Changes between Version 6 and Version 7 of research/trig Tweet

Timestamp:

Oct 14, 2011, 12:16:42 PM (11 years ago)

Author:

sam

Comment:

--

research/trig

@@ -14 +14 @@
 }}}
 
-We know sin(x) is an odd function, so instead we look for a polynomial Q(x) such that P(x) = xQ(x²):
+We know sin(x) is an odd function, so instead we look for a polynomial Q(x) such that P(x) = xQ(x²), and we reduce the range to positive values:
 
 {{{
 #!latex
-\[\max_{x \in [-\pi/2, \pi/2]}{\big\vert\sin(x) - xQ(x^2)\big\vert} = E\]
+\[\max_{x \in [0, \pi/2]}{\big\vert\sin(x) - xQ(x^2)\big\vert} = E\]
 }}}
 
-Substitute y for x² and reduce the range to positive values:
+Substitute y for x²:
 
 {{{
@@ -53 +53 @@
 {{{
 #!cpp
-
 static real myfun(real const &y)
 {
@@ -92 +91 @@
 a6 = -7.36458957326227991327065122848667046e-13;
 }}}
+
+=== Relative error ===
+
+Searching for '''relative error''' instead:
+
+{{{
+#!latex
+\[\max_{x \in [-\pi/2, \pi/2]}{\dfrac{\big\vert\sin(x) - P(x)\big\vert}{|\sin(x)|}} = E\]
+}}}
+
+Using the same method as for absolute error, we get:
+
+{{{
+#!latex
+\[\max_{x \in [0, \pi^2/4]}{\dfrac{\bigg\lvert\dfrac{\sin(\sqrt{y})-\sqrt{y}}{y\sqrt{y}} - R(y)\bigg\rvert}{\bigg\lvert\dfrac{\sin(y)}{y\sqrt{y}}\bigg\rvert}} = E\]
+}}}
+
+{{{
+#!cpp
+static real myfun(real const &y)
+{
+    real x = sqrt(y);
+    return (sin(x) - x) / (x * y);
+}
+
+static real myerr(real const &y)
+{
+    real x = sqrt(y);
+    return sin(x) / (x * y);
+}
+
+RemezSolver<6> solver;
+solver.Run(real::R_1 >> 400, real::R_PI_2 * real::R_PI_2, myfun, myerr, 15);
+}}}
+
+{{{
+#!cpp
+a0 = -1.666666666666666587374325845020415990185e-1;
+a1 = +8.333333333333133768001243698120735518527e-3;
+a2 = -1.984126984109960366729319073763957206143e-4;
+a3 = +2.755731915499171528179303925040423384803e-6;
+a4 = -2.505209340355388148617179634180834358690e-8;
+a5 = +1.605725287696319345779134635418774782711e-10;
+a6 = -7.535968124281960435283756562793611388136e-13;
+}}}