diff options
| -rw-r--r-- | Include/pymath.h | 22 | ||||
| -rw-r--r-- | Modules/Setup.dist | 2 | ||||
| -rw-r--r-- | Modules/_math.c | 199 | ||||
| -rw-r--r-- | Modules/_math.h | 36 | ||||
| -rw-r--r-- | Modules/cmathmodule.c | 11 | ||||
| -rw-r--r-- | Modules/mathmodule.c | 8 | ||||
| -rw-r--r-- | Python/pymath.c | 199 | ||||
| -rw-r--r-- | setup.py | 4 |
8 files changed, 250 insertions, 231 deletions
diff --git a/Include/pymath.h b/Include/pymath.h index dc2c427..e3cf22b 100644 --- a/Include/pymath.h +++ b/Include/pymath.h @@ -8,9 +8,9 @@ Symbols and macros to supply platform-independent interfaces to mathematical 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 */ @@ -22,22 +22,6 @@ extern double copysign(double, double); 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 diff --git a/Modules/Setup.dist b/Modules/Setup.dist index e1cb235..45bf531 100644 --- a/Modules/Setup.dist +++ b/Modules/Setup.dist @@ -157,7 +157,7 @@ _symtable symtablemodule.c # 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 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 @@ _Py_expm1(double x) 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); + } +} diff --git a/Modules/_math.h b/Modules/_math.h index 69c96b5..c0ceece 100644 --- a/Modules/_math.h +++ b/Modules/_math.h @@ -1,4 +1,32 @@ +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 @@ -7,3 +35,11 @@ double _Py_expm1(double x); 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 diff --git a/Modules/cmathmodule.c b/Modules/cmathmodule.c index fbf6ece..4d13e58 100644 --- a/Modules/cmathmodule.c +++ b/Modules/cmathmodule.c @@ -3,6 +3,7 @@ /* 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> @@ -149,7 +150,7 @@ c_acos(Py_complex z) 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; @@ -181,7 +182,7 @@ c_acosh(Py_complex z) 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; @@ -238,7 +239,7 @@ c_asinh(Py_complex z) 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; @@ -342,7 +343,7 @@ c_atanh(Py_complex z) 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; } @@ -552,7 +553,7 @@ c_log(Py_complex z) 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); } diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c index 92f5f42..a057cac 100644 --- a/Modules/mathmodule.c +++ b/Modules/mathmodule.c @@ -816,18 +816,18 @@ math_2(PyObject *args, double (*func) (double, double), char *funcname) 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.") static PyObject * math_ceil(PyObject *self, PyObject *number) { @@ -895,7 +895,7 @@ FUNC1A(gamma, m_tgamma, "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, diff --git a/Python/pymath.c b/Python/pymath.c index db2920c..83105f2 100644 --- a/Python/pymath.c +++ b/Python/pymath.c @@ -77,202 +77,3 @@ round(double x) 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 */ @@ -394,9 +394,9 @@ class PyBuildExt(build_ext): # 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'], |