Merged revisions 76978 via svnmerge from
svn+ssh://pythondev@svn.python.org/python/trunk
........
r76978 | mark.dickinson | 2009-12-21 15:22:00 +0000 (Mon, 21 Dec 2009) | 3 lines
Issue #7518: Move substitute definitions of C99 math functions from
pymath.c to Modules/_math.c.
........
diff --git a/Modules/_math.c b/Modules/_math.c
index 9d330aa..e27c100 100644
--- a/Modules/_math.c
+++ b/Modules/_math.c
@@ -1,8 +1,161 @@
/* Definitions of some C99 math library functions, for those platforms
that don't implement these functions already. */
+#include "Python.h"
#include <float.h>
-#include <math.h>
+
+/* The following copyright notice applies to the original
+ implementations of acosh, asinh and atanh. */
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+static const double ln2 = 6.93147180559945286227E-01;
+static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */
+static const double two_pow_p28 = 268435456.0; /* 2**28 */
+static const double zero = 0.0;
+
+/* acosh(x)
+ * Method :
+ * Based on
+ * acosh(x) = log [ x + sqrt(x*x-1) ]
+ * we have
+ * acosh(x) := log(x)+ln2, if x is large; else
+ * acosh(x) := log(2x-1/(sqrt(x*x-1)+x)) if x>2; else
+ * acosh(x) := log1p(t+sqrt(2.0*t+t*t)); where t=x-1.
+ *
+ * Special cases:
+ * acosh(x) is NaN with signal if x<1.
+ * acosh(NaN) is NaN without signal.
+ */
+
+double
+_Py_acosh(double x)
+{
+ if (Py_IS_NAN(x)) {
+ return x+x;
+ }
+ if (x < 1.) { /* x < 1; return a signaling NaN */
+ errno = EDOM;
+#ifdef Py_NAN
+ return Py_NAN;
+#else
+ return (x-x)/(x-x);
+#endif
+ }
+ else if (x >= two_pow_p28) { /* x > 2**28 */
+ if (Py_IS_INFINITY(x)) {
+ return x+x;
+ } else {
+ return log(x)+ln2; /* acosh(huge)=log(2x) */
+ }
+ }
+ else if (x == 1.) {
+ return 0.0; /* acosh(1) = 0 */
+ }
+ else if (x > 2.) { /* 2 < x < 2**28 */
+ double t = x*x;
+ return log(2.0*x - 1.0 / (x + sqrt(t - 1.0)));
+ }
+ else { /* 1 < x <= 2 */
+ double t = x - 1.0;
+ return log1p(t + sqrt(2.0*t + t*t));
+ }
+}
+
+
+/* asinh(x)
+ * Method :
+ * Based on
+ * asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
+ * we have
+ * asinh(x) := x if 1+x*x=1,
+ * := sign(x)*(log(x)+ln2)) for large |x|, else
+ * := sign(x)*log(2|x|+1/(|x|+sqrt(x*x+1))) if|x|>2, else
+ * := sign(x)*log1p(|x| + x^2/(1 + sqrt(1+x^2)))
+ */
+
+double
+_Py_asinh(double x)
+{
+ double w;
+ double absx = fabs(x);
+
+ if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) {
+ return x+x;
+ }
+ if (absx < two_pow_m28) { /* |x| < 2**-28 */
+ return x; /* return x inexact except 0 */
+ }
+ if (absx > two_pow_p28) { /* |x| > 2**28 */
+ w = log(absx)+ln2;
+ }
+ else if (absx > 2.0) { /* 2 < |x| < 2**28 */
+ w = log(2.0*absx + 1.0 / (sqrt(x*x + 1.0) + absx));
+ }
+ else { /* 2**-28 <= |x| < 2= */
+ double t = x*x;
+ w = log1p(absx + t / (1.0 + sqrt(1.0 + t)));
+ }
+ return copysign(w, x);
+
+}
+
+/* atanh(x)
+ * Method :
+ * 1.Reduced x to positive by atanh(-x) = -atanh(x)
+ * 2.For x>=0.5
+ * 1 2x x
+ * atanh(x) = --- * log(1 + -------) = 0.5 * log1p(2 * --------)
+ * 2 1 - x 1 - x
+ *
+ * For x<0.5
+ * atanh(x) = 0.5*log1p(2x+2x*x/(1-x))
+ *
+ * Special cases:
+ * atanh(x) is NaN if |x| >= 1 with signal;
+ * atanh(NaN) is that NaN with no signal;
+ *
+ */
+
+double
+_Py_atanh(double x)
+{
+ double absx;
+ double t;
+
+ if (Py_IS_NAN(x)) {
+ return x+x;
+ }
+ absx = fabs(x);
+ if (absx >= 1.) { /* |x| >= 1 */
+ errno = EDOM;
+#ifdef Py_NAN
+ return Py_NAN;
+#else
+ return x/zero;
+#endif
+ }
+ if (absx < two_pow_m28) { /* |x| < 2**-28 */
+ return x;
+ }
+ if (absx < 0.5) { /* |x| < 0.5 */
+ t = absx+absx;
+ t = 0.5 * log1p(t + t*absx / (1.0 - absx));
+ }
+ else { /* 0.5 <= |x| <= 1.0 */
+ t = 0.5 * log1p((absx + absx) / (1.0 - absx));
+ }
+ return copysign(t, x);
+}
/* Mathematically, expm1(x) = exp(x) - 1. The expm1 function is designed
to avoid the significant loss of precision that arises from direct
@@ -29,3 +182,47 @@
else
return exp(x) - 1.0;
}
+
+/* log1p(x) = log(1+x). The log1p function is designed to avoid the
+ significant loss of precision that arises from direct evaluation when x is
+ small. */
+
+double
+_Py_log1p(double x)
+{
+ /* For x small, we use the following approach. Let y be the nearest float
+ to 1+x, then
+
+ 1+x = y * (1 - (y-1-x)/y)
+
+ so log(1+x) = log(y) + log(1-(y-1-x)/y). Since (y-1-x)/y is tiny, the
+ second term is well approximated by (y-1-x)/y. If abs(x) >=
+ DBL_EPSILON/2 or the rounding-mode is some form of round-to-nearest
+ then y-1-x will be exactly representable, and is computed exactly by
+ (y-1)-x.
+
+ If abs(x) < DBL_EPSILON/2 and the rounding mode is not known to be
+ round-to-nearest then this method is slightly dangerous: 1+x could be
+ rounded up to 1+DBL_EPSILON instead of down to 1, and in that case
+ y-1-x will not be exactly representable any more and the result can be
+ off by many ulps. But this is easily fixed: for a floating-point
+ number |x| < DBL_EPSILON/2., the closest floating-point number to
+ log(1+x) is exactly x.
+ */
+
+ double y;
+ if (fabs(x) < DBL_EPSILON/2.) {
+ return x;
+ } else if (-0.5 <= x && x <= 1.) {
+ /* WARNING: it's possible than an overeager compiler
+ will incorrectly optimize the following two lines
+ to the equivalent of "return log(1.+x)". If this
+ happens, then results from log1p will be inaccurate
+ for small x. */
+ y = 1.+x;
+ return log(y)-((y-1.)-x)/y;
+ } else {
+ /* NaNs and infinities should end up here */
+ return log(1.+x);
+ }
+}