pymath.c to Modules/_math.c.
functions and constants
**************************************************************************/
-/* Python provides implementations for copysign, acosh, asinh, atanh,
- * log1p and hypot in Python/pymath.c just in case your math library doesn't
- * provide the functions.
+/* Python provides implementations for copysign, round and hypot in
+ * Python/pymath.c just in case your math library doesn't provide the
+ * functions.
*
*Note: PC/pyconfig.h defines copysign as _copysign
*/
extern double round(double);
#endif
-#ifndef HAVE_ACOSH
-extern double acosh(double);
-#endif
-
-#ifndef HAVE_ASINH
-extern double asinh(double);
-#endif
-
-#ifndef HAVE_ATANH
-extern double atanh(double);
-#endif
-
-#ifndef HAVE_LOG1P
-extern double log1p(double);
-#endif
-
#ifndef HAVE_HYPOT
extern double hypot(double, double);
#endif
# Modules that should always be present (non UNIX dependent):
#array arraymodule.c # array objects
-#cmath cmathmodule.c # -lm # complex math library functions
+#cmath cmathmodule.c _math.c # -lm # complex math library functions
#math mathmodule.c _math.c # -lm # math library functions, e.g. sin()
#_struct _struct.c # binary structure packing/unpacking
#time timemodule.c # -lm # time operations and variables
/* 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
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);
+ }
+}
+double _Py_acosh(double x);
+double _Py_asinh(double x);
+double _Py_atanh(double x);
double _Py_expm1(double x);
+double _Py_log1p(double x);
+
+#ifdef HAVE_ACOSH
+#define m_acosh acosh
+#else
+/* if the system doesn't have acosh, use the substitute
+ function defined in Modules/_math.c. */
+#define m_acosh _Py_acosh
+#endif
+
+#ifdef HAVE_ASINH
+#define m_asinh asinh
+#else
+/* if the system doesn't have asinh, use the substitute
+ function defined in Modules/_math.c. */
+#define m_asinh _Py_asinh
+#endif
+
+#ifdef HAVE_ATANH
+#define m_atanh atanh
+#else
+/* if the system doesn't have atanh, use the substitute
+ function defined in Modules/_math.c. */
+#define m_atanh _Py_atanh
+#endif
#ifdef HAVE_EXPM1
#define m_expm1 expm1
function defined in Modules/_math.c. */
#define m_expm1 _Py_expm1
#endif
+
+#ifdef HAVE_LOG1P
+#define m_log1p log1p
+#else
+/* if the system doesn't have log1p, use the substitute
+ function defined in Modules/_math.c. */
+#define m_log1p _Py_log1p
+#endif
/* much code borrowed from mathmodule.c */
#include "Python.h"
+#include "_math.h"
/* we need DBL_MAX, DBL_MIN, DBL_EPSILON, DBL_MANT_DIG and FLT_RADIX from
float.h. We assume that FLT_RADIX is either 2 or 16. */
#include <float.h>
s2.imag = z.imag;
s2 = c_sqrt(s2);
r.real = 2.*atan2(s1.real, s2.real);
- r.imag = asinh(s2.real*s1.imag - s2.imag*s1.real);
+ r.imag = m_asinh(s2.real*s1.imag - s2.imag*s1.real);
}
errno = 0;
return r;
s2.real = z.real + 1.;
s2.imag = z.imag;
s2 = c_sqrt(s2);
- r.real = asinh(s1.real*s2.real + s1.imag*s2.imag);
+ r.real = m_asinh(s1.real*s2.real + s1.imag*s2.imag);
r.imag = 2.*atan2(s1.imag, s2.real);
}
errno = 0;
s2.real = 1.-z.imag;
s2.imag = z.real;
s2 = c_sqrt(s2);
- r.real = asinh(s1.real*s2.imag-s2.real*s1.imag);
+ r.real = m_asinh(s1.real*s2.imag-s2.real*s1.imag);
r.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag);
}
errno = 0;
errno = 0;
}
} else {
- r.real = log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.;
+ r.real = m_log1p(4.*z.real/((1-z.real)*(1-z.real) + ay*ay))/4.;
r.imag = -atan2(-2.*z.imag, (1-z.real)*(1+z.real) - ay*ay)/2.;
errno = 0;
}
if (0.71 <= h && h <= 1.73) {
am = ax > ay ? ax : ay; /* max(ax, ay) */
an = ax > ay ? ay : ax; /* min(ax, ay) */
- r.real = log1p((am-1)*(am+1)+an*an)/2.;
+ r.real = m_log1p((am-1)*(am+1)+an*an)/2.;
} else {
r.real = log(h);
}
FUNC1(acos, acos, 0,
"acos(x)\n\nReturn the arc cosine (measured in radians) of x.")
-FUNC1(acosh, acosh, 0,
+FUNC1(acosh, m_acosh, 0,
"acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.")
FUNC1(asin, asin, 0,
"asin(x)\n\nReturn the arc sine (measured in radians) of x.")
-FUNC1(asinh, asinh, 0,
+FUNC1(asinh, m_asinh, 0,
"asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.")
FUNC1(atan, atan, 0,
"atan(x)\n\nReturn the arc tangent (measured in radians) of x.")
FUNC2(atan2, m_atan2,
"atan2(y, x)\n\nReturn the arc tangent (measured in radians) of y/x.\n"
"Unlike atan(y/x), the signs of both x and y are considered.")
-FUNC1(atanh, atanh, 0,
+FUNC1(atanh, m_atanh, 0,
"atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.")
FUNC1(ceil, ceil, 0,
"ceil(x)\n\nReturn the ceiling of x as a float.\n"
"gamma(x)\n\nGamma function at x.")
FUNC1A(lgamma, m_lgamma,
"lgamma(x)\n\nNatural logarithm of absolute value of Gamma function at x.")
-FUNC1(log1p, log1p, 1,
+FUNC1(log1p, m_log1p, 1,
"log1p(x)\n\nReturn the natural logarithm of 1+x (base e).\n"
"The result is computed in a way which is accurate for x near zero.")
FUNC1(sin, sin, 0,
return copysign(y, x);
}
#endif /* HAVE_ROUND */
-
-#ifndef HAVE_LOG1P
-#include <float.h>
-
-double
-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);
- }
-}
-#endif /* HAVE_LOG1P */
-
-/*
- * ====================================================
- * 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;
-
-/* 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)))
- */
-
-#ifndef HAVE_ASINH
-double
-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);
-
-}
-#endif /* HAVE_ASINH */
-
-/* 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.
- */
-
-#ifndef HAVE_ACOSH
-double
-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));
- }
-}
-#endif /* HAVE_ACOSH */
-
-/* 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;
- *
- */
-
-#ifndef HAVE_ATANH
-double
-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);
-}
-#endif /* HAVE_ATANH */
# array objects
exts.append( Extension('array', ['arraymodule.c']) )
# complex math library functions
- exts.append( Extension('cmath', ['cmathmodule.c'],
+ exts.append( Extension('cmath', ['cmathmodule.c', '_math.c'],
+ depends=['_math.h'],
libraries=math_libs) )
-
# math library functions, e.g. sin()
exts.append( Extension('math', ['mathmodule.c', '_math.c'],
depends=['_math.h'],
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