--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/ex/Approximations.thy Fri Apr 03 21:25:55 2015 +0200
@@ -0,0 +1,38 @@
+section {* Binary Approximations to Constants *}
+
+theory Approximations
+imports "~~/src/HOL/Multivariate_Analysis/Complex_Transcendental"
+begin
+
+declare of_real_numeral [simp]
+
+subsection{*Approximation to pi*}
+
+lemma sin_pi6_straddle:
+ assumes "0 \<le> a" "a \<le> b" "b \<le> 4" "sin(a/6) \<le> 1/2" "1/2 \<le> sin(b/6)"
+ shows "a \<le> pi \<and> pi \<le> b"
+proof -
+ have *: "\<And>x::real. 0 < x & x < 7/5 \<Longrightarrow> 0 < sin x"
+ using pi_ge_two
+ by (auto intro: sin_gt_zero)
+ have ab: "(b \<le> pi * 3 \<Longrightarrow> pi \<le> b)" "(a \<le> pi * 3 \<Longrightarrow> a \<le> pi)"
+ using sin_mono_le_eq [of "pi/6" "b/6"] sin_mono_le_eq [of "a/6" "pi/6"] assms
+ by (simp_all add: sin_30 power.power_Suc norm_divide)
+ show ?thesis
+ using assms Taylor_sin [of "a/6" 0] pi_ge_two
+ by (auto simp: sin_30 power.power_Suc norm_divide intro: ab)
+qed
+
+(*32-bit approximation. SLOW simplification steps: big calculations with the rewriting engine*)
+lemma pi_approx_32: "abs(pi - 13493037705/4294967296) \<le> inverse(2 ^ 32)"
+ apply (simp only: abs_diff_le_iff)
+ apply (rule sin_pi6_straddle, simp_all)
+ using Taylor_sin [of "1686629713/3221225472" 11]
+ apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral)
+ apply (simp only: pos_le_divide_eq [symmetric])
+ using Taylor_sin [of "6746518853/12884901888" 11]
+ apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral)
+ apply (simp only: pos_le_divide_eq [symmetric] pos_divide_le_eq [symmetric])
+ done
+
+end