805
  1835
proof (cases "x < Float 1 1")
hoelzl@31468
  1836
  case True
hoelzl@31468
  1837
  hence "real (x - 1) < 1" and "real x < 2" unfolding less_float_def Float_num by auto
hoelzl@29805
  1838
  have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
hoelzl@31098
  1839
  hence "0 \<le> real (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
hoelzl@31468
  1840
hoelzl@31468
  1841
  have [simp]: "real (Float 3 -1) = 3 / 2" by (simp add: real_of_float_def pow2_def)
hoelzl@31468
  1842
hoelzl@31468
  1843
  show ?thesis
hoelzl@31468
  1844
  proof (cases "x \<le> Float 3 -1")
hoelzl@31468
  1845
    case True
hoelzl@31468
  1846
    show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
hoelzl@31468
  1847
      using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
hoelzl@31468
  1848
      by auto
hoelzl@31468
  1849
  next
hoelzl@31468
  1850
    case False hence *: "3 / 2 < real x" by (auto simp add: le_float_def)
hoelzl@31468
  1851
hoelzl@31468
  1852
    with ln_add[of "3 / 2" "real x - 3 / 2"]
hoelzl@31468
  1853
    have add: "ln (real x) = ln (3 / 2) + ln (real x * 2 / 3)"
hoelzl@31468
  1854
      by (auto simp add: algebra_simps diff_divide_distrib)
hoelzl@31468
  1855
hoelzl@31468
  1856
    let "?ub_horner x" = "x * ub_ln_horner prec (get_odd prec) 1 x"
hoelzl@31468
  1857
    let "?lb_horner x" = "x * lb_ln_horner prec (get_even prec) 1 x"
hoelzl@31468
  1858
hoelzl@31468
  1859
    { have up: "real (rapprox_rat prec 2 3) \<le> 1"
hoelzl@31468
  1860
	by (rule rapprox_rat_le1) simp_all
hoelzl@31468
  1861
      have low: "2 / 3 \<le> real (rapprox_rat prec 2 3)"
hoelzl@31468
  1862
	by (rule order_trans[OF _ rapprox_rat]) simp
hoelzl@31468
  1863
      from mult_less_le_imp_less[OF * low] *
hoelzl@31468
  1864
      have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
hoelzl@31468
  1865
hoelzl@31468
  1866
      have "ln (real x * 2/3)
hoelzl@31468
  1867
	\<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
hoelzl@31468
  1868
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
hoelzl@31468
  1869
	show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
hoelzl@31468
  1870
	  using * low by auto
hoelzl@31468
  1871
	show "0 < real x * 2 / 3" using * by simp
hoelzl@31468
  1872
	show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
hoelzl@31468
  1873
      qed
hoelzl@31468
  1874
      also have "\<dots> \<le> real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
hoelzl@31468
  1875
      proof (rule ln_float_bounds(2))
hoelzl@31468
  1876
	from mult_less_le_imp_less[OF `real x < 2` up] low *
hoelzl@31468
  1877
	show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
hoelzl@31468
  1878
	show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
hoelzl@31468
  1879
      qed
hoelzl@31468
  1880
      finally have "ln (real x)
hoelzl@31468
  1881
	\<le> real (?ub_horner (Float 1 -1))
hoelzl@31468
  1882
	  + real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
hoelzl@31468
  1883
	using ln_float_bounds(2)[of "Float 1 -1" prec prec] add by auto }
hoelzl@31468
  1884
    moreover
hoelzl@31468
  1885
    { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
hoelzl@31468
  1886
hoelzl@31468
  1887
      have up: "real (lapprox_rat prec 2 3) \<le> 2/3"
hoelzl@31468
  1888
	by (rule order_trans[OF lapprox_rat], simp)
hoelzl@31468
  1889
hoelzl@31468
  1890
      have low: "0 \<le> real (lapprox_rat prec 2 3)"
hoelzl@31468
  1891
	using lapprox_rat_bottom[of 2 3 prec] by simp
hoelzl@31468
  1892
hoelzl@31468
  1893
      have "real (?lb_horner ?max)
hoelzl@31468
  1894
	\<le> ln (real ?max + 1)"
hoelzl@31468
  1895
      proof (rule ln_float_bounds(1))
hoelzl@31468
  1896
	from mult_less_le_imp_less[OF `real x < 2` up] * low
hoelzl@31468
  1897
	show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
hoelzl@31468
  1898
	  auto simp add: real_of_float_max)
hoelzl@31468
  1899
	show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
hoelzl@31468
  1900
      qed
hoelzl@31468
  1901
      also have "\<dots> \<le> ln (real x * 2/3)"
hoelzl@31468
  1902
      proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
hoelzl@31468
  1903
	show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
hoelzl@31468
  1904
	show "0 < real x * 2/3" using * by auto
hoelzl@31468
  1905
	show "real ?max + 1 \<le> real x * 2/3" using * up
hoelzl@31468
  1906
	  by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
hoelzl@31468
  1907
	      auto simp add: real_of_float_max max_def)
hoelzl@31468
  1908
      qed
hoelzl@31468
  1909
      finally have "real (?lb_horner (Float 1 -1)) + real (?lb_horner ?max)
hoelzl@31468
  1910
	\<le> ln (real x)"
hoelzl@31468
  1911
	using ln_float_bounds(1)[of "Float 1 -1" prec prec] add by auto }
hoelzl@31468
  1912
    ultimately
hoelzl@31468
  1913
    show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
hoelzl@31468
  1914
      using `\<not> x \<le> 0` `\<not> x < 1` True False by auto
hoelzl@31468
  1915
  qed
hoelzl@29805
  1916
next
hoelzl@29805
  1917
  case False
hoelzl@31468
  1918
  hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 -1"
hoelzl@31468
  1919
    using `1 \<le> x` unfolding less_float_def le_float_def real_of_float_simp pow2_def
hoelzl@31468
  1920
    by auto
hoelzl@29805
  1921
  show ?thesis
hoelzl@29805
  1922
  proof (cases x)
hoelzl@29805
  1923
    case (Float m e)
hoelzl@29805
  1924
    let ?s = "Float (e + (bitlen m - 1)) 0"
hoelzl@29805
  1925
    let ?x = "Float m (- (bitlen m - 1))"
hoelzl@29805
  1926
hoelzl@29805
  1927
    have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
hoelzl@29805
  1928
hoelzl@29805
  1929
    {
hoelzl@31098
  1930
      have "real (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
hoelzl@31098
  1931
	unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
hoelzl@29805
  1932
	using lb_ln2[of prec]
hoelzl@29805
  1933
      proof (rule mult_right_mono)
hoelzl@29805
  1934
	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
hoelzl@29805
  1935
	from float_gt1_scale[OF this]
hoelzl@29805
  1936
	show "0 \<le> real (e + (bitlen m - 1))" by auto
hoelzl@29805
  1937
      qed
hoelzl@29805
  1938
      moreover
hoelzl@29805
  1939
      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
hoelzl@31098
  1940
      have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
hoelzl@29805
  1941
      from ln_float_bounds(1)[OF this]
hoelzl@31098
  1942
      have "real ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (real ?x)" (is "?lb_horner \<le> _") by auto
hoelzl@31098
  1943
      ultimately have "?lb2 + ?lb_horner \<le> ln (real x)"
hoelzl@29805
  1944
	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@31468
  1945
    }
hoelzl@29805
  1946
    moreover
hoelzl@29805
  1947
    {
hoelzl@29805
  1948
      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
hoelzl@31098
  1949
      have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
hoelzl@29805
  1950
      from ln_float_bounds(2)[OF this]
hoelzl@31098
  1951
      have "ln (real ?x) \<le> real ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
hoelzl@29805
  1952
      moreover
hoelzl@31098
  1953
      have "ln 2 * real (e + (bitlen m - 1)) \<le> real (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
hoelzl@31098
  1954
	unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
hoelzl@31468
  1955
	using ub_ln2[of prec]
hoelzl@29805
  1956
      proof (rule mult_right_mono)
hoelzl@29805
  1957
	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
hoelzl@29805
  1958
	from float_gt1_scale[OF this]
hoelzl@29805
  1959
	show "0 \<le> real (e + (bitlen m - 1))" by auto
hoelzl@29805
  1960
      qed
hoelzl@31098
  1961
      ultimately have "ln (real x) \<le> ?ub2 + ?ub_horner"
hoelzl@29805
  1962
	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@29805
  1963
    }
hoelzl@29805
  1964
    ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
hoelzl@31468
  1965
      unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] if_not_P[OF `\<not> x \<le> Float 3 -1`] Let_def
hoelzl@31468
  1966
      unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] real_of_float_add
hoelzl@31468
  1967
      by auto
hoelzl@29805
  1968
  qed
hoelzl@29805
  1969
qed
hoelzl@29805
  1970
hoelzl@29805
  1971
lemma ub_ln_lb_ln_bounds: assumes "0 < x"
hoelzl@31098
  1972
  shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
hoelzl@29805
  1973
  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805
  1974
proof (cases "x < 1")
hoelzl@29805
  1975
  case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1976
  show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
hoelzl@29805
  1977
next
hoelzl@29805
  1978
  case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1979
hoelzl@31098
  1980
  have "0 < real x" and "real x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
hoelzl@31098
  1981
  hence A: "0 < 1 / real x" by auto
hoelzl@29805
  1982
hoelzl@29805
  1983
  {
hoelzl@29805
  1984
    let ?divl = "float_divl (max prec 1) 1 x"
hoelzl@29805
  1985
    have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
hoelzl@31098
  1986
    hence B: "0 < real ?divl" unfolding le_float_def by auto
hoelzl@31468
  1987
hoelzl@31098
  1988
    have "ln (real ?divl) \<le> ln (1 / real x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
hoelzl@31098
  1989
    hence "ln (real x) \<le> - ln (real ?divl)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
hoelzl@31468
  1990
    from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
hoelzl@31098
  1991
    have "?ln \<le> real (- the (lb_ln prec ?divl))" unfolding real_of_float_minus by (rule order_trans)
hoelzl@29805
  1992
  } moreover
hoelzl@29805
  1993
  {
hoelzl@29805
  1994
    let ?divr = "float_divr prec 1 x"
hoelzl@29805
  1995
    have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
hoelzl@31098
  1996
    hence B: "0 < real ?divr" unfolding le_float_def by auto
hoelzl@31468
  1997
hoelzl@31098
  1998
    have "ln (1 / real x) \<le> ln (real ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
hoelzl@31098
  1999
    hence "- ln (real ?divr) \<le> ln (real x)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
hoelzl@29805
  2000
    from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
hoelzl@31098
  2001
    have "real (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding real_of_float_minus by (rule order_trans)
hoelzl@29805
  2002
  }
hoelzl@29805
  2003
  ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
hoelzl@29805
  2004
    unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
hoelzl@29805
  2005
qed
hoelzl@29805
  2006
hoelzl@29805
  2007
lemma lb_ln: assumes "Some y = lb_ln prec x"
hoelzl@31098
  2008
  shows "real y \<le> ln (real x)" and "0 < real x"
hoelzl@29805
  2009
proof -
hoelzl@29805
  2010
  have "0 < x"
hoelzl@29805
  2011
  proof (rule ccontr)
hoelzl@29805
  2012
    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
hoelzl@29805
  2013
    thus False using assms by auto
hoelzl@29805
  2014
  qed
hoelzl@31098
  2015
  thus "0 < real x" unfolding less_float_def by auto
hoelzl@31098
  2016
  have "real (the (lb_ln prec x)) \<le> ln (real x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@31098
  2017
  thus "real y \<le> ln (real x)" unfolding assms[symmetric] by auto
hoelzl@29805
  2018
qed
hoelzl@29805
  2019
hoelzl@29805
  2020
lemma ub_ln: assumes "Some y = ub_ln prec x"
hoelzl@31098
  2021
  shows "ln (real x) \<le> real y" and "0 < real x"
hoelzl@29805
  2022
proof -
hoelzl@29805
  2023
  have "0 < x"
hoelzl@29805
  2024
  proof (rule ccontr)
hoelzl@29805
  2025
    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
hoelzl@29805
  2026
    thus False using assms by auto
hoelzl@29805
  2027
  qed
hoelzl@31098
  2028
  thus "0 < real x" unfolding less_float_def by auto
hoelzl@31098
  2029
  have "ln (real x) \<le> real (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@31098
  2030
  thus "ln (real x) \<le> real y" unfolding assms[symmetric] by auto
hoelzl@29805
  2031
qed
hoelzl@29805
  2032
hoelzl@31098
  2033
lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> ln x \<and> ln x \<le> real u"
hoelzl@29805
  2034
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  2035
  fix x lx ux
hoelzl@31098
  2036
  assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux}"
hoelzl@31098
  2037
  hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {real lx .. real ux}" by auto
hoelzl@29805
  2038
hoelzl@31098
  2039
  have "ln (real ux) \<le> real u" and "0 < real ux" using ub_ln u by auto
hoelzl@31098
  2040
  have "real l \<le> ln (real lx)" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
hoelzl@29805
  2041
hoelzl@31467
  2042
  from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `real l \<le> ln (real lx)`
hoelzl@31098
  2043
  have "real l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
hoelzl@29805
  2044
  moreover
hoelzl@31467
  2045
  from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln (real ux) \<le> real u`
hoelzl@31098
  2046
  have "ln x \<le> real u" using x unfolding atLeastAtMost_iff by auto
hoelzl@31098
  2047
  ultimately show "real l \<le> ln x \<and> ln x \<le> real u" ..
hoelzl@29805
  2048
qed
hoelzl@29805
  2049
hoelzl@29805
  2050
section "Implement floatarith"
hoelzl@29805
  2051
hoelzl@29805
  2052
subsection "Define syntax and semantics"
hoelzl@29805
  2053
hoelzl@29805
  2054
datatype floatarith
hoelzl@29805
  2055
  = Add floatarith floatarith
hoelzl@29805
  2056
  | Minus floatarith
hoelzl@29805
  2057
  | Mult floatarith floatarith
hoelzl@29805
  2058
  | Inverse floatarith
hoelzl@29805
  2059
  | Cos floatarith
hoelzl@29805
  2060
  | Arctan floatarith
hoelzl@29805
  2061
  | Abs floatarith
hoelzl@29805
  2062
  | Max floatarith floatarith
hoelzl@29805
  2063
  | Min floatarith floatarith
hoelzl@29805
  2064
  | Pi
hoelzl@29805
  2065
  | Sqrt floatarith
hoelzl@29805
  2066
  | Exp floatarith
hoelzl@29805
  2067
  | Ln floatarith
hoelzl@29805
  2068
  | Power floatarith nat
hoelzl@29805
  2069
  | Atom nat
hoelzl@29805
  2070
  | Num float
hoelzl@29805
  2071
hoelzl@31098
  2072
fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
hoelzl@29805
  2073
where
hoelzl@31098
  2074
"interpret_floatarith (Add a b) vs   = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
hoelzl@31098
  2075
"interpret_floatarith (Minus a) vs    = - (interpret_floatarith a vs)" |
hoelzl@31098
  2076
"interpret_floatarith (Mult a b) vs   = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
hoelzl@31098
  2077
"interpret_floatarith (Inverse a) vs  = inverse (interpret_floatarith a vs)" |
hoelzl@31098
  2078
"interpret_floatarith (Cos a) vs      = cos (interpret_floatarith a vs)" |
hoelzl@31098
  2079
"interpret_floatarith (Arctan a) vs   = arctan (interpret_floatarith a vs)" |
hoelzl@31098
  2080
"interpret_floatarith (Min a b) vs    = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
hoelzl@31098
  2081
"interpret_floatarith (Max a b) vs    = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
hoelzl@31098
  2082
"interpret_floatarith (Abs a) vs      = abs (interpret_floatarith a vs)" |
hoelzl@31098
  2083
"interpret_floatarith Pi vs           = pi" |
hoelzl@31098
  2084
"interpret_floatarith (Sqrt a) vs     = sqrt (interpret_floatarith a vs)" |
hoelzl@31098
  2085
"interpret_floatarith (Exp a) vs      = exp (interpret_floatarith a vs)" |
hoelzl@31098
  2086
"interpret_floatarith (Ln a) vs       = ln (interpret_floatarith a vs)" |
hoelzl@31098
  2087
"interpret_floatarith (Power a n) vs  = (interpret_floatarith a vs)^n" |
hoelzl@31098
  2088
"interpret_floatarith (Num f) vs      = real f" |
hoelzl@31098
  2089
"interpret_floatarith (Atom n) vs     = vs ! n"
hoelzl@29805
  2090
hoelzl@31811
  2091
lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
hoelzl@31811
  2092
  unfolding real_divide_def interpret_floatarith.simps ..
hoelzl@31811
  2093
hoelzl@31811
  2094
lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
hoelzl@31811
  2095
  unfolding real_diff_def interpret_floatarith.simps ..
hoelzl@31811
  2096
hoelzl@31811
  2097
lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) vs =
hoelzl@31811
  2098
  sin (interpret_floatarith a vs)"
hoelzl@31811
  2099
  unfolding sin_cos_eq interpret_floatarith.simps
hoelzl@31811
  2100
            interpret_floatarith_divide interpret_floatarith_diff real_diff_def
hoelzl@31811
  2101
  by auto
hoelzl@31811
  2102
hoelzl@31811
  2103
lemma interpret_floatarith_tan:
hoelzl@31811
  2104
  "interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (Inverse (Cos a))) vs =
hoelzl@31811
  2105
   tan (interpret_floatarith a vs)"
hoelzl@31811
  2106
  unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def real_divide_def
hoelzl@31811
  2107
  by auto
hoelzl@31811
  2108
hoelzl@31811
  2109
lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
hoelzl@31811
  2110
  unfolding powr_def interpret_floatarith.simps ..
hoelzl@31811
  2111
hoelzl@31811
  2112
lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
hoelzl@31811
  2113
  unfolding log_def interpret_floatarith.simps real_divide_def ..
hoelzl@31811
  2114
hoelzl@31811
  2115
lemma interpret_floatarith_num:
hoelzl@31811
  2116
  shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
hoelzl@31811
  2117
  and "interpret_floatarith (Num (Float 1 0)) vs = 1"
hoelzl@31811
  2118
  and "interpret_floatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
hoelzl@31811
  2119
hoelzl@31811
  2120
hoelzl@29805
  2121
subsection "Implement approximation function"
hoelzl@29805
  2122
hoelzl@29805
  2123
fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
hoelzl@29805
  2124
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
hoelzl@29805
  2125
"lift_bin' a b f = None"
hoelzl@29805
  2126
hoelzl@29805
  2127
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
hoelzl@29805
  2128
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
hoelzl@29805
  2129
                                             | t \<Rightarrow> None)" |
hoelzl@29805
  2130
"lift_un b f = None"
hoelzl@29805
  2131
hoelzl@29805
  2132
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
hoelzl@29805
  2133
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
hoelzl@29805
  2134
"lift_un' b f = None"
hoelzl@29805
  2135
hoelzl@31811
  2136
definition
hoelzl@31811
  2137
"bounded_by xs vs \<longleftrightarrow>
hoelzl@31811
  2138
  (\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
hoelzl@31811
  2139
         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"
hoelzl@31811
  2140
hoelzl@31811
  2141
lemma bounded_byE:
hoelzl@31811
  2142
  assumes "bounded_by xs vs"
hoelzl@31811
  2143
  shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
hoelzl@31811
  2144
         | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
hoelzl@31811
  2145
  using assms bounded_by_def by blast
hoelzl@31811
  2146
hoelzl@31811
  2147
lemma bounded_by_update:
hoelzl@31811
  2148
  assumes "bounded_by xs vs"
hoelzl@31811
  2149
  and bnd: "xs ! i \<in> { real l .. real u }"
hoelzl@31811
  2150
  shows "bounded_by xs (vs[i := Some (l,u)])"
hoelzl@31811
  2151
proof -
hoelzl@31811
  2152
{ fix j
hoelzl@31811
  2153
  let ?vs = "vs[i := Some (l,u)]"
hoelzl@31811
  2154
  assume "j < length ?vs" hence [simp]: "j < length vs" by simp
hoelzl@31811
  2155
  have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
hoelzl@31811
  2156
  proof (cases "?vs ! j")
hoelzl@31811
  2157
    case (Some b)
hoelzl@31811
  2158
    thus ?thesis
hoelzl@31811
  2159
    proof (cases "i = j")
hoelzl@31811
  2160
      case True
hoelzl@31811
  2161
      thus ?thesis using `?vs ! j = Some b` and bnd by auto
hoelzl@31811
  2162
    next
hoelzl@31811
  2163
      case False
hoelzl@31811
  2164
      thus ?thesis using `bounded_by xs vs` unfolding bounded_by_def by auto
hoelzl@31811
  2165
    qed
hoelzl@31811
  2166
  qed auto }
hoelzl@31811
  2167
  thus ?thesis unfolding bounded_by_def by auto
hoelzl@31811
  2168
qed
hoelzl@31811
  2169
hoelzl@31811
  2170
lemma bounded_by_None:
hoelzl@31811
  2171
  shows "bounded_by xs (replicate (length xs) None)"
hoelzl@31811
  2172
  unfolding bounded_by_def by auto
hoelzl@31811
  2173
hoelzl@31811
  2174
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where
hoelzl@29805
  2175
"approx' prec a bs          = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (round_down prec l, round_up prec u) | None \<Rightarrow> None)" |
hoelzl@31811
  2176
"approx prec (Add a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (l1 + l2, u1 + u2))" |
hoelzl@29805
  2177
"approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
hoelzl@29805
  2178
"approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
hoelzl@31809
  2179
                                    (\<lambda> a1 a2 b1 b2. (float_nprt a1 * float_pprt b2 + float_nprt a2 * float_nprt b2 + float_pprt a1 * float_pprt b1 + float_pprt a2 * float_nprt b1,
hoelzl@29805
  2180
                                                     float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" |
hoelzl@29805
  2181
"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" |
hoelzl@29805
  2182
"approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
hoelzl@29805
  2183
"approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
hoelzl@29805
  2184
"approx prec (Min a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" |
hoelzl@29805
  2185
"approx prec (Max a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" |
hoelzl@29805
  2186
"approx prec (Abs a) bs     = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" |
hoelzl@29805
  2187
"approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
hoelzl@31467
  2188
"approx prec (Sqrt a) bs    = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
hoelzl@29805
  2189
"approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
hoelzl@29805
  2190
"approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
hoelzl@29805
  2191
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
hoelzl@29805
  2192
"approx prec (Num f) bs     = Some (f, f)" |
hoelzl@31811
  2193
"approx prec (Atom i) bs    = (if i < length bs then bs ! i else None)"
hoelzl@29805
  2194
hoelzl@29805
  2195
lemma lift_bin'_ex:
hoelzl@29805
  2196
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
hoelzl@29805
  2197
  shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
hoelzl@29805