Index: Doc/library/cmath.rst =================================================================== --- Doc/library/cmath.rst (revision 60151) +++ Doc/library/cmath.rst (working copy) @@ -14,6 +14,15 @@ floating-point number, respectively, and the function is then applied to the result of the conversion. +.. note:: + + On platforms with hardware and system-level support for signed + zeros, functions involving branch cuts are continuous on *both* + sides of the branch cut: the sign of the zero distinguishes one + side of the branch cut from the other. On platforms that do not + support signed zeros the continuity is as specified below. + + The functions are: @@ -37,32 +46,37 @@ .. function:: asinh(x) - Return the hyperbolic arc sine of *x*. There are two branch cuts, extending - left from ``±1j`` to ``±∞j``, both continuous from above. These branch cuts - should be considered a bug to be corrected in a future release. The correct - branch cuts should extend along the imaginary axis, one from ``1j`` up to - ``∞j`` and continuous from the right, and one from ``-1j`` down to ``-∞j`` - and continuous from the left. + Return the hyperbolic arc sine of *x*. There are two branch cuts: + One extends from ``1j`` along the imaginary axis to ``∞j``, + continuous from the right. The other extends from ``-1j`` along + the imaginary axis to ``-∞j``, continuous from the left. + .. versionchanged:: 2.6 + branch cuts moved to match those recommended by the C99 standard + .. function:: atan(x) Return the arc tangent of *x*. There are two branch cuts: One extends from - ``1j`` along the imaginary axis to ``∞j``, continuous from the left. The + ``1j`` along the imaginary axis to ``∞j``, continuous from the right. The other extends from ``-1j`` along the imaginary axis to ``-∞j``, continuous - from the left. (This should probably be changed so the upper cut becomes - continuous from the other side.) + from the left. + .. versionchanged:: 2.6 + direction of continuity of upper cut reversed + .. function:: atanh(x) Return the hyperbolic arc tangent of *x*. There are two branch cuts: One - extends from ``1`` along the real axis to ``∞``, continuous from above. The + extends from ``1`` along the real axis to ``∞``, continuous from below. The other extends from ``-1`` along the real axis to ``-∞``, continuous from - above. (This should probably be changed so the right cut becomes continuous - from the other side.) + above. + .. versionchanged:: 2.6 + direction of continuity of right cut reversed + .. function:: cos(x) Return the cosine of *x*. @@ -154,3 +168,4 @@ nothing's sign bit. In Iserles, A., and Powell, M. (eds.), The state of the art in numerical analysis. Clarendon Press (1987) pp165-211. + Index: Doc/library/math.rst =================================================================== --- Doc/library/math.rst (revision 60151) +++ Doc/library/math.rst (working copy) @@ -131,6 +131,15 @@ *base* argument added. +.. function:: log1p(x[, base]) + + Return the logarithm of *1+x* to the given *base*. If the *base* is not specified, + return the natural logarithm of *1+x* (that is, the logarithm to base *e*). The + result is calculated in a way which is accurate for *x* near zero. + + .. versionadded:: 2.6 + + .. function:: log10(x) Return the base-10 logarithm of *x*. @@ -189,6 +198,13 @@ Return the sine of *x* radians. +.. function:: asinh(x) + + Return the inverse hyperbolic sine of *x*, in radians. + + .. versionadded:: 2.6 + + .. function:: tan(x) Return the tangent of *x* radians. @@ -213,6 +229,13 @@ Return the hyperbolic cosine of *x*. +.. function:: acosh(x) + + Return the inverse hyperbolic cosine of *x*, in radians. + + .. versionadded:: 2.6 + + .. function:: sinh(x) Return the hyperbolic sine of *x*. @@ -222,6 +245,14 @@ Return the hyperbolic tangent of *x*. + +.. function:: atanh(x) + + Return the inverse hyperbolic tangent of *x*, in radians. + + .. versionadded:: 2.6 + + The module also defines two mathematical constants: @@ -234,6 +265,7 @@ The mathematical constant *e*. + .. note:: The :mod:`math` module consists mostly of thin wrappers around the platform C Index: Doc/license.rst =================================================================== --- Doc/license.rst (revision 60151) +++ Doc/license.rst (working copy) @@ -645,3 +645,15 @@ ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. + +Python/pymath.c +--------------- + +The :file:`Python/pymath.c` module contains the following notice:: + + 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. Index: Include/pymath.h =================================================================== --- Include/pymath.h (revision 0) +++ Include/pymath.h (revision 0) @@ -0,0 +1,175 @@ +#ifndef Py_PYMATH_H +#define Py_PYMATH_H + +#include "pyconfig.h" /* include for defines */ + +#ifdef HAVE_STDINT_H +#include +#endif + +/************************************************************************** +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. + * + *Note: PC/pyconfig.h defines copysign as _copysign + */ +#ifndef HAVE_COPYSIGN +extern double copysign(doube, 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 + +/* extra declarations */ +#ifndef _MSC_VER +#ifndef __STDC__ +extern double fmod (double, double); +extern double frexp (double, int *); +extern double ldexp (double, int); +extern double modf (double, double *); +#endif /* __STDC__ */ +#endif /* _MSC_VER */ + +/* High precision defintion of pi and e (Euler) + * The values are taken from libc6's math.h. + */ +#ifndef Py_MATH_PIl +#define Py_MATH_PIl 3.1415926535897932384626433832795029L +#endif +#ifndef Py_MATH_PI +#define Py_MATH_PI 3.14159265358979323846 +#endif + +#ifndef Py_MATH_El +#define Py_MATH_El 2.7182818284590452353602874713526625L +#endif + +#ifndef Py_MATH_E +#define Py_MATH_E 2.7182818284590452354 +#endif + +/* Py_IS_NAN(X) + * Return 1 if float or double arg is a NaN, else 0. + * Caution: + * X is evaluated more than once. + * This may not work on all platforms. Each platform has *some* + * way to spell this, though -- override in pyconfig.h if you have + * a platform where it doesn't work. + * Note: PC/pyconfig.h defines Py_IS_NAN as _isnan + */ +#ifndef Py_IS_NAN +#ifdef HAVE_ISNAN +#define Py_IS_NAN(X) isnan(X) +#else +#define Py_IS_NAN(X) ((X) != (X)) +#endif +#endif + +/* Py_IS_INFINITY(X) + * Return 1 if float or double arg is an infinity, else 0. + * Caution: + * X is evaluated more than once. + * This implementation may set the underflow flag if |X| is very small; + * it really can't be implemented correctly (& easily) before C99. + * Override in pyconfig.h if you have a better spelling on your platform. + * Note: PC/pyconfig.h defines Py_IS_INFINITY as _isinf + */ +#ifndef Py_IS_INFINITY +#ifdef HAVE_ISINF +#define Py_IS_INFINITY(X) isinf(X) +#else +#define Py_IS_INFINITY(X) ((X) && (X)*0.5 == (X)) +#endif +#endif + +/* Py_IS_FINITE(X) + * Return 1 if float or double arg is neither infinite nor NAN, else 0. + * Some compilers (e.g. VisualStudio) have intrisics for this, so a special + * macro for this particular test is useful + * Note: PC/pyconfig.h defines Py_IS_FINITE as _finite + */ +#ifndef Py_IS_FINITE +#ifdef HAVE_FINITE +#define Py_IS_FINITE(X) finite(X) +#else +#define Py_IS_FINITE(X) (!Py_IS_INFINITY(X) && !Py_IS_NAN(X)) +#endif +#endif + +/* HUGE_VAL is supposed to expand to a positive double infinity. Python + * uses Py_HUGE_VAL instead because some platforms are broken in this + * respect. We used to embed code in pyport.h to try to worm around that, + * but different platforms are broken in conflicting ways. If you're on + * a platform where HUGE_VAL is defined incorrectly, fiddle your Python + * config to #define Py_HUGE_VAL to something that works on your platform. + */ +#ifndef Py_HUGE_VAL +#define Py_HUGE_VAL HUGE_VAL +#endif + +/* Py_NAN + * A value that evaluates to a NaN. On IEEE 754 platforms INF*0 or + * INF/INF works. Define Py_NO_NAN in pyconfig.h if your platform + * doesn't support NaNs. + */ +#if !defined(Py_NAN) && !defined(Py_NO_NAN) +#define Py_NAN (Py_HUGE_VAL * 0.) +#endif + +/* Py_OVERFLOWED(X) + * Return 1 iff a libm function overflowed. Set errno to 0 before calling + * a libm function, and invoke this macro after, passing the function + * result. + * Caution: + * This isn't reliable. C99 no longer requires libm to set errno under + * any exceptional condition, but does require +- HUGE_VAL return + * values on overflow. A 754 box *probably* maps HUGE_VAL to a + * double infinity, and we're cool if that's so, unless the input + * was an infinity and an infinity is the expected result. A C89 + * system sets errno to ERANGE, so we check for that too. We're + * out of luck if a C99 754 box doesn't map HUGE_VAL to +Inf, or + * if the returned result is a NaN, or if a C89 box returns HUGE_VAL + * in non-overflow cases. + * X is evaluated more than once. + * Some platforms have better way to spell this, so expect some #ifdef'ery. + * + * OpenBSD uses 'isinf()' because a compiler bug on that platform causes + * the longer macro version to be mis-compiled. This isn't optimal, and + * should be removed once a newer compiler is available on that platform. + * The system that had the failure was running OpenBSD 3.2 on Intel, with + * gcc 2.95.3. + * + * According to Tim's checkin, the FreeBSD systems use isinf() to work + * around a FPE bug on that platform. + */ +#if defined(__FreeBSD__) || defined(__OpenBSD__) +#define Py_OVERFLOWED(X) isinf(X) +#else +#define Py_OVERFLOWED(X) ((X) != 0.0 && (errno == ERANGE || \ + (X) == Py_HUGE_VAL || \ + (X) == -Py_HUGE_VAL)) +#endif + +#endif /* Py_PYMATH_H */ Property changes on: Include\pymath.h ___________________________________________________________________ Name: svn:keywords + Id Name: svn:eol-style + native Index: Include/pyport.h =================================================================== --- Include/pyport.h (revision 60151) +++ Include/pyport.h (working copy) @@ -349,123 +349,6 @@ #define Py_SAFE_DOWNCAST(VALUE, WIDE, NARROW) (NARROW)(VALUE) #endif -/* High precision defintion of pi and e (Euler) - * The values are taken from libc6's math.h. - */ -#ifndef Py_MATH_PIl -#define Py_MATH_PIl 3.1415926535897932384626433832795029L -#endif -#ifndef Py_MATH_PI -#define Py_MATH_PI 3.14159265358979323846 -#endif - -#ifndef Py_MATH_El -#define Py_MATH_El 2.7182818284590452353602874713526625L -#endif - -#ifndef Py_MATH_E -#define Py_MATH_E 2.7182818284590452354 -#endif - -/* Py_IS_NAN(X) - * Return 1 if float or double arg is a NaN, else 0. - * Caution: - * X is evaluated more than once. - * This may not work on all platforms. Each platform has *some* - * way to spell this, though -- override in pyconfig.h if you have - * a platform where it doesn't work. - */ -#ifndef Py_IS_NAN -#ifdef HAVE_ISNAN -#define Py_IS_NAN(X) isnan(X) -#else -#define Py_IS_NAN(X) ((X) != (X)) -#endif -#endif - -/* Py_IS_INFINITY(X) - * Return 1 if float or double arg is an infinity, else 0. - * Caution: - * X is evaluated more than once. - * This implementation may set the underflow flag if |X| is very small; - * it really can't be implemented correctly (& easily) before C99. - * Override in pyconfig.h if you have a better spelling on your platform. - */ -#ifndef Py_IS_INFINITY -#ifdef HAVE_ISINF -#define Py_IS_INFINITY(X) isinf(X) -#else -#define Py_IS_INFINITY(X) ((X) && (X)*0.5 == (X)) -#endif -#endif - -/* Py_IS_FINITE(X) - * Return 1 if float or double arg is neither infinite nor NAN, else 0. - * Some compilers (e.g. VisualStudio) have intrisics for this, so a special - * macro for this particular test is useful - */ -#ifndef Py_IS_FINITE -#ifdef HAVE_FINITE -#define Py_IS_FINITE(X) finite -#else -#define Py_IS_FINITE(X) (!Py_IS_INFINITY(X) && !Py_IS_NAN(X)) -#endif -#endif - -/* HUGE_VAL is supposed to expand to a positive double infinity. Python - * uses Py_HUGE_VAL instead because some platforms are broken in this - * respect. We used to embed code in pyport.h to try to worm around that, - * but different platforms are broken in conflicting ways. If you're on - * a platform where HUGE_VAL is defined incorrectly, fiddle your Python - * config to #define Py_HUGE_VAL to something that works on your platform. - */ -#ifndef Py_HUGE_VAL -#define Py_HUGE_VAL HUGE_VAL -#endif - -/* Py_NAN - * A value that evaluates to a NaN. On IEEE 754 platforms INF*0 or - * INF/INF works. Define Py_NO_NAN in pyconfig.h if your platform - * doesn't support NaNs. - */ -#if !defined(Py_NAN) && !defined(Py_NO_NAN) -#define Py_NAN (Py_HUGE_VAL * 0.) -#endif - -/* Py_OVERFLOWED(X) - * Return 1 iff a libm function overflowed. Set errno to 0 before calling - * a libm function, and invoke this macro after, passing the function - * result. - * Caution: - * This isn't reliable. C99 no longer requires libm to set errno under - * any exceptional condition, but does require +- HUGE_VAL return - * values on overflow. A 754 box *probably* maps HUGE_VAL to a - * double infinity, and we're cool if that's so, unless the input - * was an infinity and an infinity is the expected result. A C89 - * system sets errno to ERANGE, so we check for that too. We're - * out of luck if a C99 754 box doesn't map HUGE_VAL to +Inf, or - * if the returned result is a NaN, or if a C89 box returns HUGE_VAL - * in non-overflow cases. - * X is evaluated more than once. - * Some platforms have better way to spell this, so expect some #ifdef'ery. - * - * OpenBSD uses 'isinf()' because a compiler bug on that platform causes - * the longer macro version to be mis-compiled. This isn't optimal, and - * should be removed once a newer compiler is available on that platform. - * The system that had the failure was running OpenBSD 3.2 on Intel, with - * gcc 2.95.3. - * - * According to Tim's checkin, the FreeBSD systems use isinf() to work - * around a FPE bug on that platform. - */ -#if defined(__FreeBSD__) || defined(__OpenBSD__) -#define Py_OVERFLOWED(X) isinf(X) -#else -#define Py_OVERFLOWED(X) ((X) != 0.0 && (errno == ERANGE || \ - (X) == Py_HUGE_VAL || \ - (X) == -Py_HUGE_VAL)) -#endif - /* Py_SET_ERRNO_ON_MATH_ERROR(x) * If a libm function did not set errno, but it looks like the result * overflowed or not-a-number, set errno to ERANGE or EDOM. Set errno @@ -597,15 +480,6 @@ #endif /* 0 */ -/************************ - * WRAPPER FOR * - ************************/ - -#ifndef HAVE_HYPOT -extern double hypot(double, double); -#endif - - /* On 4.4BSD-descendants, ctype functions serves the whole range of * wchar_t character set rather than single byte code points only. * This characteristic can break some operations of string object Index: Include/Python.h =================================================================== --- Include/Python.h (revision 60151) +++ Include/Python.h (working copy) @@ -73,6 +73,7 @@ #if defined(PYMALLOC_DEBUG) && !defined(WITH_PYMALLOC) #error "PYMALLOC_DEBUG requires WITH_PYMALLOC" #endif +#include "pymath.h" #include "pymem.h" #include "object.h" Index: Lib/test/cmath.ctest =================================================================== --- Lib/test/cmath.ctest (revision 0) +++ Lib/test/cmath.ctest (revision 0) @@ -0,0 +1,1276 @@ +-- Testcases for functions in cmath. +-- +-- Each line takes the form: +-- +-- -> +-- +-- where: +-- is a short name identifying the test, +-- is the function to be tested (exp, cos, asinh, ...), +-- is a pair of floats separated by whitespace +-- representing real and imaginary parts of a complex number, and +-- is the expected (ideal) output value, again +-- represented as a pair of floats. +-- +-- Lines beginning with '--' (like this one) start a comment, and are +-- ignored. Blank lines, or lines containing only whitespace, are also +-- ignored. +-- + +-------------------------- +-- acos: Inverse cosine -- +-------------------------- + +-- zeros +acos0000 acos 0.0 0.0 -> 1.5707963267948966 -0.0 +acos0001 acos 0.0 -0.0 -> 1.5707963267948966 0.0 +acos0002 acos -0.0 0.0 -> 1.5707963267948966 -0.0 +acos0003 acos -0.0 -0.0 -> 1.5707963267948966 0.0 + +-- branch points: +/-1 +acos0010 acos 1.0 0.0 -> 0.0 -0.0 +acos0011 acos 1.0 -0.0 -> 0.0 0.0 +acos0012 acos -1.0 0.0 -> 3.1415926535897931 -0.0 +acos0013 acos -1.0 -0.0 -> 3.1415926535897931 0.0 + +-- values along both sides of real axis +acos0020 acos -9.8813129168249309e-324 0.0 -> 1.5707963267948966 -0.0 +acos0021 acos -9.8813129168249309e-324 -0.0 -> 1.5707963267948966 0.0 +acos0022 acos -1e-305 0.0 -> 1.5707963267948966 -0.0 +acos0023 acos -1e-305 -0.0 -> 1.5707963267948966 0.0 +acos0024 acos -1e-150 0.0 -> 1.5707963267948966 -0.0 +acos0025 acos -1e-150 -0.0 -> 1.5707963267948966 0.0 +acos0026 acos -9.9999999999999998e-17 0.0 -> 1.5707963267948968 -0.0 +acos0027 acos -9.9999999999999998e-17 -0.0 -> 1.5707963267948968 0.0 +acos0028 acos -0.001 0.0 -> 1.5717963269615634 -0.0 +acos0029 acos -0.001 -0.0 -> 1.5717963269615634 0.0 +acos0030 acos -0.57899999999999996 0.0 -> 2.1882979816120667 -0.0 +acos0031 acos -0.57899999999999996 -0.0 -> 2.1882979816120667 0.0 +acos0032 acos -0.99999999999999989 0.0 -> 3.1415926386886319 -0.0 +acos0033 acos -0.99999999999999989 -0.0 -> 3.1415926386886319 0.0 +acos0034 acos -1.0000000000000002 0.0 -> 3.1415926535897931 -2.1073424255447014e-08 +acos0035 acos -1.0000000000000002 -0.0 -> 3.1415926535897931 2.1073424255447014e-08 +acos0036 acos -1.0009999999999999 0.0 -> 3.1415926535897931 -0.044717633608306849 +acos0037 acos -1.0009999999999999 -0.0 -> 3.1415926535897931 0.044717633608306849 +acos0038 acos -2.0 0.0 -> 3.1415926535897931 -1.3169578969248168 +acos0039 acos -2.0 -0.0 -> 3.1415926535897931 1.3169578969248168 +acos0040 acos -23.0 0.0 -> 3.1415926535897931 -3.8281684713331012 +acos0041 acos -23.0 -0.0 -> 3.1415926535897931 3.8281684713331012 +acos0042 acos -10000000000000000.0 0.0 -> 3.1415926535897931 -37.534508668464674 +acos0043 acos -10000000000000000.0 -0.0 -> 3.1415926535897931 37.534508668464674 +acos0044 acos -9.9999999999999998e+149 0.0 -> 3.1415926535897931 -346.08091112966679 +acos0045 acos -9.9999999999999998e+149 -0.0 -> 3.1415926535897931 346.08091112966679 +acos0046 acos -1.0000000000000001e+299 0.0 -> 3.1415926535897931 -689.16608998577965 +acos0047 acos -1.0000000000000001e+299 -0.0 -> 3.1415926535897931 689.16608998577965 +acos0048 acos 9.8813129168249309e-324 0.0 -> 1.5707963267948966 -0.0 +acos0049 acos 9.8813129168249309e-324 -0.0 -> 1.5707963267948966 0.0 +acos0050 acos 1e-305 0.0 -> 1.5707963267948966 -0.0 +acos0051 acos 1e-305 -0.0 -> 1.5707963267948966 0.0 +acos0052 acos 1e-150 0.0 -> 1.5707963267948966 -0.0 +acos0053 acos 1e-150 -0.0 -> 1.5707963267948966 0.0 +acos0054 acos 9.9999999999999998e-17 0.0 -> 1.5707963267948966 -0.0 +acos0055 acos 9.9999999999999998e-17 -0.0 -> 1.5707963267948966 0.0 +acos0056 acos 0.001 0.0 -> 1.56979632662823 -0.0 +acos0057 acos 0.001 -0.0 -> 1.56979632662823 0.0 +acos0058 acos 0.57899999999999996 0.0 -> 0.95329467197772655 -0.0 +acos0059 acos 0.57899999999999996 -0.0 -> 0.95329467197772655 0.0 +acos0060 acos 0.99999999999999989 0.0 -> 1.4901161193847656e-08 -0.0 +acos0061 acos 0.99999999999999989 -0.0 -> 1.4901161193847656e-08 0.0 +acos0062 acos 1.0000000000000002 0.0 -> 0.0 -2.1073424255447014e-08 +acos0063 acos 1.0000000000000002 -0.0 -> 0.0 2.1073424255447014e-08 +acos0064 acos 1.0009999999999999 0.0 -> 0.0 -0.044717633608306849 +acos0065 acos 1.0009999999999999 -0.0 -> 0.0 0.044717633608306849 +acos0066 acos 2.0 0.0 -> 0.0 -1.3169578969248168 +acos0067 acos 2.0 -0.0 -> 0.0 1.3169578969248168 +acos0068 acos 23.0 0.0 -> 0.0 -3.8281684713331012 +acos0069 acos 23.0 -0.0 -> 0.0 3.8281684713331012 +acos0070 acos 10000000000000000.0 0.0 -> 0.0 -37.534508668464674 +acos0071 acos 10000000000000000.0 -0.0 -> 0.0 37.534508668464674 +acos0072 acos 9.9999999999999998e+149 0.0 -> 0.0 -346.08091112966679 +acos0073 acos 9.9999999999999998e+149 -0.0 -> 0.0 346.08091112966679 +acos0074 acos 1.0000000000000001e+299 0.0 -> 0.0 -689.16608998577965 +acos0075 acos 1.0000000000000001e+299 -0.0 -> 0.0 689.16608998577965 + +-- random inputs +acos0100 acos -3.3307113324596682 -10.732007530863266 -> 1.8706085694482339 3.113986806554613 +acos0101 acos -2863.952991743291 -2681013315.2571239 -> 1.5707973950301699 22.402607843274758 +acos0102 acos -0.33072639793220088 -0.85055464658253055 -> 1.8219426895922601 0.79250166729311966 +acos0103 acos -2.5722325842097802 -12.703940809821574 -> 1.7699942413107408 3.2565170156527325 +acos0104 acos -42.495233785459583 -0.54039320751337161 -> 3.1288732573153304 4.4424815519735601 +acos0105 acos -1.1363818625856401 9641.1325498630376 -> 1.5709141948820049 -9.8669410553254284 +acos0106 acos -2.4398426824157866e-11 0.33002051890266165 -> 1.570796326818066 -0.32430578041578667 +acos0107 acos -1.3521340428186552 2.9369737912076772 -> 1.9849059192339338 -1.8822893674117942 +acos0108 acos -1.827364706477915 1.0355459232147557 -> 2.5732246307960032 -1.4090688267854969 +acos0109 acos -0.25978373706403546 10.09712669185833 -> 1.5963940386378306 -3.0081673050196063 +acos0110 acos 0.33561778471072551 -4587350.6823999118 -> 1.5707962536333251 16.031960402579539 +acos0111 acos 0.49133444610998445 -0.8071422362990015 -> 1.1908761712801788 0.78573345813187867 +acos0112 acos 0.42196734507823974 -2.4812965431745115 -> 1.414091186100692 1.651707260988172 +acos0113 acos 2.961426210100655 -219.03295695248664 -> 1.5572768319822778 6.0824659885827304 +acos0114 acos 2.886209063652641 -20.38011207220606 -> 1.4302765252297889 3.718201853147642 +acos0115 acos 0.4180568075276509 1.4833433990823484 -> 1.3393834558303042 -1.2079847758301576 +acos0116 acos 52.376111405924718 0.013930429001941001 -> 0.00026601761804024188 -4.6515066691204714 +acos0117 acos 41637948387.625969 1.563418292894041 -> 3.7547918507883548e-11 -25.145424989809381 +acos0118 acos 0.061226659122249526 0.8447234394615154 -> 1.5240280306367315 -0.76791798971140812 +acos0119 acos 2.4480466420442959e+26 0.18002339201384662 -> 7.353756620564798e-28 -61.455650015996376 + +-- values near infinity +acos0200 acos 1.6206860518683021e+308 1.0308426226285283e+308 -> 0.56650826093826223 -710.54206874241561 +acos0201 acos 1.2067735875070062e+308 -1.3429173724390276e+308 -> 0.83874369390864889 710.48017794027498 +acos0202 acos -7.4130145132549047e+307 1.1759130543927645e+308 -> 2.1332729346478536 -710.21871115698752 +acos0203 acos -8.6329426442257249e+307 -1.2316282952184133e+308 -> 2.1821511032444838 710.29752145697148 +acos0204 acos 0.0 1.4289713855849746e+308 -> 1.5707963267948966 -710.24631069738996 +acos0205 acos -0.0 1.3153524545987432e+308 -> 1.5707963267948966 -710.1634604787539 +acos0206 acos 0.0 -9.6229037669269321e+307 -> 1.5707963267948966 709.85091679573691 +acos0207 acos -0.0 -4.9783616421107088e+307 -> 1.5707963267948966 709.19187157911233 +acos0208 acos 1.3937541925739389e+308 0.0 -> 0.0 -710.22135678707264 +acos0209 acos 9.1362388967371536e+307 -0.0 -> 0.0 709.79901953124613 +acos0210 acos -1.3457361220697436e+308 0.0 -> 3.1415926535897931 -710.18629698871848 +acos0211 acos -5.4699090056144284e+307 -0.0 -> 3.1415926535897931 709.28603271085649 +acos0212 acos 1.5880716932358901e+308 5.5638401252339929 -> 3.503519487773873e-308 -710.35187633140583 +acos0213 acos 1.2497211663463164e+308 -3.0456477717911024 -> 2.4370618453197486e-308 710.11227628223412 +acos0214 acos -9.9016224006029528e+307 4.9570427340789056 -> 3.1415926535897931 -709.87946935229468 +acos0215 acos -1.5854071066874139e+308 -4.4233577741497783 -> 3.1415926535897931 710.35019704672004 +acos0216 acos 9.3674623083647628 1.5209559051877979e+308 -> 1.5707963267948966 -710.30869484491086 +acos0217 acos 8.1773832021784383 -6.6093445795000056e+307 -> 1.5707963267948966 709.4752552227792 +acos0218 acos -3.1845935000665104 1.5768856396650893e+308 -> 1.5707963267948966 -710.34480761042687 +acos0219 acos -1.0577303880953903 -6.4574626815735613e+307 -> 1.5707963267948966 709.45200719662046 + +-- values near 0 +acos0220 acos 1.8566986970714045e-320 3.1867234156760402e-321 -> 1.5707963267948966 -3.1867234156760402e-321 +acos0221 acos 7.9050503334599447e-323 -8.8931816251424378e-323 -> 1.5707963267948966 8.8931816251424378e-323 +acos0222 acos -4.4465908125712189e-323 2.4654065097222727e-311 -> 1.5707963267948966 -2.4654065097222727e-311 +acos0223 acos -6.1016916408192619e-311 -2.4703282292062327e-323 -> 1.5707963267948966 2.4703282292062327e-323 +acos0224 acos 0.0 3.4305783621842729e-311 -> 1.5707963267948966 -3.4305783621842729e-311 +acos0225 acos -0.0 1.6117409498633145e-319 -> 1.5707963267948966 -1.6117409498633145e-319 +acos0226 acos 0.0 -4.9900630229965901e-322 -> 1.5707963267948966 4.9900630229965901e-322 +acos0227 acos -0.0 -4.4889279210592818e-311 -> 1.5707963267948966 4.4889279210592818e-311 +acos0228 acos 5.3297678681477214e-312 0.0 -> 1.5707963267948966 -0.0 +acos0229 acos 6.2073425897211614e-313 -0.0 -> 1.5707963267948966 0.0 +acos0230 acos -4.9406564584124654e-324 0.0 -> 1.5707963267948966 -0.0 +acos0231 acos -1.7107517052899003e-318 -0.0 -> 1.5707963267948966 0.0 + +-------------------------------------- +-- acosh: Inverse hyperbolic cosine -- +-------------------------------------- + +-- zeros +acosh0000 acosh 0.0 0.0 -> 0.0 1.5707963267948966 +acosh0001 acosh 0.0 -0.0 -> 0.0 -1.5707963267948966 +acosh0002 acosh -0.0 0.0 -> 0.0 1.5707963267948966 +acosh0003 acosh -0.0 -0.0 -> 0.0 -1.5707963267948966 + +-- branch points: +/-1 +acosh0010 acosh 1.0 0.0 -> 0.0 0.0 +acosh0011 acosh 1.0 -0.0 -> 0.0 -0.0 +acosh0012 acosh -1.0 0.0 -> 0.0 3.1415926535897931 +acosh0013 acosh -1.0 -0.0 -> 0.0 -3.1415926535897931 + +-- values along both sides of real axis +acosh0020 acosh -9.8813129168249309e-324 0.0 -> 0.0 1.5707963267948966 +acosh0021 acosh -9.8813129168249309e-324 -0.0 -> 0.0 -1.5707963267948966 +acosh0022 acosh -1e-305 0.0 -> 0.0 1.5707963267948966 +acosh0023 acosh -1e-305 -0.0 -> 0.0 -1.5707963267948966 +acosh0024 acosh -1e-150 0.0 -> 0.0 1.5707963267948966 +acosh0025 acosh -1e-150 -0.0 -> 0.0 -1.5707963267948966 +acosh0026 acosh -9.9999999999999998e-17 0.0 -> 0.0 1.5707963267948968 +acosh0027 acosh -9.9999999999999998e-17 -0.0 -> 0.0 -1.5707963267948968 +acosh0028 acosh -0.001 0.0 -> 0.0 1.5717963269615634 +acosh0029 acosh -0.001 -0.0 -> 0.0 -1.5717963269615634 +acosh0030 acosh -0.57899999999999996 0.0 -> 0.0 2.1882979816120667 +acosh0031 acosh -0.57899999999999996 -0.0 -> 0.0 -2.1882979816120667 +acosh0032 acosh -0.99999999999999989 0.0 -> 0.0 3.1415926386886319 +acosh0033 acosh -0.99999999999999989 -0.0 -> 0.0 -3.1415926386886319 +acosh0034 acosh -1.0000000000000002 0.0 -> 2.1073424255447014e-08 3.1415926535897931 +acosh0035 acosh -1.0000000000000002 -0.0 -> 2.1073424255447014e-08 -3.1415926535897931 +acosh0036 acosh -1.0009999999999999 0.0 -> 0.044717633608306849 3.1415926535897931 +acosh0037 acosh -1.0009999999999999 -0.0 -> 0.044717633608306849 -3.1415926535897931 +acosh0038 acosh -2.0 0.0 -> 1.3169578969248168 3.1415926535897931 +acosh0039 acosh -2.0 -0.0 -> 1.3169578969248168 -3.1415926535897931 +acosh0040 acosh -23.0 0.0 -> 3.8281684713331012 3.1415926535897931 +acosh0041 acosh -23.0 -0.0 -> 3.8281684713331012 -3.1415926535897931 +acosh0042 acosh -10000000000000000.0 0.0 -> 37.534508668464674 3.1415926535897931 +acosh0043 acosh -10000000000000000.0 -0.0 -> 37.534508668464674 -3.1415926535897931 +acosh0044 acosh -9.9999999999999998e+149 0.0 -> 346.08091112966679 3.1415926535897931 +acosh0045 acosh -9.9999999999999998e+149 -0.0 -> 346.08091112966679 -3.1415926535897931 +acosh0046 acosh -1.0000000000000001e+299 0.0 -> 689.16608998577965 3.1415926535897931 +acosh0047 acosh -1.0000000000000001e+299 -0.0 -> 689.16608998577965 -3.1415926535897931 +acosh0048 acosh 9.8813129168249309e-324 0.0 -> 0.0 1.5707963267948966 +acosh0049 acosh 9.8813129168249309e-324 -0.0 -> 0.0 -1.5707963267948966 +acosh0050 acosh 1e-305 0.0 -> 0.0 1.5707963267948966 +acosh0051 acosh 1e-305 -0.0 -> 0.0 -1.5707963267948966 +acosh0052 acosh 1e-150 0.0 -> 0.0 1.5707963267948966 +acosh0053 acosh 1e-150 -0.0 -> 0.0 -1.5707963267948966 +acosh0054 acosh 9.9999999999999998e-17 0.0 -> 0.0 1.5707963267948966 +acosh0055 acosh 9.9999999999999998e-17 -0.0 -> 0.0 -1.5707963267948966 +acosh0056 acosh 0.001 0.0 -> 0.0 1.56979632662823 +acosh0057 acosh 0.001 -0.0 -> 0.0 -1.56979632662823 +acosh0058 acosh 0.57899999999999996 0.0 -> 0.0 0.95329467197772655 +acosh0059 acosh 0.57899999999999996 -0.0 -> 0.0 -0.95329467197772655 +acosh0060 acosh 0.99999999999999989 0.0 -> 0.0 1.4901161193847656e-08 +acosh0061 acosh 0.99999999999999989 -0.0 -> 0.0 -1.4901161193847656e-08 +acosh0062 acosh 1.0000000000000002 0.0 -> 2.1073424255447014e-08 0.0 +acosh0063 acosh 1.0000000000000002 -0.0 -> 2.1073424255447014e-08 -0.0 +acosh0064 acosh 1.0009999999999999 0.0 -> 0.044717633608306849 0.0 +acosh0065 acosh 1.0009999999999999 -0.0 -> 0.044717633608306849 -0.0 +acosh0066 acosh 2.0 0.0 -> 1.3169578969248168 0.0 +acosh0067 acosh 2.0 -0.0 -> 1.3169578969248168 -0.0 +acosh0068 acosh 23.0 0.0 -> 3.8281684713331012 0.0 +acosh0069 acosh 23.0 -0.0 -> 3.8281684713331012 -0.0 +acosh0070 acosh 10000000000000000.0 0.0 -> 37.534508668464674 0.0 +acosh0071 acosh 10000000000000000.0 -0.0 -> 37.534508668464674 -0.0 +acosh0072 acosh 9.9999999999999998e+149 0.0 -> 346.08091112966679 0.0 +acosh0073 acosh 9.9999999999999998e+149 -0.0 -> 346.08091112966679 -0.0 +acosh0074 acosh 1.0000000000000001e+299 0.0 -> 689.16608998577965 0.0 +acosh0075 acosh 1.0000000000000001e+299 -0.0 -> 689.16608998577965 -0.0 + +-- random inputs +acosh0100 acosh -1.4328589581250843 -1.8370347775558309 -> 1.5526962646549587 -2.190250168435786 +acosh0101 acosh -0.31075819156220957 -1.0772555786839297 -> 0.95139168286193709 -1.7812228089636479 +acosh0102 acosh -1.9044776578070453 -20.485370158932124 -> 3.7177411088932359 -1.6633888745861227 +acosh0103 acosh -0.075642506000858742 -21965976320.873051 -> 24.505907742881991 -1.5707963267983402 +acosh0104 acosh -1.6162271181056307 -3.0369343458696099 -> 1.9407057262861227 -2.0429549461750209 +acosh0105 acosh -0.3103780280298063 0.00018054880018078987 -> 0.00018992877058761416 1.886386995096728 +acosh0106 acosh -9159468751.5897655 5.8014747664273649 -> 23.631201197959193 3.1415926529564078 +acosh0107 acosh -0.037739157550933884 0.21841357493510705 -> 0.21685844960602488 1.6076735133449402 +acosh0108 acosh -8225991.0508394297 0.28318543008913644 -> 16.615956520420287 3.1415926191641019 +acosh0109 acosh -35.620070502302639 0.31303237005015 -> 4.2658980006943965 3.1328013255541873 +acosh0110 acosh 96.729939906820917 -0.029345228372365334 -> 5.2650434775863548 -0.00030338895866972843 +acosh0111 acosh 0.59656024007966491 -2.0412294654163978 -> 1.4923002024287835 -1.312568421900338 +acosh0112 acosh 109.29384112677828 -0.00015454863061533812 -> 5.3871662961545477 -1.4141245154061214e-06 +acosh0113 acosh 8.6705651969361597 -3.6723631649787465 -> 2.9336180958363545 -0.40267362031872861 +acosh0114 acosh 1.8101646445052686 -0.012345132721855478 -> 1.1997148566285769 -0.0081813912760150265 +acosh0115 acosh 52.56897195025288 0.001113916065985443 -> 4.6551827622264135 2.1193445872040307e-05 +acosh0116 acosh 0.28336786164214739 355643992457.40485 -> 27.290343226816528 1.5707963267940999 +acosh0117 acosh 0.73876621291911437 2.8828594541104322e-20 -> 4.2774820978159067e-20 0.73955845836827927 +acosh0118 acosh 0.025865471781718878 37125746064318.492 -> 31.938478989418012 1.5707963267948959 +acosh0119 acosh 2.2047353511780132 0.074712248143489271 -> 1.4286403248698021 0.037997904971626598 + +-- values near infinity +acosh0200 acosh 8.1548592876467785e+307 9.0943779335951128e+307 -> 710.08944620800605 0.83981165425478954 +acosh0201 acosh 1.4237229680972531e+308 -1.0336966617874858e+308 -> 710.4543331094759 -0.6279972876348755 +acosh0202 acosh -1.5014526899738939e+308 1.5670700378448792e+308 -> 710.66420706795464 2.3348137299106697 +acosh0203 acosh -1.0939040375213928e+308 -1.0416960351127978e+308 -> 710.30182863115886 -2.380636147787027 +acosh0204 acosh 0.0 1.476062433559588e+308 -> 710.27873384716929 1.5707963267948966 +acosh0205 acosh -0.0 6.2077210326221094e+307 -> 709.41256457484769 1.5707963267948966 +acosh0206 acosh 0.0 -1.5621899909968308e+308 -> 710.33544449990734 -1.5707963267948966 +acosh0207 acosh -0.0 -8.3556624833839122e+307 -> 709.70971018048317 -1.5707963267948966 +acosh0208 acosh 1.3067079752499342e+308 0.0 -> 710.15686680107228 0.0 +acosh0209 acosh 1.5653640340214026e+308 -0.0 -> 710.33747422926706 -0.0 +acosh0210 acosh -6.9011375992290636e+307 0.0 -> 709.51845699719922 3.1415926535897931 +acosh0211 acosh -9.9539576809926973e+307 -0.0 -> 709.88474095870185 -3.1415926535897931 +acosh0212 acosh 7.6449598518914925e+307 9.5706540768268358 -> 709.62081731754802 1.2518906916769345e-307 +acosh0213 acosh 5.4325410972602197e+307 -7.8064807816522706 -> 709.279177727925 -1.4369851312471974e-307 +acosh0214 acosh -1.1523626112360465e+308 7.0617510038869336 -> 710.03117010216909 3.1415926535897931 +acosh0215 acosh -1.1685027786862599e+308 -5.1568558357925625 -> 710.04507907571417 -3.1415926535897931 +acosh0216 acosh 3.0236370339788721 1.7503248720096417e+308 -> 710.44915723458064 1.5707963267948966 +acosh0217 acosh 6.6108007926031149 -9.1469968225806149e+307 -> 709.80019633903328 -1.5707963267948966 +acosh0218 acosh -5.1096262905623959 6.4484926785412395e+307 -> 709.45061713997973 1.5707963267948966 +acosh0219 acosh -2.8080920608735846 -1.7716118836519368e+308 -> 710.46124562363445 -1.5707963267948966 + +-- values near 0 +acosh0220 acosh 4.5560530326699304e-317 7.3048989121436657e-318 -> 7.3048989121436657e-318 1.5707963267948966 +acosh0221 acosh 4.8754274133585331e-314 -9.8469794897684199e-315 -> 9.8469794897684199e-315 -1.5707963267948966 +acosh0222 acosh -4.6748876009960097e-312 9.7900342887557606e-318 -> 9.7900342887557606e-318 1.5707963267948966 +acosh0223 acosh -4.3136871538399236e-320 -4.9406564584124654e-323 -> 4.9406564584124654e-323 -1.5707963267948966 +acosh0224 acosh 0.0 4.3431013866496774e-314 -> 4.3431013866496774e-314 1.5707963267948966 +acosh0225 acosh -0.0 6.0147334335829184e-317 -> 6.0147334335829184e-317 1.5707963267948966 +acosh0226 acosh 0.0 -1.2880291387081297e-320 -> 1.2880291387081297e-320 -1.5707963267948966 +acosh0227 acosh -0.0 -1.4401563976534621e-317 -> 1.4401563976534621e-317 -1.5707963267948966 +acosh0228 acosh 1.3689680570863091e-313 0.0 -> 0.0 1.5707963267948966 +acosh0229 acosh 1.5304346893494371e-312 -0.0 -> 0.0 -1.5707963267948966 +acosh0230 acosh -3.7450175954766488e-320 0.0 -> 0.0 1.5707963267948966 +acosh0231 acosh -8.4250563080885801e-311 -0.0 -> 0.0 -1.5707963267948966 + +------------------------ +-- asin: Inverse sine -- +------------------------ + +-- zeros +asin0000 asin 0.0 0.0 -> 0.0 0.0 +asin0001 asin 0.0 -0.0 -> 0.0 -0.0 +asin0002 asin -0.0 0.0 -> -0.0 0.0 +asin0003 asin -0.0 -0.0 -> -0.0 -0.0 + +-- branch points: +/-1 +asin0010 asin 1.0 0.0 -> 1.5707963267948966 0.0 +asin0011 asin 1.0 -0.0 -> 1.5707963267948966 -0.0 +asin0012 asin -1.0 0.0 -> -1.5707963267948966 0.0 +asin0013 asin -1.0 -0.0 -> -1.5707963267948966 -0.0 + +-- values along both sides of real axis +asin0020 asin -9.8813129168249309e-324 0.0 -> -9.8813129168249309e-324 0.0 +asin0021 asin -9.8813129168249309e-324 -0.0 -> -9.8813129168249309e-324 -0.0 +asin0022 asin -1e-305 0.0 -> -1e-305 0.0 +asin0023 asin -1e-305 -0.0 -> -1e-305 -0.0 +asin0024 asin -1e-150 0.0 -> -1e-150 0.0 +asin0025 asin -1e-150 -0.0 -> -1e-150 -0.0 +asin0026 asin -9.9999999999999998e-17 0.0 -> -9.9999999999999998e-17 0.0 +asin0027 asin -9.9999999999999998e-17 -0.0 -> -9.9999999999999998e-17 -0.0 +asin0028 asin -0.001 0.0 -> -0.0010000001666667416 0.0 +asin0029 asin -0.001 -0.0 -> -0.0010000001666667416 -0.0 +asin0030 asin -0.57899999999999996 0.0 -> -0.61750165481717001 0.0 +asin0031 asin -0.57899999999999996 -0.0 -> -0.61750165481717001 -0.0 +asin0032 asin -0.99999999999999989 0.0 -> -1.5707963118937354 0.0 +asin0033 asin -0.99999999999999989 -0.0 -> -1.5707963118937354 -0.0 +asin0034 asin -1.0000000000000002 0.0 -> -1.5707963267948966 2.1073424255447014e-08 +asin0035 asin -1.0000000000000002 -0.0 -> -1.5707963267948966 -2.1073424255447014e-08 +asin0036 asin -1.0009999999999999 0.0 -> -1.5707963267948966 0.044717633608306849 +asin0037 asin -1.0009999999999999 -0.0 -> -1.5707963267948966 -0.044717633608306849 +asin0038 asin -2.0 0.0 -> -1.5707963267948966 1.3169578969248168 +asin0039 asin -2.0 -0.0 -> -1.5707963267948966 -1.3169578969248168 +asin0040 asin -23.0 0.0 -> -1.5707963267948966 3.8281684713331012 +asin0041 asin -23.0 -0.0 -> -1.5707963267948966 -3.8281684713331012 +asin0042 asin -10000000000000000.0 0.0 -> -1.5707963267948966 37.534508668464674 +asin0043 asin -10000000000000000.0 -0.0 -> -1.5707963267948966 -37.534508668464674 +asin0044 asin -9.9999999999999998e+149 0.0 -> -1.5707963267948966 346.08091112966679 +asin0045 asin -9.9999999999999998e+149 -0.0 -> -1.5707963267948966 -346.08091112966679 +asin0046 asin -1.0000000000000001e+299 0.0 -> -1.5707963267948966 689.16608998577965 +asin0047 asin -1.0000000000000001e+299 -0.0 -> -1.5707963267948966 -689.16608998577965 +asin0048 asin 9.8813129168249309e-324 0.0 -> 9.8813129168249309e-324 0.0 +asin0049 asin 9.8813129168249309e-324 -0.0 -> 9.8813129168249309e-324 -0.0 +asin0050 asin 1e-305 0.0 -> 1e-305 0.0 +asin0051 asin 1e-305 -0.0 -> 1e-305 -0.0 +asin0052 asin 1e-150 0.0 -> 1e-150 0.0 +asin0053 asin 1e-150 -0.0 -> 1e-150 -0.0 +asin0054 asin 9.9999999999999998e-17 0.0 -> 9.9999999999999998e-17 0.0 +asin0055 asin 9.9999999999999998e-17 -0.0 -> 9.9999999999999998e-17 -0.0 +asin0056 asin 0.001 0.0 -> 0.0010000001666667416 0.0 +asin0057 asin 0.001 -0.0 -> 0.0010000001666667416 -0.0 +asin0058 asin 0.57899999999999996 0.0 -> 0.61750165481717001 0.0 +asin0059 asin 0.57899999999999996 -0.0 -> 0.61750165481717001 -0.0 +asin0060 asin 0.99999999999999989 0.0 -> 1.5707963118937354 0.0 +asin0061 asin 0.99999999999999989 -0.0 -> 1.5707963118937354 -0.0 +asin0062 asin 1.0000000000000002 0.0 -> 1.5707963267948966 2.1073424255447014e-08 +asin0063 asin 1.0000000000000002 -0.0 -> 1.5707963267948966 -2.1073424255447014e-08 +asin0064 asin 1.0009999999999999 0.0 -> 1.5707963267948966 0.044717633608306849 +asin0065 asin 1.0009999999999999 -0.0 -> 1.5707963267948966 -0.044717633608306849 +asin0066 asin 2.0 0.0 -> 1.5707963267948966 1.3169578969248168 +asin0067 asin 2.0 -0.0 -> 1.5707963267948966 -1.3169578969248168 +asin0068 asin 23.0 0.0 -> 1.5707963267948966 3.8281684713331012 +asin0069 asin 23.0 -0.0 -> 1.5707963267948966 -3.8281684713331012 +asin0070 asin 10000000000000000.0 0.0 -> 1.5707963267948966 37.534508668464674 +asin0071 asin 10000000000000000.0 -0.0 -> 1.5707963267948966 -37.534508668464674 +asin0072 asin 9.9999999999999998e+149 0.0 -> 1.5707963267948966 346.08091112966679 +asin0073 asin 9.9999999999999998e+149 -0.0 -> 1.5707963267948966 -346.08091112966679 +asin0074 asin 1.0000000000000001e+299 0.0 -> 1.5707963267948966 689.16608998577965 +asin0075 asin 1.0000000000000001e+299 -0.0 -> 1.5707963267948966 -689.16608998577965 + +-- random inputs +asin0100 asin -1.5979555835086083 -0.15003009814595247 -> -1.4515369557405788 -1.0544476399790823 +asin0101 asin -0.57488225895317679 -9.6080397838952743e-13 -> -0.61246024460412851 -1.174238005400403e-12 +asin0102 asin -3.6508087930516249 -0.36027527093220152 -> -1.4685890605305874 -1.9742273007152038 +asin0103 asin -1.5238659792326819 -1.1360813516996364 -> -0.86080051691147275 -1.3223742205689195 +asin0104 asin -1592.0639045555306 -0.72362427935018236 -> -1.5703418071175179 -8.0659336918729228 +asin0105 asin -0.19835471371312019 4.2131508416697709 -> -0.045777831019935149 2.1461732751933171 +asin0106 asin -1.918471054430213 0.40603305079779234 -> -1.3301396585791556 1.30263642314981 +asin0107 asin -254495.01623373642 0.71084414434470822 -> -1.5707935336394359 13.140183712762321 +asin0108 asin -0.31315882715691157 3.9647994288429866 -> -0.076450403840916004 2.0889762138713457 +asin0109 asin -0.90017064284720816 1.2530659485907105 -> -0.53466509741943447 1.1702811557577 +asin0110 asin 2.1615181696571075 -0.14058647488229523 -> 1.4976166323896871 -1.4085811039334604 +asin0111 asin 1.2104749210707795 -0.85732484485298999 -> 0.83913071588343924 -1.0681719250525901 +asin0112 asin 1.7059733185128891 -0.84032966373156581 -> 1.0510900815816229 -1.2967979791361652 +asin0113 asin 9.9137085017290687 -1.4608383970250893 -> 1.4237704820128891 -2.995414677560686 +asin0114 asin 117.12344751041495 -5453908091.5334015 -> 2.1475141411392012e-08 -23.112745450217066 +asin0115 asin 0.081041187798029227 0.067054349860173196 -> 0.080946786856771813 0.067223991060639698 +asin0116 asin 46.635472322049949 2.3835190718056678 -> 1.5197194940010779 4.5366989600972083 +asin0117 asin 3907.0687961127105 19.144021886390181 -> 1.5658965233083235 8.9637018715924217 +asin0118 asin 1.0889312322308273 509.01577883554768 -> 0.0021392803817829316 6.9256294494524706 +asin0119 asin 0.10851518277509224 1.5612510908217476 -> 0.058491014243902621 1.2297075725621327 + +-- values near infinity +asin0200 asin 1.5230241998821499e+308 5.5707228994084525e+307 -> 1.2201446370892068 710.37283486535966 +asin0201 asin 8.1334317698672204e+307 -9.2249425197872451e+307 -> 0.72259991284020042 -710.0962453049026 +asin0202 asin -9.9138506659241768e+307 6.701544526434995e+307 -> -0.97637511742194594 710.06887486671371 +asin0203 asin -1.4141298868173842e+308 -5.401505134514191e+307 -> -1.2059319055160587 -710.30396478954628 +asin0204 asin 0.0 9.1618092977897431e+307 -> 0.0 709.80181441050593 +asin0205 asin -0.0 6.8064342551939755e+307 -> -0.0 709.50463910853489 +asin0206 asin 0.0 -6.4997516454798215e+307 -> 0.0 -709.45853469751592 +asin0207 asin -0.0 -1.6767449053345242e+308 -> -0.0 -710.4062101803022 +asin0208 asin 5.4242749957378916e+307 0.0 -> 1.5707963267948966 709.27765497888902 +asin0209 asin 9.5342145121164749e+307 -0.0 -> 1.5707963267948966 -709.84165758595907 +asin0210 asin -7.0445698006201847e+307 0.0 -> -1.5707963267948966 709.53902780872136 +asin0211 asin -1.0016025569769706e+308 -0.0 -> -1.5707963267948966 -709.89095709697881 +asin0212 asin 1.6552203778877204e+308 0.48761543336249491 -> 1.5707963267948966 710.39328998153474 +asin0213 asin 1.2485712830384869e+308 -4.3489311161278899 -> 1.5707963267948966 -710.1113557467786 +asin0214 asin -1.5117842813353125e+308 5.123452666102434 -> -1.5707963267948966 710.30264641923031 +asin0215 asin -1.3167634313008016e+308 -0.52939679793528982 -> -1.5707963267948966 -710.16453260239768 +asin0216 asin 0.80843929176985907 1.0150851827767876e+308 -> 7.9642507396113875e-309 709.90432835561637 +asin0217 asin 8.2544809829680901 -1.7423548140539474e+308 -> 4.7375430746865733e-308 -710.44459336242164 +asin0218 asin -5.2499000118824295 4.6655578977512214e+307 -> -1.1252459249113292e-307 709.1269781491103 +asin0219 asin -5.9904782760833433 -4.7315689314781163e+307 -> -1.2660659419394637e-307 -709.14102757522312 + +------------------------------------ +-- asinh: Inverse hyperbolic sine -- +------------------------------------ + +-- zeros +asinh0000 asinh 0.0 0.0 -> 0.0 0.0 +asinh0001 asinh 0.0 -0.0 -> 0.0 -0.0 +asinh0002 asinh -0.0 0.0 -> -0.0 0.0 +asinh0003 asinh -0.0 -0.0 -> -0.0 -0.0 + +-- branch points: +/-i +asinh0010 asinh 0.0 1.0 -> 0.0 1.5707963267948966 +asinh0011 asinh 0.0 -1.0 -> 0.0 -1.5707963267948966 +asinh0012 asinh -0.0 1.0 -> -0.0 1.5707963267948966 +asinh0013 asinh -0.0 -1.0 -> -0.0 -1.5707963267948966 + +-- values along both sides of imaginary axis +asinh0020 asinh 0.0 -9.8813129168249309e-324 -> 0.0 -9.8813129168249309e-324 +asinh0021 asinh -0.0 -9.8813129168249309e-324 -> -0.0 -9.8813129168249309e-324 +asinh0022 asinh 0.0 -1e-305 -> 0.0 -1e-305 +asinh0023 asinh -0.0 -1e-305 -> -0.0 -1e-305 +asinh0024 asinh 0.0 -1e-150 -> 0.0 -1e-150 +asinh0025 asinh -0.0 -1e-150 -> -0.0 -1e-150 +asinh0026 asinh 0.0 -9.9999999999999998e-17 -> 0.0 -9.9999999999999998e-17 +asinh0027 asinh -0.0 -9.9999999999999998e-17 -> -0.0 -9.9999999999999998e-17 +asinh0028 asinh 0.0 -0.001 -> 0.0 -0.0010000001666667416 +asinh0029 asinh -0.0 -0.001 -> -0.0 -0.0010000001666667416 +asinh0030 asinh 0.0 -0.57899999999999996 -> 0.0 -0.61750165481717001 +asinh0031 asinh -0.0 -0.57899999999999996 -> -0.0 -0.61750165481717001 +asinh0032 asinh 0.0 -0.99999999999999989 -> 0.0 -1.5707963118937354 +asinh0033 asinh -0.0 -0.99999999999999989 -> -0.0 -1.5707963118937354 +asinh0034 asinh 0.0 -1.0000000000000002 -> 2.1073424255447014e-08 -1.5707963267948966 +asinh0035 asinh -0.0 -1.0000000000000002 -> -2.1073424255447014e-08 -1.5707963267948966 +asinh0036 asinh 0.0 -1.0009999999999999 -> 0.044717633608306849 -1.5707963267948966 +asinh0037 asinh -0.0 -1.0009999999999999 -> -0.044717633608306849 -1.5707963267948966 +asinh0038 asinh 0.0 -2.0 -> 1.3169578969248168 -1.5707963267948966 +asinh0039 asinh -0.0 -2.0 -> -1.3169578969248168 -1.5707963267948966 +asinh0040 asinh 0.0 -20.0 -> 3.6882538673612966 -1.5707963267948966 +asinh0041 asinh -0.0 -20.0 -> -3.6882538673612966 -1.5707963267948966 +asinh0042 asinh 0.0 -10000000000000000.0 -> 37.534508668464674 -1.5707963267948966 +asinh0043 asinh -0.0 -10000000000000000.0 -> -37.534508668464674 -1.5707963267948966 +asinh0044 asinh 0.0 -9.9999999999999998e+149 -> 346.08091112966679 -1.5707963267948966 +asinh0045 asinh -0.0 -9.9999999999999998e+149 -> -346.08091112966679 -1.5707963267948966 +asinh0046 asinh 0.0 -1.0000000000000001e+299 -> 689.16608998577965 -1.5707963267948966 +asinh0047 asinh -0.0 -1.0000000000000001e+299 -> -689.16608998577965 -1.5707963267948966 +asinh0048 asinh 0.0 9.8813129168249309e-324 -> 0.0 9.8813129168249309e-324 +asinh0049 asinh -0.0 9.8813129168249309e-324 -> -0.0 9.8813129168249309e-324 +asinh0050 asinh 0.0 1e-305 -> 0.0 1e-305 +asinh0051 asinh -0.0 1e-305 -> -0.0 1e-305 +asinh0052 asinh 0.0 1e-150 -> 0.0 1e-150 +asinh0053 asinh -0.0 1e-150 -> -0.0 1e-150 +asinh0054 asinh 0.0 9.9999999999999998e-17 -> 0.0 9.9999999999999998e-17 +asinh0055 asinh -0.0 9.9999999999999998e-17 -> -0.0 9.9999999999999998e-17 +asinh0056 asinh 0.0 0.001 -> 0.0 0.0010000001666667416 +asinh0057 asinh -0.0 0.001 -> -0.0 0.0010000001666667416 +asinh0058 asinh 0.0 0.57899999999999996 -> 0.0 0.61750165481717001 +asinh0059 asinh -0.0 0.57899999999999996 -> -0.0 0.61750165481717001 +asinh0060 asinh 0.0 0.99999999999999989 -> 0.0 1.5707963118937354 +asinh0061 asinh -0.0 0.99999999999999989 -> -0.0 1.5707963118937354 +asinh0062 asinh 0.0 1.0000000000000002 -> 2.1073424255447014e-08 1.5707963267948966 +asinh0063 asinh -0.0 1.0000000000000002 -> -2.1073424255447014e-08 1.5707963267948966 +asinh0064 asinh 0.0 1.0009999999999999 -> 0.044717633608306849 1.5707963267948966 +asinh0065 asinh -0.0 1.0009999999999999 -> -0.044717633608306849 1.5707963267948966 +asinh0066 asinh 0.0 2.0 -> 1.3169578969248168 1.5707963267948966 +asinh0067 asinh -0.0 2.0 -> -1.3169578969248168 1.5707963267948966 +asinh0068 asinh 0.0 20.0 -> 3.6882538673612966 1.5707963267948966 +asinh0069 asinh -0.0 20.0 -> -3.6882538673612966 1.5707963267948966 +asinh0070 asinh 0.0 10000000000000000.0 -> 37.534508668464674 1.5707963267948966 +asinh0071 asinh -0.0 10000000000000000.0 -> -37.534508668464674 1.5707963267948966 +asinh0072 asinh 0.0 9.9999999999999998e+149 -> 346.08091112966679 1.5707963267948966 +asinh0073 asinh -0.0 9.9999999999999998e+149 -> -346.08091112966679 1.5707963267948966 +asinh0074 asinh 0.0 1.0000000000000001e+299 -> 689.16608998577965 1.5707963267948966 +asinh0075 asinh -0.0 1.0000000000000001e+299 -> -689.16608998577965 1.5707963267948966 + +-- random inputs +asinh0100 asinh -0.5946402853710423 -0.044506548910000145 -> -0.56459775392653022 -0.038256221441536356 +asinh0101 asinh -0.19353958046180916 -0.017489624793193454 -> -0.19237926804196651 -0.017171741895336792 +asinh0102 asinh -0.033117585138955893 -8.5256414015933757 -> -2.8327758348650969 -1.5668848791092411 +asinh0103 asinh -1.5184043184035716 -0.73491245339073275 -> -1.2715891419764005 -0.39204624408542355 +asinh0104 asinh -0.60716120271208818 -0.28900743958436542 -> -0.59119299421187232 -0.24745931678118135 +asinh0105 asinh -0.0237177865112429 2.8832601052166313 -> -1.7205820772413236 1.5620261702963094 +asinh0106 asinh -2.3906812342743979 2.6349216848574013 -> -1.9609636249445124 0.8142142660574706 +asinh0107 asinh -0.0027605019787620517 183.85588476550555 -> -5.9072920005445066 1.5707813120847871 +asinh0108 asinh -0.99083661164404713 0.028006797051617648 -> -0.8750185251283995 0.019894099615994653 +asinh0109 asinh -3.0362951937986393 0.86377266758504867 -> -1.8636030714685221 0.26475058859950168 +asinh0110 asinh 0.34438464536152769 -0.71603790174885029 -> 0.43985415690734164 -0.71015037409294324 +asinh0111 asinh 4.4925124413876256 -60604595352.871613 -> 25.520783738612078 -1.5707963267207683 +asinh0112 asinh 2.3213991428170337 -7.5459667007307258 -> 2.7560464993451643 -1.270073210856117 +asinh0113 asinh 0.21291939741682028 -1.2720428814784408 -> 0.77275088137338266 -1.3182099250896895 +asinh0114 asinh 6.6447359379455957 -0.97196191666946996 -> 2.602830695139672 -0.14368247412319965 +asinh0115 asinh 7.1326256655083746 2.1516360452706857 -> 2.7051146374367212 0.29051701669727581 +asinh0116 asinh 0.18846550905063442 3.4705348585339832 -> 1.917697875799296 1.514155593347924 +asinh0117 asinh 0.19065075303281598 0.26216814548222012 -> 0.19603050785932474 0.26013422809614117 +asinh0118 asinh 2.0242004665739719 0.70510281647495787 -> 1.4970366212896002 0.30526007200481453 +asinh0119 asinh 37.336596461576057 717.29157391678234 -> 7.269981997945294 1.5187910219576033 + +-- values near infinity +asinh0200 asinh 1.0760517500874541e+308 1.1497786241240167e+308 -> 710.34346055651815 0.81850936961793475 +asinh0201 asinh 1.1784839328845529e+308 -1.6478429586716638e+308 -> 710.59536255783678 -0.94996311735607697 +asinh0202 asinh -4.8777682248909193e+307 1.4103736217538474e+308 -> -710.28970147376992 1.2378239519096443 +asinh0203 asinh -1.2832478903233108e+308 -1.5732392613155698e+308 -> -710.59750164290745 -0.88657181439322452 +asinh0204 asinh 0.0 6.8431383856345372e+307 -> 709.51001718444604 1.5707963267948966 +asinh0205 asinh -0.0 8.601822432238051e+307 -> -709.73874482126689 1.5707963267948966 +asinh0206 asinh 0.0 -5.5698396067303782e+307 -> 709.30413698733742 -1.5707963267948966 +asinh0207 asinh -0.0 -7.1507777734621804e+307 -> -709.55399186002705 -1.5707963267948966 +asinh0208 asinh 1.6025136110019349e+308 0.0 -> 710.3609292261076 0.0 +asinh0209 asinh 1.3927819858239114e+308 -0.0 -> 710.22065899832899 -0.0 +asinh0210 asinh -6.0442994056210995e+307 0.0 -> -709.38588631057621 0.0 +asinh0211 asinh -1.2775271979042634e+308 -0.0 -> -710.13428215553972 -0.0 +asinh0212 asinh 1.0687496260268489e+308 1.0255615699476961 -> 709.95584521407841 9.5959010882679093e-309 +asinh0213 asinh 1.0050967333370962e+308 -0.87668970117333433 -> 709.89443961168183 -8.7224410556242882e-309 +asinh0214 asinh -5.7161452814862392e+307 8.2377808413450122 -> -709.33006540611166 1.4411426644501116e-307 +asinh0215 asinh -8.2009040727653315e+307 -6.407409526654976 -> -709.69101513070109 -7.8130526461510088e-308 +asinh0216 asinh 6.4239368496483982 1.6365990821551427e+308 -> 710.38197618101287 1.5707963267948966 +asinh0217 asinh 5.4729111423315882 -1.1227237438144211e+308 -> 710.00511346983546 -1.5707963267948966 +asinh0218 asinh -8.3455818297412723 1.443172020182019e+308 -> -710.25619930551818 1.5707963267948966 +asinh0219 asinh -2.6049726230372441 -1.7952291144022702e+308 -> -710.47448847685644 -1.5707963267948966 + +-- values near 0 +asinh0220 asinh 1.2940113339664088e-314 6.9169190417774516e-323 -> 1.2940113339664088e-314 6.9169190417774516e-323 +asinh0221 asinh 2.3848478863874649e-315 -3.1907655025717717e-310 -> 2.3848478863874649e-315 -3.1907655025717717e-310 +asinh0222 asinh -3.0097643679641622e-316 4.6936236354918422e-322 -> -3.0097643679641622e-316 4.6936236354918422e-322 +asinh0223 asinh -1.787997087755751e-308 -8.5619622834902341e-310 -> -1.787997087755751e-308 -8.5619622834902341e-310 +asinh0224 asinh 0.0 1.2491433448427325e-314 -> 0.0 1.2491433448427325e-314 +asinh0225 asinh -0.0 2.5024072154538062e-308 -> -0.0 2.5024072154538062e-308 +asinh0226 asinh 0.0 -2.9643938750474793e-323 -> 0.0 -2.9643938750474793e-323 +asinh0227 asinh -0.0 -2.9396905927554169e-320 -> -0.0 -2.9396905927554169e-320 +asinh0228 asinh 5.64042930029359e-317 0.0 -> 5.64042930029359e-317 0.0 +asinh0229 asinh 3.3833911866596068e-318 -0.0 -> 3.3833911866596068e-318 -0.0 +asinh0230 asinh -4.9406564584124654e-324 0.0 -> -4.9406564584124654e-324 0.0 +asinh0231 asinh -2.221137922799485e-308 -0.0 -> -2.2211379227994845e-308 -0.0 + +--------------------------- +-- atan: Inverse tangent -- +--------------------------- + +-- zeros +atan0000 atan 0.0 0.0 -> 0.0 0.0 +atan0001 atan 0.0 -0.0 -> 0.0 0.0 +atan0002 atan -0.0 0.0 -> -0.0 0.0 +atan0003 atan -0.0 -0.0 -> -0.0 0.0 + +-- values along both sides of imaginary axis +atan0010 atan 0.0 -9.8813129168249309e-324 -> 0.0 -9.8813129168249309e-324 +atan0011 atan -0.0 -9.8813129168249309e-324 -> -0.0 -9.8813129168249309e-324 +atan0012 atan 0.0 -1e-305 -> 0.0 -1e-305 +atan0013 atan -0.0 -1e-305 -> -0.0 -1e-305 +atan0014 atan 0.0 -1e-150 -> 0.0 -1e-150 +atan0015 atan -0.0 -1e-150 -> -0.0 -1e-150 +atan0016 atan 0.0 -9.9999999999999998e-17 -> 0.0 -9.9999999999999998e-17 +atan0017 atan -0.0 -9.9999999999999998e-17 -> -0.0 -9.9999999999999998e-17 +atan0018 atan 0.0 -0.001 -> 0.0 -0.0010000003333335333 +atan0019 atan -0.0 -0.001 -> -0.0 -0.0010000003333335333 +atan0020 atan 0.0 -0.57899999999999996 -> 0.0 -0.6609570902866303 +atan0021 atan -0.0 -0.57899999999999996 -> -0.0 -0.6609570902866303 +atan0022 atan 0.0 -0.99999999999999989 -> 0.0 -18.714973875118524 +atan0023 atan -0.0 -0.99999999999999989 -> -0.0 -18.714973875118524 +atan0024 atan 0.0 -1.0000000000000002 -> 1.5707963267948966 -18.36840028483855 +atan0025 atan -0.0 -1.0000000000000002 -> -1.5707963267948966 -18.36840028483855 +atan0026 atan 0.0 -1.0009999999999999 -> 1.5707963267948966 -3.8007011672919218 +atan0027 atan -0.0 -1.0009999999999999 -> -1.5707963267948966 -3.8007011672919218 +atan0028 atan 0.0 -2.0 -> 1.5707963267948966 -0.54930614433405489 +atan0029 atan -0.0 -2.0 -> -1.5707963267948966 -0.54930614433405489 +atan0030 atan 0.0 -20.0 -> 1.5707963267948966 -0.050041729278491265 +atan0031 atan -0.0 -20.0 -> -1.5707963267948966 -0.050041729278491265 +atan0032 atan 0.0 -10000000000000000.0 -> 1.5707963267948966 -9.9999999999999998e-17 +atan0033 atan -0.0 -10000000000000000.0 -> -1.5707963267948966 -9.9999999999999998e-17 +atan0034 atan 0.0 -9.9999999999999998e+149 -> 1.5707963267948966 -1e-150 +atan0035 atan -0.0 -9.9999999999999998e+149 -> -1.5707963267948966 -1e-150 +atan0036 atan 0.0 -1.0000000000000001e+299 -> 1.5707963267948966 -9.9999999999999999e-300 +atan0037 atan -0.0 -1.0000000000000001e+299 -> -1.5707963267948966 -9.9999999999999999e-300 +atan0038 atan 0.0 9.8813129168249309e-324 -> 0.0 9.8813129168249309e-324 +atan0039 atan -0.0 9.8813129168249309e-324 -> -0.0 9.8813129168249309e-324 +atan0040 atan 0.0 1e-305 -> 0.0 1e-305 +atan0041 atan -0.0 1e-305 -> -0.0 1e-305 +atan0042 atan 0.0 1e-150 -> 0.0 1e-150 +atan0043 atan -0.0 1e-150 -> -0.0 1e-150 +atan0044 atan 0.0 9.9999999999999998e-17 -> 0.0 9.9999999999999998e-17 +atan0045 atan -0.0 9.9999999999999998e-17 -> -0.0 9.9999999999999998e-17 +atan0046 atan 0.0 0.001 -> 0.0 0.0010000003333335333 +atan0047 atan -0.0 0.001 -> -0.0 0.0010000003333335333 +atan0048 atan 0.0 0.57899999999999996 -> 0.0 0.6609570902866303 +atan0049 atan -0.0 0.57899999999999996 -> -0.0 0.6609570902866303 +atan0050 atan 0.0 0.99999999999999989 -> 0.0 18.714973875118524 +atan0051 atan -0.0 0.99999999999999989 -> -0.0 18.714973875118524 +atan0052 atan 0.0 1.0000000000000002 -> 1.5707963267948966 18.36840028483855 +atan0053 atan -0.0 1.0000000000000002 -> -1.5707963267948966 18.36840028483855 +atan0054 atan 0.0 1.0009999999999999 -> 1.5707963267948966 3.8007011672919218 +atan0055 atan -0.0 1.0009999999999999 -> -1.5707963267948966 3.8007011672919218 +atan0056 atan 0.0 2.0 -> 1.5707963267948966 0.54930614433405489 +atan0057 atan -0.0 2.0 -> -1.5707963267948966 0.54930614433405489 +atan0058 atan 0.0 20.0 -> 1.5707963267948966 0.050041729278491265 +atan0059 atan -0.0 20.0 -> -1.5707963267948966 0.050041729278491265 +atan0060 atan 0.0 10000000000000000.0 -> 1.5707963267948966 9.9999999999999998e-17 +atan0061 atan -0.0 10000000000000000.0 -> -1.5707963267948966 9.9999999999999998e-17 +atan0062 atan 0.0 9.9999999999999998e+149 -> 1.5707963267948966 1e-150 +atan0063 atan -0.0 9.9999999999999998e+149 -> -1.5707963267948966 1e-150 +atan0064 atan 0.0 1.0000000000000001e+299 -> 1.5707963267948966 9.9999999999999999e-300 +atan0065 atan -0.0 1.0000000000000001e+299 -> -1.5707963267948966 9.9999999999999999e-300 + +-- random inputs +atan0100 atan -0.32538873661060214 -1.5530461550412578 -> -1.3682728427554227 -0.69451401598762041 +atan0101 atan -0.45863393495197929 -4799.1747094903594 -> -1.5707963068820623 -0.00020836916050636145 +atan0102 atan -8.3006999685976162 -2.6788890251790938 -> -1.4619862771810199 -0.034811669653327826 +atan0103 atan -1.8836307682985314 -1.1441976638861771 -> -1.1839984370871612 -0.20630956157312796 +atan0104 atan -0.00063230482407491669 -4.9312520961829485 -> -1.5707692093223147 -0.20563867743008304 +atan0105 atan -0.84278137150065946 179012.37493146997 -> -1.5707963267685969 5.5862059836425272e-06 +atan0106 atan -0.95487853984049287 14.311334539886177 -> -1.5661322859434561 0.069676024526232005 +atan0107 atan -1.3513252539663239 6.0500727021632198e-08 -> -0.93371676315220975 2.140800269742656e-08 +atan0108 atan -0.20566254458595795 0.11933771944159823 -> -0.20556463711174916 0.11493405387141732 +atan0109 atan -0.58563718795408559 0.64438965423212868 -> -0.68361089300233124 0.46759762751800249 +atan0110 atan 48.479267751948292 -78.386382460112543 -> 1.5650888770910523 -0.0092276811373297584 +atan0111 atan 1.0575373914056061 -0.75988012377296987 -> 0.94430886722043594 -0.31915698126703118 +atan0112 atan 4444810.4314677203 -0.56553404593942558 -> 1.5707961018134231 -2.8625446437701909e-14 +atan0113 atan 0.010101405082520009 -0.032932668550282478 -> 0.01011202676646334 -0.032941214776834996 +atan0114 atan 1.5353585300154911 -2.1947099346796519 -> 1.3400310739206394 -0.29996003607449045 +atan0115 atan 0.21869457055670882 9.9915684254007093 -> 1.5685846078876444 0.1003716881759439 +atan0116 atan 0.17783290150246836 0.064334689863650957 -> 0.17668728064286277 0.062435808728873846 +atan0117 atan 15.757474087615918 383.57262142534 -> 1.5706894060369621 0.0026026817278826603 +atan0118 atan 10.587017408533317 0.21720238081843438 -> 1.4766594681336236 0.0019199097383010061 +atan0119 atan 0.86026078678781204 0.1230148609359502 -> 0.7147259322534929 0.070551221954286605 + +-- values near infinity +atan0200 atan 7.8764397011195798e+307 8.1647921137746308e+307 -> 1.5707963267948966 6.3439446939604493e-309 +atan0201 atan 1.5873698696131487e+308 -1.0780367422960641e+308 -> 1.5707963267948966 -2.9279309368530781e-309 +atan0202 atan -1.5844551864825834e+308 1.0290657809098675e+308 -> -1.5707963267948966 2.8829614736961417e-309 +atan0203 atan -1.3168792562524032e+308 -9.088432341614825e+307 -> -1.5707963267948966 -3.5499373057390056e-309 +atan0204 atan 0.0 1.0360465742258337e+308 -> 1.5707963267948966 9.6520757355646018e-309 +atan0205 atan -0.0 1.0045063210373196e+308 -> -1.5707963267948966 9.955138947929503e-309 +atan0206 atan 0.0 -9.5155296715763696e+307 -> 1.5707963267948966 -1.050913648020118e-308 +atan0207 atan -0.0 -1.5565700490496501e+308 -> -1.5707963267948966 -6.4243816114189071e-309 +atan0208 atan 1.2956339389525244e+308 0.0 -> 1.5707963267948966 0.0 +atan0209 atan 1.4408126243772151e+308 -0.0 -> 1.5707963267948966 0.0 +atan0210 atan -1.0631786461936417e+308 0.0 -> -1.5707963267948966 0.0 +atan0211 atan -1.0516056964171069e+308 -0.0 -> -1.5707963267948966 0.0 +atan0212 atan 1.236162319603838e+308 4.6827953496242936 -> 1.5707963267948966 0.0 +atan0213 atan 7.000516472897218e+307 -5.8631608017844163 -> 1.5707963267948966 -0.0 +atan0214 atan -1.5053444003338508e+308 5.1199197268420313 -> -1.5707963267948966 0.0 +atan0215 atan -1.399172518147259e+308 -3.5687766472913673 -> -1.5707963267948966 -0.0 +atan0216 atan 8.1252833070803021 6.2782953917343822e+307 -> 1.5707963267948966 1.5927890256908564e-308 +atan0217 atan 2.8034285947515167 -1.3378049775753878e+308 -> 1.5707963267948966 -7.4749310756219562e-309 +atan0218 atan -1.4073509988974953 1.6776381785968355e+308 -> -1.5707963267948966 5.9607608646364569e-309 +atan0219 atan -2.7135551527592119 -1.281567445525738e+308 -> -1.5707963267948966 -7.8029447727565326e-309 + +-- imaginary part = +/-1, real part tiny +atan0300 atan -1e-150 -1.0 -> -0.78539816339744828 -173.04045556483339 +atan0301 atan 1e-155 1.0 -> 0.78539816339744828 178.79691829731851 +atan0302 atan 9.9999999999999999e-161 -1.0 -> 0.78539816339744828 -184.55338102980363 +atan0303 atan -1e-165 1.0 -> -0.78539816339744828 190.30984376228875 +atan0304 atan -9.9998886718268301e-321 -1.0 -> -0.78539816339744828 -368.76019403576692 + +--------------------------------------- +-- atanh: Inverse hyperbolic tangent -- +--------------------------------------- + +-- zeros +atanh0000 atanh 0.0 0.0 -> 0.0 0.0 +atanh0001 atanh 0.0 -0.0 -> 0.0 -0.0 +atanh0002 atanh -0.0 0.0 -> 0.0 0.0 +atanh0003 atanh -0.0 -0.0 -> 0.0 -0.0 + +-- values along both sides of real axis +atanh0010 atanh -9.8813129168249309e-324 0.0 -> -9.8813129168249309e-324 0.0 +atanh0011 atanh -9.8813129168249309e-324 -0.0 -> -9.8813129168249309e-324 -0.0 +atanh0012 atanh -1e-305 0.0 -> -1e-305 0.0 +atanh0013 atanh -1e-305 -0.0 -> -1e-305 -0.0 +atanh0014 atanh -1e-150 0.0 -> -1e-150 0.0 +atanh0015 atanh -1e-150 -0.0 -> -1e-150 -0.0 +atanh0016 atanh -9.9999999999999998e-17 0.0 -> -9.9999999999999998e-17 0.0 +atanh0017 atanh -9.9999999999999998e-17 -0.0 -> -9.9999999999999998e-17 -0.0 +atanh0018 atanh -0.001 0.0 -> -0.0010000003333335333 0.0 +atanh0019 atanh -0.001 -0.0 -> -0.0010000003333335333 -0.0 +atanh0020 atanh -0.57899999999999996 0.0 -> -0.6609570902866303 0.0 +atanh0021 atanh -0.57899999999999996 -0.0 -> -0.6609570902866303 -0.0 +atanh0022 atanh -0.99999999999999989 0.0 -> -18.714973875118524 0.0 +atanh0023 atanh -0.99999999999999989 -0.0 -> -18.714973875118524 -0.0 +atanh0024 atanh -1.0000000000000002 0.0 -> -18.36840028483855 1.5707963267948966 +atanh0025 atanh -1.0000000000000002 -0.0 -> -18.36840028483855 -1.5707963267948966 +atanh0026 atanh -1.0009999999999999 0.0 -> -3.8007011672919218 1.5707963267948966 +atanh0027 atanh -1.0009999999999999 -0.0 -> -3.8007011672919218 -1.5707963267948966 +atanh0028 atanh -2.0 0.0 -> -0.54930614433405489 1.5707963267948966 +atanh0029 atanh -2.0 -0.0 -> -0.54930614433405489 -1.5707963267948966 +atanh0030 atanh -23.0 0.0 -> -0.043505688494814884 1.5707963267948966 +atanh0031 atanh -23.0 -0.0 -> -0.043505688494814884 -1.5707963267948966 +atanh0032 atanh -10000000000000000.0 0.0 -> -9.9999999999999998e-17 1.5707963267948966 +atanh0033 atanh -10000000000000000.0 -0.0 -> -9.9999999999999998e-17 -1.5707963267948966 +atanh0034 atanh -9.9999999999999998e+149 0.0 -> -1e-150 1.5707963267948966 +atanh0035 atanh -9.9999999999999998e+149 -0.0 -> -1e-150 -1.5707963267948966 +atanh0036 atanh -1.0000000000000001e+299 0.0 -> -9.9999999999999999e-300 1.5707963267948966 +atanh0037 atanh -1.0000000000000001e+299 -0.0 -> -9.9999999999999999e-300 -1.5707963267948966 +atanh0038 atanh 9.8813129168249309e-324 0.0 -> 9.8813129168249309e-324 0.0 +atanh0039 atanh 9.8813129168249309e-324 -0.0 -> 9.8813129168249309e-324 -0.0 +atanh0040 atanh 1e-305 0.0 -> 1e-305 0.0 +atanh0041 atanh 1e-305 -0.0 -> 1e-305 -0.0 +atanh0042 atanh 1e-150 0.0 -> 1e-150 0.0 +atanh0043 atanh 1e-150 -0.0 -> 1e-150 -0.0 +atanh0044 atanh 9.9999999999999998e-17 0.0 -> 9.9999999999999998e-17 0.0 +atanh0045 atanh 9.9999999999999998e-17 -0.0 -> 9.9999999999999998e-17 -0.0 +atanh0046 atanh 0.001 0.0 -> 0.0010000003333335333 0.0 +atanh0047 atanh 0.001 -0.0 -> 0.0010000003333335333 -0.0 +atanh0048 atanh 0.57899999999999996 0.0 -> 0.6609570902866303 0.0 +atanh0049 atanh 0.57899999999999996 -0.0 -> 0.6609570902866303 -0.0 +atanh0050 atanh 0.99999999999999989 0.0 -> 18.714973875118524 0.0 +atanh0051 atanh 0.99999999999999989 -0.0 -> 18.714973875118524 -0.0 +atanh0052 atanh 1.0000000000000002 0.0 -> 18.36840028483855 1.5707963267948966 +atanh0053 atanh 1.0000000000000002 -0.0 -> 18.36840028483855 -1.5707963267948966 +atanh0054 atanh 1.0009999999999999 0.0 -> 3.8007011672919218 1.5707963267948966 +atanh0055 atanh 1.0009999999999999 -0.0 -> 3.8007011672919218 -1.5707963267948966 +atanh0056 atanh 2.0 0.0 -> 0.54930614433405489 1.5707963267948966 +atanh0057 atanh 2.0 -0.0 -> 0.54930614433405489 -1.5707963267948966 +atanh0058 atanh 23.0 0.0 -> 0.043505688494814884 1.5707963267948966 +atanh0059 atanh 23.0 -0.0 -> 0.043505688494814884 -1.5707963267948966 +atanh0060 atanh 10000000000000000.0 0.0 -> 9.9999999999999998e-17 1.5707963267948966 +atanh0061 atanh 10000000000000000.0 -0.0 -> 9.9999999999999998e-17 -1.5707963267948966 +atanh0062 atanh 9.9999999999999998e+149 0.0 -> 1e-150 1.5707963267948966 +atanh0063 atanh 9.9999999999999998e+149 -0.0 -> 1e-150 -1.5707963267948966 +atanh0064 atanh 1.0000000000000001e+299 0.0 -> 9.9999999999999999e-300 1.5707963267948966 +atanh0065 atanh 1.0000000000000001e+299 -0.0 -> 9.9999999999999999e-300 -1.5707963267948966 + +-- random inputs +atanh0100 atanh -0.54460925980633501 -0.54038050126721027 -> -0.41984265808446974 -0.60354153938352828 +atanh0101 atanh -1.6934614269829051 -0.48807386108113621 -> -0.58592769102243281 -1.3537837470975898 +atanh0102 atanh -1.3467293985501207 -0.47868354895395876 -> -0.69961624370709985 -1.1994450156570076 +atanh0103 atanh -5.6142232418984888 -544551613.39307702 -> -1.8932657550925744e-17 -1.5707963249585235 +atanh0104 atanh -0.011841460381263651 -3.259978899823385 -> -0.0010183936547405188 -1.2731614020743838 +atanh0105 atanh -0.0073345736950029532 0.35821949670922248 -> -0.0065004869024682466 0.34399359971920895 +atanh0106 atanh -13.866782244320014 0.9541129545860273 -> -0.071896852055058899 1.5658322704631409 +atanh0107 atanh -708.59964982780775 21.984802159266675 -> -0.0014098779074189741 1.5707525842838959 +atanh0108 atanh -30.916832076030602 1.3691897138829843 -> -0.032292682045743676 1.5693652094847115 +atanh0109 atanh -0.57461806339861754 0.29534797443913063 -> -0.56467464472482765 0.39615612824172625 +atanh0110 atanh 0.40089246737415685 -1.632285984300659 -> 0.1063832707890608 -1.0402821335326482 +atanh0111 atanh 2119.6167688262176 -1.5383653437377242e+17 -> 8.9565008518382049e-32 -1.5707963267948966 +atanh0112 atanh 756.86017850941641 -6.6064087133223817 -> 0.0013211481136820046 -1.5707847948702234 +atanh0113 atanh 4.0490617718041602 -2.5784456791040652e-12 -> 0.25218425538553618 -1.5707963267947291 +atanh0114 atanh 10.589254957173523 -0.13956391149624509 -> 0.094700890282197664 -1.5695407140217623 +atanh0115 atanh 1.0171187553160499 0.70766113465354019 -> 0.55260251975367791 0.96619711116641682 +atanh0116 atanh 0.031645502527750849 0.067319983726544394 -> 0.031513018344086742 0.067285437670549036 +atanh0117 atanh 0.13670177624994517 0.43240089361857947 -> 0.11538933151017253 0.41392008145336212 +atanh0118 atanh 0.64173899243596688 2.9008577686695256 -> 0.065680142424134405 1.2518535724053921 +atanh0119 atanh 0.19313813528025942 38.799619150741869 -> 0.00012820765917366644 1.5450292202823612 + +-- values near infinity +atanh0200 atanh 5.3242646831347954e+307 1.3740396080084153e+308 -> 2.4519253616695576e-309 1.5707963267948966 +atanh0201 atanh 1.158701641241358e+308 -6.5579268873375853e+307 -> 6.5365375267795098e-309 -1.5707963267948966 +atanh0202 atanh -1.3435325735762247e+308 9.8947369259601547e+307 -> -4.8256680906589956e-309 1.5707963267948966 +atanh0203 atanh -1.4359857522598942e+308 -9.4701204702391004e+307 -> -4.8531282262872645e-309 -1.5707963267948966 +atanh0204 atanh 0.0 5.6614181068098497e+307 -> 0.0 1.5707963267948966 +atanh0205 atanh -0.0 6.9813212721450139e+307 -> 0.0 1.5707963267948966 +atanh0206 atanh 0.0 -7.4970613060311453e+307 -> 0.0 -1.5707963267948966 +atanh0207 atanh -0.0 -1.5280601880314068e+308 -> 0.0 -1.5707963267948966 +atanh0208 atanh 8.2219472336000745e+307 0.0 -> 1.2162568933954813e-308 1.5707963267948966 +atanh0209 atanh 1.4811519617280899e+308 -0.0 -> 6.7515017083951325e-309 -1.5707963267948966 +atanh0210 atanh -1.2282016263598785e+308 0.0 -> -8.1419856360537615e-309 1.5707963267948966 +atanh0211 atanh -1.0616427760154426e+308 -0.0 -> -9.4193642399489563e-309 -1.5707963267948966 +atanh0212 atanh 1.2971536510180682e+308 5.2847948452333293 -> 7.7091869510998328e-309 1.5707963267948966 +atanh0213 atanh 1.1849860977411851e+308 -7.9781906447459949 -> 8.4389175696339014e-309 -1.5707963267948966 +atanh0214 atanh -1.4029969422586635e+308 0.93891986543663375 -> -7.127599283218073e-309 1.5707963267948966 +atanh0215 atanh -4.7508098912248211e+307 -8.2702421247039908 -> -2.1049042645278043e-308 -1.5707963267948966 +atanh0216 atanh 8.2680742115769998 8.1153898410918065e+307 -> 0.0 1.5707963267948966 +atanh0217 atanh 1.2575325146218885 -1.4746679147661649e+308 -> 0.0 -1.5707963267948966 +atanh0218 atanh -2.4618803682310899 1.3781522717005568e+308 -> -0.0 1.5707963267948966 +atanh0219 atanh -4.0952386694788112 -1.231083376353703e+308 -> -0.0 -1.5707963267948966 + +-- values near 0 +atanh0220 atanh 3.8017563659811628e-314 2.6635484239074319e-312 -> 3.8017563659811628e-314 2.6635484239074319e-312 +atanh0221 atanh 1.7391110733611878e-321 -4.3547800672541419e-313 -> 1.7391110733611878e-321 -4.3547800672541419e-313 +atanh0222 atanh -5.9656816081325078e-317 9.9692253555416263e-313 -> -5.9656816081325078e-317 9.9692253555416263e-313 +atanh0223 atanh -6.5606671178400239e-313 -2.1680936406357335e-309 -> -6.5606671178400239e-313 -2.1680936406357335e-309 +atanh0224 atanh 0.0 2.5230944401820779e-319 -> 0.0 2.5230944401820779e-319 +atanh0225 atanh -0.0 5.6066569490064658e-320 -> 0.0 5.6066569490064658e-320 +atanh0226 atanh 0.0 -2.4222487249468377e-317 -> 0.0 -2.4222487249468377e-317 +atanh0227 atanh -0.0 -3.0861101089206037e-316 -> 0.0 -3.0861101089206037e-316 +atanh0228 atanh 3.1219222884393986e-310 0.0 -> 3.1219222884393986e-310 0.0 +atanh0229 atanh 9.8926337564976196e-309 -0.0 -> 9.8926337564976196e-309 -0.0 +atanh0230 atanh -1.5462535092918154e-312 0.0 -> -1.5462535092918154e-312 0.0 +atanh0231 atanh -9.8813129168249309e-324 -0.0 -> -9.8813129168249309e-324 -0.0 + +-- real part = +/-1, imaginary part tiny +atanh0300 atanh 1.0 1e-153 -> 176.49433320432448 0.78539816339744828 +atanh0301 atanh 1.0 9.9999999999999997e-155 -> 177.64562575082149 0.78539816339744828 +atanh0302 atanh -1.0 1e-161 -> -185.70467357630065 0.78539816339744828 +atanh0303 atanh 1.0 -1e-165 -> 190.30984376228875 -0.78539816339744828 +atanh0304 atanh -1.0 -9.8813129168249309e-324 -> -372.22003596069061 -0.78539816339744828 + +---------------------------- +-- log: Natural logarithm -- +---------------------------- + +log0000 log 1 0.0 -> 0.0 0.0 +log0001 log 1 -0.0 -> 0.0 -0.0 +log0002 log -1 0.0 -> 0.0 3.1415926535897931 +log0003 log -1 -0.0 -> 0.0 -3.1415926535897931 +-- values along both sides of real axis +log0010 log -9.8813129168249309e-324 0.0 -> -743.74692474082133 3.1415926535897931 +log0011 log -9.8813129168249309e-324 -0.0 -> -743.74692474082133 -3.1415926535897931 +log0012 log -1e-305 0.0 -> -702.28845336318398 3.1415926535897931 +log0013 log -1e-305 -0.0 -> -702.28845336318398 -3.1415926535897931 +log0014 log -1e-150 0.0 -> -345.38776394910684 3.1415926535897931 +log0015 log -1e-150 -0.0 -> -345.38776394910684 -3.1415926535897931 +log0016 log -9.9999999999999998e-17 0.0 -> -36.841361487904734 3.1415926535897931 +log0017 log -9.9999999999999998e-17 -0.0 -> -36.841361487904734 -3.1415926535897931 +log0018 log -0.001 0.0 -> -6.9077552789821368 3.1415926535897931 +log0019 log -0.001 -0.0 -> -6.9077552789821368 -3.1415926535897931 +log0020 log -0.57899999999999996 0.0 -> -0.54645280140914188 3.1415926535897931 +log0021 log -0.57899999999999996 -0.0 -> -0.54645280140914188 -3.1415926535897931 +log0022 log -0.99999999999999989 0.0 -> -1.1102230246251565e-16 3.1415926535897931 +log0023 log -0.99999999999999989 -0.0 -> -1.1102230246251565e-16 -3.1415926535897931 +log0024 log -1.0000000000000002 0.0 -> 2.2204460492503128e-16 3.1415926535897931 +log0025 log -1.0000000000000002 -0.0 -> 2.2204460492503128e-16 -3.1415926535897931 +log0026 log -1.0009999999999999 0.0 -> 0.00099950033308342321 3.1415926535897931 +log0027 log -1.0009999999999999 -0.0 -> 0.00099950033308342321 -3.1415926535897931 +log0028 log -2.0 0.0 -> 0.69314718055994529 3.1415926535897931 +log0029 log -2.0 -0.0 -> 0.69314718055994529 -3.1415926535897931 +log0030 log -23.0 0.0 -> 3.1354942159291497 3.1415926535897931 +log0031 log -23.0 -0.0 -> 3.1354942159291497 -3.1415926535897931 +log0032 log -10000000000000000.0 0.0 -> 36.841361487904734 3.1415926535897931 +log0033 log -10000000000000000.0 -0.0 -> 36.841361487904734 -3.1415926535897931 +log0034 log -9.9999999999999998e+149 0.0 -> 345.38776394910684 3.1415926535897931 +log0035 log -9.9999999999999998e+149 -0.0 -> 345.38776394910684 -3.1415926535897931 +log0036 log -1.0000000000000001e+299 0.0 -> 688.47294280521965 3.1415926535897931 +log0037 log -1.0000000000000001e+299 -0.0 -> 688.47294280521965 -3.1415926535897931 +log0038 log 9.8813129168249309e-324 0.0 -> -743.74692474082133 0.0 +log0039 log 9.8813129168249309e-324 -0.0 -> -743.74692474082133 -0.0 +log0040 log 1e-305 0.0 -> -702.28845336318398 0.0 +log0041 log 1e-305 -0.0 -> -702.28845336318398 -0.0 +log0042 log 1e-150 0.0 -> -345.38776394910684 0.0 +log0043 log 1e-150 -0.0 -> -345.38776394910684 -0.0 +log0044 log 9.9999999999999998e-17 0.0 -> -36.841361487904734 0.0 +log0045 log 9.9999999999999998e-17 -0.0 -> -36.841361487904734 -0.0 +log0046 log 0.001 0.0 -> -6.9077552789821368 0.0 +log0047 log 0.001 -0.0 -> -6.9077552789821368 -0.0 +log0048 log 0.57899999999999996 0.0 -> -0.54645280140914188 0.0 +log0049 log 0.57899999999999996 -0.0 -> -0.54645280140914188 -0.0 +log0050 log 0.99999999999999989 0.0 -> -1.1102230246251565e-16 0.0 +log0051 log 0.99999999999999989 -0.0 -> -1.1102230246251565e-16 -0.0 +log0052 log 1.0000000000000002 0.0 -> 2.2204460492503128e-16 0.0 +log0053 log 1.0000000000000002 -0.0 -> 2.2204460492503128e-16 -0.0 +log0054 log 1.0009999999999999 0.0 -> 0.00099950033308342321 0.0 +log0055 log 1.0009999999999999 -0.0 -> 0.00099950033308342321 -0.0 +log0056 log 2.0 0.0 -> 0.69314718055994529 0.0 +log0057 log 2.0 -0.0 -> 0.69314718055994529 -0.0 +log0058 log 23.0 0.0 -> 3.1354942159291497 0.0 +log0059 log 23.0 -0.0 -> 3.1354942159291497 -0.0 +log0060 log 10000000000000000.0 0.0 -> 36.841361487904734 0.0 +log0061 log 10000000000000000.0 -0.0 -> 36.841361487904734 -0.0 +log0062 log 9.9999999999999998e+149 0.0 -> 345.38776394910684 0.0 +log0063 log 9.9999999999999998e+149 -0.0 -> 345.38776394910684 -0.0 +log0064 log 1.0000000000000001e+299 0.0 -> 688.47294280521965 0.0 +log0065 log 1.0000000000000001e+299 -0.0 -> 688.47294280521965 -0.0 + +-- random inputs +log0066 log -1.9830454945186191e-16 -2.0334448025673346 -> 0.70973130194329803 -1.5707963267948968 +log0067 log -0.96745853024741857 -0.84995816228299692 -> 0.25292811398722387 -2.4207570438536905 +log0068 log -0.1603644313948418 -0.2929942111041835 -> -1.0965857872427374 -2.0715870859971419 +log0069 log -0.15917913168438699 -0.25238799251132177 -> -1.2093477313249901 -2.1334784232033863 +log0070 log -0.68907818535078802 -3.0693105853476346 -> 1.1460398629184565 -1.7916403813913211 +log0071 log -17.268133447565589 6.8165120014604756 -> 2.9212694465974836 2.7656245081603164 +log0072 log -1.7153894479690328 26.434055372802636 -> 3.2767542953718003 1.6355986276341734 +log0073 log -8.0456794648936578e-06 0.19722758057570208 -> -1.6233969848296075 1.5708371206810101 +log0074 log -2.4306442691323173 0.6846919750700996 -> 0.92633592001969589 2.8670160576718331 +log0075 log -3.5488049250888194 0.45324040643185254 -> 1.2747008374256426 3.0145640007885111 +log0076 log 0.18418516851510189 -0.26062518836212617 -> -1.1421287121940344 -0.95558440841183434 +log0077 log 2.7124837795638399 -13.148769067133387 -> 2.5971659975706802 -1.3673583045209439 +log0078 log 3.6521275476169149e-13 -3.7820543023170673e-05 -> -10.182658136741569 -1.5707963171384316 +log0079 log 5.0877545813862239 -1.2834978326786852 -> 1.6576856213076328 -0.24711583497738485 +log0080 log 0.26477986808461512 -0.67659001194187429 -> -0.31944085207999973 -1.197773671987121 +log0081 log 0.0014754261398071962 5.3514691608205442 -> 1.6773711707153829 1.5705206219261802 +log0082 log 0.29667334462157885 0.00020056045042584795 -> -1.2151233667079588 0.00067603114168689204 +log0083 log 0.82104233671099425 3.9005387130133102 -> 1.3827918965299593 1.3633304701848363 +log0084 log 0.27268135358180667 124.42088110945804 -> 4.8236724223559229 1.5686047258789015 +log0085 log 0.0026286959168267485 0.47795808180573013 -> -0.73821712137809126 1.5652965360960087 + +-- values near infinity +log0100 log 1.0512025744003172e+308 7.2621669750664611e+307 -> 709.44123967814494 0.60455434048332968 +log0101 log 5.5344249034372126e+307 -1.2155859158431275e+308 -> 709.48562300345679 -1.143553056717973 +log0102 log -1.3155575403469408e+308 1.1610793541663864e+308 -> 709.75847809546428 2.41848796504974 +log0103 log -1.632366720973235e+308 -1.54299446211448e+308 -> 710.00545236515586 -2.3843326028455087 +log0104 log 0.0 5.9449276692327712e+307 -> 708.67616191258526 1.5707963267948966 +log0105 log -0.0 1.1201850459025692e+308 -> 709.30970253338171 1.5707963267948966 +log0106 log 0.0 -1.6214225933466528e+308 -> 709.6795125501086 -1.5707963267948966 +log0107 log -0.0 -1.7453269791591058e+308 -> 709.75315056087379 -1.5707963267948966 +log0108 log 1.440860577601428e+308 0.0 -> 709.56144920058262 0.0 +log0109 log 1.391515176148282e+308 -0.0 -> 709.52660185041327 -0.0 +log0110 log -1.201354401295296e+308 0.0 -> 709.37965823023956 3.1415926535897931 +log0111 log -1.6704337825976804e+308 -0.0 -> 709.70929198492399 -3.1415926535897931 +log0112 log 7.2276974655190223e+307 7.94879711369164 -> 708.87154406512104 1.0997689307850458e-307 +log0113 log 1.1207859593716076e+308 -6.1956200868221147 -> 709.31023883080104 -5.5279244310803286e-308 +log0114 log -4.6678933874471045e+307 9.947107893220382 -> 708.43433142431388 3.1415926535897931 +log0115 log -1.5108012453950142e+308 -5.3117197179375619 -> 709.60884877835008 -3.1415926535897931 +log0116 log 7.4903750871504435 1.5320703776626352e+308 -> 709.62282865085137 1.5707963267948966 +log0117 log 5.9760325525654778 -8.0149473997349123e+307 -> 708.97493177248396 -1.5707963267948966 +log0118 log -7.880194206386629 1.7861845814767441e+308 -> 709.77629046837137 1.5707963267948966 +log0119 log -9.886438993852865 -6.19235781080747e+307 -> 708.71693946977302 -1.5707963267948966 + +-- values near 0 +log0120 log 2.2996867579227779e-308 6.7861840770939125e-312 -> -708.36343567717392 0.00029509166223339815 +log0121 log 6.9169190417774516e-323 -9.0414013188948118e-322 -> -739.22766796468386 -1.4944423210001669 +log0122 log -1.5378064962914011e-316 1.8243628389354635e-310 -> -713.20014803142965 1.5707971697228842 +log0123 log -2.3319898483706837e-321 -2.2358763941866371e-313 -> -719.9045008332522 -1.570796337224766 +log0124 log 0.0 3.872770101081121e-315 -> -723.96033425374401 1.5707963267948966 +log0125 log -0.0 9.6342800939043076e-322 -> -739.16707236281752 1.5707963267948966 +log0126 log 0.0 -2.266099393427834e-308 -> -708.37814861757965 -1.5707963267948966 +log0127 log -0.0 -2.1184695673766626e-315 -> -724.56361036731812 -1.5707963267948966 +log0128 log 1.1363509854348671e-322 0.0 -> -741.30457770545206 0.0 +log0129 log 3.5572726500569751e-322 -0.0 -> -740.16340580236522 -0.0 +log0130 log -2.3696071074040593e-310 0.0 -> -712.93865466421641 3.1415926535897931 +log0131 log -2.813283897266934e-317 -0.0 -> -728.88512203138862 -3.1415926535897931 + +-- values near the unit circle +log0200 log -0.59999999999999998 0.80000000000000004 -> 2.2204460492503132e-17 2.2142974355881808 +log0201 log 0.79999999999999993 0.60000000000000009 -> 6.1629758220391547e-33 0.64350110879328448 + +----------------------- +-- sqrt: Square root -- +----------------------- + +-- zeros +sqrt0000 sqrt 0.0 0.0 -> 0.0 0.0 +sqrt0001 sqrt 0.0 -0.0 -> 0.0 -0.0 +sqrt0002 sqrt -0.0 0.0 -> 0.0 0.0 +sqrt0003 sqrt -0.0 -0.0 -> 0.0 -0.0 + +-- values along both sides of real axis +sqrt0010 sqrt -9.8813129168249309e-324 0.0 -> 0.0 3.1434555694052576e-162 +sqrt0011 sqrt -9.8813129168249309e-324 -0.0 -> 0.0 -3.1434555694052576e-162 +sqrt0012 sqrt -1e-305 0.0 -> 0.0 3.1622776601683791e-153 +sqrt0013 sqrt -1e-305 -0.0 -> 0.0 -3.1622776601683791e-153 +sqrt0014 sqrt -1e-150 0.0 -> 0.0 9.9999999999999996e-76 +sqrt0015 sqrt -1e-150 -0.0 -> 0.0 -9.9999999999999996e-76 +sqrt0016 sqrt -9.9999999999999998e-17 0.0 -> 0.0 1e-08 +sqrt0017 sqrt -9.9999999999999998e-17 -0.0 -> 0.0 -1e-08 +sqrt0018 sqrt -0.001 0.0 -> 0.0 0.031622776601683791 +sqrt0019 sqrt -0.001 -0.0 -> 0.0 -0.031622776601683791 +sqrt0020 sqrt -0.57899999999999996 0.0 -> 0.0 0.76092049518987193 +sqrt0021 sqrt -0.57899999999999996 -0.0 -> 0.0 -0.76092049518987193 +sqrt0022 sqrt -0.99999999999999989 0.0 -> 0.0 0.99999999999999989 +sqrt0023 sqrt -0.99999999999999989 -0.0 -> 0.0 -0.99999999999999989 +sqrt0024 sqrt -1.0000000000000002 0.0 -> 0.0 1.0 +sqrt0025 sqrt -1.0000000000000002 -0.0 -> 0.0 -1.0 +sqrt0026 sqrt -1.0009999999999999 0.0 -> 0.0 1.000499875062461 +sqrt0027 sqrt -1.0009999999999999 -0.0 -> 0.0 -1.000499875062461 +sqrt0028 sqrt -2.0 0.0 -> 0.0 1.4142135623730951 +sqrt0029 sqrt -2.0 -0.0 -> 0.0 -1.4142135623730951 +sqrt0030 sqrt -23.0 0.0 -> 0.0 4.7958315233127191 +sqrt0031 sqrt -23.0 -0.0 -> 0.0 -4.7958315233127191 +sqrt0032 sqrt -10000000000000000.0 0.0 -> 0.0 100000000.0 +sqrt0033 sqrt -10000000000000000.0 -0.0 -> 0.0 -100000000.0 +sqrt0034 sqrt -9.9999999999999998e+149 0.0 -> 0.0 9.9999999999999993e+74 +sqrt0035 sqrt -9.9999999999999998e+149 -0.0 -> 0.0 -9.9999999999999993e+74 +sqrt0036 sqrt -1.0000000000000001e+299 0.0 -> 0.0 3.1622776601683796e+149 +sqrt0037 sqrt -1.0000000000000001e+299 -0.0 -> 0.0 -3.1622776601683796e+149 +sqrt0038 sqrt 9.8813129168249309e-324 0.0 -> 3.1434555694052576e-162 0.0 +sqrt0039 sqrt 9.8813129168249309e-324 -0.0 -> 3.1434555694052576e-162 -0.0 +sqrt0040 sqrt 1e-305 0.0 -> 3.1622776601683791e-153 0.0 +sqrt0041 sqrt 1e-305 -0.0 -> 3.1622776601683791e-153 -0.0 +sqrt0042 sqrt 1e-150 0.0 -> 9.9999999999999996e-76 0.0 +sqrt0043 sqrt 1e-150 -0.0 -> 9.9999999999999996e-76 -0.0 +sqrt0044 sqrt 9.9999999999999998e-17 0.0 -> 1e-08 0.0 +sqrt0045 sqrt 9.9999999999999998e-17 -0.0 -> 1e-08 -0.0 +sqrt0046 sqrt 0.001 0.0 -> 0.031622776601683791 0.0 +sqrt0047 sqrt 0.001 -0.0 -> 0.031622776601683791 -0.0 +sqrt0048 sqrt 0.57899999999999996 0.0 -> 0.76092049518987193 0.0 +sqrt0049 sqrt 0.57899999999999996 -0.0 -> 0.76092049518987193 -0.0 +sqrt0050 sqrt 0.99999999999999989 0.0 -> 0.99999999999999989 0.0 +sqrt0051 sqrt 0.99999999999999989 -0.0 -> 0.99999999999999989 -0.0 +sqrt0052 sqrt 1.0000000000000002 0.0 -> 1.0 0.0 +sqrt0053 sqrt 1.0000000000000002 -0.0 -> 1.0 -0.0 +sqrt0054 sqrt 1.0009999999999999 0.0 -> 1.000499875062461 0.0 +sqrt0055 sqrt 1.0009999999999999 -0.0 -> 1.000499875062461 -0.0 +sqrt0056 sqrt 2.0 0.0 -> 1.4142135623730951 0.0 +sqrt0057 sqrt 2.0 -0.0 -> 1.4142135623730951 -0.0 +sqrt0058 sqrt 23.0 0.0 -> 4.7958315233127191 0.0 +sqrt0059 sqrt 23.0 -0.0 -> 4.7958315233127191 -0.0 +sqrt0060 sqrt 10000000000000000.0 0.0 -> 100000000.0 0.0 +sqrt0061 sqrt 10000000000000000.0 -0.0 -> 100000000.0 -0.0 +sqrt0062 sqrt 9.9999999999999998e+149 0.0 -> 9.9999999999999993e+74 0.0 +sqrt0063 sqrt 9.9999999999999998e+149 -0.0 -> 9.9999999999999993e+74 -0.0 +sqrt0064 sqrt 1.0000000000000001e+299 0.0 -> 3.1622776601683796e+149 0.0 +sqrt0065 sqrt 1.0000000000000001e+299 -0.0 -> 3.1622776601683796e+149 -0.0 + +-- random inputs +sqrt0100 sqrt -0.34252542541549913 -223039880.15076211 -> 10560.300180587592 -10560.300196805192 +sqrt0101 sqrt -0.88790791393018909 -5.3307751730827402 -> 1.5027154613689004 -1.7737140896343291 +sqrt0102 sqrt -113916.89291310767 -0.018143374626153858 -> 2.6877817875351178e-05 -337.51576691038952 +sqrt0103 sqrt -0.63187172386197121 -0.26293913366617694 -> 0.16205707495266153 -0.81125471918761971 +sqrt0104 sqrt -0.058185169308906215 -2.3548312990430991 -> 1.0717660342420072 -1.0985752598086966 +sqrt0105 sqrt -1.0580584765935896 0.14400319259151736 -> 0.069837489270111242 1.030987755262468 +sqrt0106 sqrt -1.1667595947504932 0.11159711473953678 -> 0.051598531319315251 1.0813981705111229 +sqrt0107 sqrt -0.5123728411449906 0.026175433648339085 -> 0.018278026262418718 0.71603556293597614 +sqrt0108 sqrt -3.7453400060067228 1.0946500314809635 -> 0.27990088541692498 1.9554243814742367 +sqrt0109 sqrt -0.0027736121575097673 1.0367943000839817 -> 0.71903560338719175 0.72096172651250545 +sqrt0110 sqrt 1501.2559699453188 -1.1997325207283589 -> 38.746047664730959 -0.015481998720355024 +sqrt0111 sqrt 1.4830075326850578 -0.64100878436755349 -> 1.244712815741096 -0.25749264258434584 +sqrt0112 sqrt 0.095395618499734602 -0.48226565701639595 -> 0.54175904053472879 -0.44509239434231551 +sqrt0113 sqrt 0.50109185681863277 -0.54054037379892561 -> 0.7868179858332387 -0.34349772344520979 +sqrt0114 sqrt 0.98779807595367897 -0.00019848758437225191 -> 0.99388031770665153 -9.9854872279921968e-05 +sqrt0115 sqrt 11.845472380792259 0.0010051104581506761 -> 3.4417252072345397 0.00014601840612346451 +sqrt0116 sqrt 2.3558249686735975 0.25605157371744403 -> 1.5371278477386647 0.083288964575761404 +sqrt0117 sqrt 0.77584894123159098 1.0496420627016076 -> 1.0200744386390885 0.51449287568756552 +sqrt0118 sqrt 1.8961715669604893 0.34940793467158854 -> 1.3827991781411615 0.12634080935066902 +sqrt0119 sqrt 0.96025378316565801 0.69573224860140515 -> 1.0358710342209998 0.33581991658093457 + +-- values near 0 +sqrt0120 sqrt 7.3577938365086866e-313 8.1181408465112743e-319 -> 8.5777583531543516e-157 4.732087634251168e-163 +sqrt0121 sqrt 1.2406883874892108e-310 -5.1210133324269776e-312 -> 1.1140990057468052e-155 -2.2982756945349973e-157 +sqrt0122 sqrt -7.1145453001139502e-322 2.9561379244703735e-314 -> 1.2157585807480286e-157 1.2157586100077242e-157 +sqrt0123 sqrt -4.9963244206801218e-314 -8.4718424423690227e-319 -> 1.8950582312540437e-162 -2.2352459419578971e-157 +sqrt0124 sqrt 0.0 7.699553609385195e-318 -> 1.9620848107797476e-159 1.9620848107797476e-159 +sqrt0125 sqrt -0.0 3.3900826606499415e-309 -> 4.1170879639922327e-155 4.1170879639922327e-155 +sqrt0126 sqrt 0.0 -9.8907989772250828e-319 -> 7.032353438652342e-160 -7.032353438652342e-160 +sqrt0127 sqrt -0.0 -1.3722939367590908e-315 -> 2.6194407196566702e-158 -2.6194407196566702e-158 +sqrt0128 sqrt 7.9050503334599447e-323 0.0 -> 8.8910349979403099e-162 0.0 +sqrt0129 sqrt 1.8623241768349486e-309 -0.0 -> 4.3154654173506579e-155 -0.0 +sqrt0130 sqrt -2.665971134499887e-308 0.0 -> 0.0 1.6327801856036491e-154 +sqrt0131 sqrt -1.5477066694467245e-310 -0.0 -> 0.0 -1.2440685951533077e-155 + +-- inputs whose absolute value overflows +sqrt0140 sqrt 1.6999999999999999e+308 -1.6999999999999999e+308 -> 1.4325088230154573e+154 -5.9336458271212207e+153 +sqrt0141 sqrt -1.797e+308 -9.9999999999999999e+306 -> 3.7284476432057307e+152 -1.3410406899802901e+154 + +-- For exp, cosh, sinh, tanh we limit tests to arguments whose +-- imaginary part is less than 10 in absolute value: most math +-- libraries have poor accuracy for (real) sine and cosine for +-- large arguments, and the accuracy of these complex functions +-- suffer correspondingly. +-- +-- Similarly, for cos, sin and tan we limit tests to arguments +-- with relatively small real part. + +------------------------------- +-- exp: Exponential function -- +------------------------------- + +-- zeros +exp0000 exp 0.0 0.0 -> 1.0 0.0 +exp0001 exp 0.0 -0.0 -> 1.0 -0.0 +exp0002 exp -0.0 0.0 -> 1.0 0.0 +exp0003 exp -0.0 -0.0 -> 1.0 -0.0 + +-- random inputs +exp0004 exp -17.957359009564684 -1.108613895795274 -> 7.0869292576226611e-09 -1.4225929202377833e-08 +exp0005 exp -1.4456149663368642e-15 -0.75359817331772239 -> 0.72923148323917997 -0.68426708517419033 +exp0006 exp -0.76008654883512661 -0.46657235480105019 -> 0.41764393109928666 -0.21035108396792854 +exp0007 exp -5.7071614697735731 -2.3744161818115816e-11 -> 0.0033220890242068356 -7.8880219364953578e-14 +exp0008 exp -0.4653981327927097 -5.2236706667445587e-21 -> 0.62788507378216663 -3.2798648420026468e-21 +exp0009 exp -3.2444565242295518 1.1535625304243959 -> 0.015799936931457641 0.035644950380024749 +exp0010 exp -3.0651456337977727 0.87765086532391878 -> 0.029805595629855953 0.035882775180855669 +exp0011 exp -0.11080823753233926 0.96486386300873106 -> 0.50979112534376314 0.73575512419561562 +exp0012 exp -2.5629722598928648 0.019636235754708079 -> 0.077060452853917397 0.0015133717341137684 +exp0013 exp -3.3201709957983357e-10 1.2684017344487268 -> 0.29780699855434889 0.95462610007689186 +exp0014 exp 0.88767276057993272 -0.18953422986895557 -> 2.3859624049858095 -0.45771559132044426 +exp0015 exp 1.5738333486794742 -2.2576803075544328e-11 -> 4.8251091132458654 -1.0893553826776623e-10 +exp0016 exp 1.6408702341813795 -1.438879484380837 -> 0.6786733590689048 -5.1148284173168825 +exp0017 exp 1.820279424202033 -0.020812040370785722 -> 6.1722462896420902 -0.1284755888435051 +exp0018 exp 1.7273965735945873 -0.61140621328954947 -> 4.6067931898799976 -3.2294267694441308 +exp0019 exp 2.5606034306862995 0.098153136008435504 -> 12.881325889966629 1.2684184812864494 +exp0020 exp 10.280368619483029 3.4564622559748535 -> -27721.283321551502 -9028.9663215568835 +exp0021 exp 1.104007405129741e-155 0.21258803067317278 -> 0.97748813933531764 0.21099037290544478 +exp0022 exp 0.027364777809295172 0.00059226603500623363 -> 1.0277424518451876 0.0006086970181346579 +exp0023 exp 0.94356313429255245 3.418530463518592 -> -2.4712285695346194 -0.70242654900218349 + +-- cases where exp(z) representable, exp(z.real) not +exp0030 exp 710.0 0.78500000000000003 -> 1.5803016909637158e+308 1.5790437551806911e+308 +exp0031 exp 710.0 -0.78500000000000003 -> 1.5803016909637158e+308 -1.5790437551806911e+308 + +-- values for which exp(x) underflows +exp0040 exp -735.0 0.78500000000000003 -> 4.3976783136329355e-320 4.3942198541120468e-320 +exp0041 exp -735.0 -2.3559999999999999 -> -4.3952079854037293e-320 -4.396690182341253e-320 + +----------------------------- +-- cosh: Hyperbolic Cosine -- +----------------------------- + +-- zeros +cosh0000 cosh 0.0 0.0 -> 1.0 0.0 +cosh0001 cosh 0.0 -0.0 -> 1.0 -0.0 +cosh0002 cosh -0.0 0.0 -> 1.0 -0.0 +cosh0003 cosh -0.0 -0.0 -> 1.0 0.0 + +-- random inputs +cosh0004 cosh -0.85395264297414253 -8.8553756148671958 -> -1.1684340348021185 0.51842195359787435 +cosh0005 cosh -19.584904237211223 -0.066582627994906177 -> 159816812.23336992 10656776.050406246 +cosh0006 cosh -0.11072618401130772 -1.484820215073247 -> 0.086397164744949503 0.11054275637717284 +cosh0007 cosh -3.4764840250681752 -0.48440348288275276 -> 14.325931955190844 7.5242053548737955 +cosh0008 cosh -0.52047063604524602 -0.3603805382775585 -> 1.0653940354683802 0.19193293606252473 +cosh0009 cosh -1.39518962975995 0.0074738604700702906 -> 2.1417031027235969 -0.01415518712296308 +cosh0010 cosh -0.37107064757653541 0.14728085307856609 -> 1.0580601496776991 -0.055712531964568587 +cosh0011 cosh -5.8470200958739653 4.0021722388336292 -> -112.86220667618285 131.24734033545013 +cosh0012 cosh -0.1700261444851883 0.97167540135354513 -> 0.57208748253577946 -0.1410904820240203 +cosh0013 cosh -0.44042397902648783 1.0904791964139742 -> 0.50760322393058133 -0.40333966652010816 +cosh0014 cosh 0.052267552491867299 -3.8889011430644174 -> -0.73452303414639297 0.035540704833537134 +cosh0015 cosh 0.98000764177127453 -1.2548829247784097 -> 0.47220747341416142 -1.0879421432180316 +cosh0016 cosh 0.083594701222644008 -0.88847899930181284 -> 0.63279782419312613 -0.064954566816002285 +cosh0017 cosh 1.38173531783776 -0.43185040816732229 -> 1.9221663374671647 -0.78073830858849347 +cosh0018 cosh 0.57315681120148465 -0.22255760951027942 -> 1.1399733125173004 -0.1335512343605956 +cosh0019 cosh 1.8882512333062347 4.5024932182383797 -> -0.7041602065362691 -3.1573822131964615 +cosh0020 cosh 0.5618219206858317 0.92620452129575348 -> 0.69822380405378381 0.47309067471054522 +cosh0021 cosh 0.54361442847062591 0.64176483583018462 -> 0.92234462074193491 0.34167906495845501 +cosh0022 cosh 0.0014777403107920331 1.3682028122677661 -> 0.2012106963899549 0.001447518137863219 +cosh0023 cosh 2.218885944363501 2.0015727395883687 -> -1.94294321081968 4.1290269176083196 + +-- large real part +cosh0030 cosh 710.5 2.3519999999999999 -> -1.2967465239355998e+308 1.3076707908857333e+308 +cosh0031 cosh -710.5 0.69999999999999996 -> 1.4085466381392499e+308 -1.1864024666450239e+308 + +--------------------------- +-- sinh: Hyperbolic Sine -- +--------------------------- + +-- zeros +sinh0000 sinh 0.0 0.0 -> 0.0 0.0 +sinh0001 sinh 0.0 -0.0 -> 0.0 -0.0 +sinh0002 sinh -0.0 0.0 -> -0.0 0.0 +sinh0003 sinh -0.0 -0.0 -> -0.0 -0.0 + +-- random inputs +sinh0004 sinh -17.282588091462742 -0.38187948694103546 -> -14867386.857248396 -5970648.6553516639 +sinh0005 sinh -343.91971203143208 -5.0172868877771525e-22 -> -1.1518691776521735e+149 -5.7792581214689021e+127 +sinh0006 sinh -14.178122253300922 -1.9387157579351293 -> 258440.37909034826 -670452.58500946441 +sinh0007 sinh -1.0343810581686239 -1.0970235266369905 -> -0.56070858278092739 -1.4098883258046697 +sinh0008 sinh -0.066126561416368204 -0.070461584169961872 -> -0.066010558700938124 -0.070557276738637542 +sinh0009 sinh -0.37630149150308484 3.3621734692162173 -> 0.37591118119332617 -0.23447115926369383 +sinh0010 sinh -0.049941960978670055 0.40323767020414625 -> -0.045955482136329009 0.3928878494430646 +sinh0011 sinh -16.647852603903715 0.0026852219129082098 -> -8492566.5739382561 22804.480671133562 +sinh0012 sinh -1.476625314303694 0.89473773116683386 -> -1.2982943334382224 1.7966593367791204 +sinh0013 sinh -422.36429577556913 0.10366634502307912 -> -1.3400321008920044e+183 1.3941600948045599e+182 +sinh0014 sinh 0.09108340745641981 -0.40408227416070353 -> 0.083863724802237902 -0.39480716553935602 +sinh0015 sinh 2.036064132067386 -2.6831729961386239 -> -3.37621124363175 -1.723868330002817 +sinh0016 sinh 2.5616717223063317 -0.0078978498622717767 -> 6.4399415853815869 -0.051472264400722133 +sinh0017 sinh 0.336804011985188 -6.5654622971649337 -> 0.32962499307574578 -0.29449170159995197 +sinh0018 sinh 0.23774603755649693 -0.92467195799232049 -> 0.14449839490603389 -0.82109449053556793 +sinh0019 sinh 0.0011388273541465494 1.9676196882949855 -> -0.00044014605389634999 0.92229398407098806 +sinh0020 sinh 3.2443870105663759 0.8054287559616895 -> 8.8702890778527426 9.2610748597042196 +sinh0021 sinh 0.040628908857054738 0.098206391190944958 -> 0.04044426841671233 0.098129544739707392 +sinh0022 sinh 4.7252283918217696e-30 9.1198155642656697 -> -4.5071980561644404e-30 0.30025730701661713 +sinh0023 sinh 0.043713693678420068 0.22512549887532657 -> 0.042624198673416713 0.22344201231217961 + +-- large real part +sinh0030 sinh 710.5 -2.3999999999999999 -> -1.3579970564885919e+308 -1.24394470907798e+308 +sinh0031 sinh -710.5 0.80000000000000004 -> -1.2830671601735164e+308 1.3210954193997678e+308 + +------------------------------ +-- tanh: Hyperbolic Tangent -- +------------------------------ + +-- zeros +tanh0000 tanh 0.0 0.0 -> 0.0 0.0 +tanh0001 tanh 0.0 -0.0 -> 0.0 -0.0 +tanh0002 tanh -0.0 0.0 -> -0.0 0.0 +tanh0003 tanh -0.0 -0.0 -> -0.0 -0.0 + +-- random inputs +tanh0004 tanh -21.200500450664993 -1.6970729480342996 -> -1.0 1.9241352344849399e-19 +tanh0005 tanh -0.34158771504251928 -8.0848504951747131 -> -2.123711225855613 1.2827526782026006 +tanh0006 tanh -15.454144725193689 -0.23619582288265617 -> -0.99999999999993283 -3.4336684248260036e-14 +tanh0007 tanh -7.6103163119661952 -0.7802748320307008 -> -0.99999999497219438 -4.9064845343755437e-07 +tanh0008 tanh -0.15374717235792129 -0.6351086327306138 -> -0.23246081703561869 -0.71083467433910219 +tanh0009 tanh -0.49101115474392465 0.09723001264886301 -> -0.45844445715492133 0.077191158541805888 +tanh0010 tanh -0.10690612157664491 2.861612800856395 -> -0.11519761626257358 -0.28400488355647507 +tanh0011 tanh -0.91505774192066702 1.5431174597727007 -> -1.381109893068114 0.025160819663709356 +tanh0012 tanh -0.057433367093792223 0.35491159541246459 -> -0.065220499046696953 0.36921788332369498 +tanh0013 tanh -1.3540418621233514 0.18969415642242535 -> -0.88235642861151387 0.043764069984411721 +tanh0014 tanh 0.94864783961003529 -0.11333689578867717 -> 0.74348401861861368 -0.051271042543855221 +tanh0015 tanh 1.9591698133845488 -0.0029654444904578339 -> 0.9610270776968135 -0.00022664240049212933 +tanh0016 tanh 1.0949715796669197 -0.24706642853984456 -> 0.81636574501369386 -0.087767436914149954 +tanh0017 tanh 5770428.2113731047 -3.7160580339833165 -> 1.0 -0.0 +tanh0018 tanh 1.5576782321399629 -1.0357943787966468 -> 1.0403002384895388 -0.081126347894671463 +tanh0019 tanh 0.62378536230552961 2.3471393579560216 -> 0.85582499238960363 -0.53569473646842869 +tanh0020 tanh 17.400628602508025 9.3987059533841979 -> 0.99999999999999845 -8.0175867720530832e-17 +tanh0021 tanh 0.15026177509871896 0.50630349159505472 -> 0.19367536571827768 0.53849847858853661 +tanh0022 tanh 0.57433977530711167 1.0071604546265627 -> 1.0857848159262844 0.69139213955872214 +tanh0023 tanh 0.16291181500449456 0.006972810241567544 -> 0.16149335907551157 0.0067910772903467817 + +-- large real part +tanh0030 tanh 710 0.13 -> 1.0 0.0 +tanh0031 tanh -711 7.4000000000000004 -> -1.0 0.0 +tanh0032 tanh 1000 -2.3199999999999998 -> 1.0 0.0 +tanh0033 tanh -1.0000000000000001e+300 -9.6699999999999999 -> -1.0 -0.0 + +----------------- +-- cos: Cosine -- +----------------- + +-- zeros +cos0000 cos 0.0 0.0 -> 1.0 -0.0 +cos0001 cos 0.0 -0.0 -> 1.0 0.0 +cos0002 cos -0.0 0.0 -> 1.0 0.0 +cos0003 cos -0.0 -0.0 -> 1.0 -0.0 + +-- random inputs +cos0004 cos -2.0689194692073034 -0.0016802181751734313 -> -0.47777827208561469 -0.0014760401501695971 +cos0005 cos -0.4209627318177977 -1.8238516774258027 -> 2.9010402201444108 -1.2329207042329617 +cos0006 cos -1.9402181630694557 -2.9751857392891217 -> -3.5465459297970985 -9.1119163586282248 +cos0007 cos -3.3118320290191616 -0.87871302909286142 -> -1.3911528636565498 0.16878141517391701 +cos0008 cos -4.9540404623376872 -0.57949232239026827 -> 0.28062445586552065 0.59467861308508008 +cos0009 cos -0.45374584316245026 1.3950283448373935 -> 1.9247665574290578 0.83004572204761107 +cos0010 cos -0.42578172040176843 1.2715881615413049 -> 1.7517161459489148 0.67863902697363332 +cos0011 cos -0.13862985354300136 0.43587635877670328 -> 1.0859880290361912 0.062157548146672272 +cos0012 cos -0.11073221308966584 9.9384082307326475e-15 -> 0.99387545040722947 1.0982543264065479e-15 +cos0013 cos -1.5027633662054623e-07 0.0069668060249955498 -> 1.0000242682912412 1.0469545565660995e-09 +cos0014 cos 4.9728645490503052 -0.00027479808860952822 -> 0.25754011731975501 -0.00026552849549083186 +cos0015 cos 7.81969303486719 -0.79621523445878783 -> 0.045734882501585063 0.88253139933082991 +cos0016 cos 0.13272421880766716 -0.74668445308718201 -> 1.2806012244432847 0.10825373267437005 +cos0017 cos 4.2396521985973274 -2.2178848380884881 -> -2.1165117057056855 -4.0416492444641401 +cos0018 cos 1.1622206624927296 -0.50400115461197081 -> 0.44884072613370379 0.4823469915034318 +cos0019 cos 1.628772864620884e-08 0.58205705428979282 -> 1.1742319995791435 -1.0024839481956604e-08 +cos0020 cos 2.6385212606111241 2.9886107100937296 -> -8.7209475927161417 -4.7748352107199796 +cos0021 cos 4.8048375263775256 0.0062248852898515658 -> 0.092318702015846243 0.0061983430422306142 +cos0022 cos 7.9914515433858515 0.71659966615501436 -> -0.17375439906936566 -0.77217043527294582 +cos0023 cos 0.45124351152540226 1.6992693993812158 -> 2.543477948972237 -1.1528193694875477 + +--------------- +-- sin: Sine -- +--------------- + +-- zeros +sin0000 sin 0.0 0.0 -> 0.0 0.0 +sin0001 sin 0.0 -0.0 -> 0.0 -0.0 +sin0002 sin -0.0 0.0 -> -0.0 0.0 +sin0003 sin -0.0 -0.0 -> -0.0 -0.0 + +-- random inputs +sin0004 sin -0.18691829163163759 -0.74388741985507034 -> -0.2396636733773444 -0.80023231101856751 +sin0005 sin -0.45127453702459158 -461.81339920716164 -> -7.9722299331077877e+199 -1.6450205811004628e+200 +sin0006 sin -0.47669228345768921 -2.7369936564987514 -> -3.557238022267124 -6.8308030771226615 +sin0007 sin -0.31024285525950857 -1.4869219939188296 -> -0.70972676047175209 -1.9985029635426839 +sin0008 sin -4.4194573407025608 -1.405999210989288 -> 2.0702480800802685 0.55362250792180601 +sin0009 sin -1.7810832046434898e-05 0.0016439555384379083 -> -1.7810856113185261e-05 0.0016439562786668375 +sin0010 sin -0.8200017874897666 0.61724876887771929 -> -0.8749078195948865 0.44835295550987758 +sin0011 sin -1.4536502806107114 0.63998575534150415 -> -1.2035709929437679 0.080012187489163708 +sin0012 sin -2.2653412155506079 0.13172760685583729 -> -0.77502093809190431 -0.084554426868229532 +sin0013 sin -0.02613983069491858 0.18404766597776073 -> -0.026580778863127943 0.18502525396735642 +sin0014 sin 1.5743065001054617 -0.53125574272642029 -> 1.1444596332092725 0.0019537598099352077 +sin0015 sin 7.3833101791283289e-20 -0.16453221324236217 -> 7.4834720674379429e-20 -0.16527555646466915 +sin0016 sin 0.34763834641254038 -2.8377416421089565 -> 2.918883541504663 -8.0002718053250224 +sin0017 sin 0.077105785180421563 -0.090056027316200674 -> 0.077341973814471304 -0.089909869380524587 +sin0018 sin 3.9063227798142329e-17 -0.05954098654295524 -> 3.9132490348956512e-17 -0.059576172859837351 +sin0019 sin 0.57333917932544598 8.7785221430594696e-06 -> 0.54244029338302935 7.3747869125301368e-06 +sin0020 sin 0.024861722816513169 0.33044620756118515 -> 0.026228801369651 0.3363889671570689 +sin0021 sin 1.4342727387492671 0.81361889790284347 -> 1.3370960060947923 0.12336137961387163 +sin0022 sin 1.1518087354403725 4.8597235966150558 -> 58.919141989603041 26.237003403758852 +sin0023 sin 0.00087773078406649192 34.792379211312095 -> 565548145569.38245 644329685822700.62 + +------------------ +-- tan: Tangent -- +------------------ + +-- zeros +tan0000 tan 0.0 0.0 -> 0.0 0.0 +tan0001 tan 0.0 -0.0 -> 0.0 -0.0 +tan0002 tan -0.0 0.0 -> -0.0 0.0 +tan0003 tan -0.0 -0.0 -> -0.0 -0.0 + +-- random inputs +tan0004 tan -0.56378561833861074 -1.7110276237187664e+73 -> -0.0 -1.0 +tan0005 tan -3.5451633993471915e-12 -2.855471863564059 -> -4.6622441304889575e-14 -0.99340273843093951 +tan0006 tan -2.502442719638696 -0.26742234390504221 -> 0.66735215252994995 -0.39078997935420956 +tan0007 tan -0.87639597720371365 -55.586225523280206 -> -1.0285264565948176e-48 -1.0 +tan0008 tan -0.015783869596427243 -520.05944436039272 -> -0.0 -1.0 +tan0009 tan -0.84643549990725164 2.0749097935396343 -> -0.031412661676959573 1.0033548479526764 +tan0010 tan -0.43613792248559646 8.1082741629458059 -> -1.3879848444644593e-07 0.99999988344224011 +tan0011 tan -1.0820906367833114 0.28571868992480248 -> -1.3622485737936536 0.99089269377971245 +tan0012 tan -1.1477859580220084 1.9021637002708041 -> -0.034348450042071196 1.0293954097901687 +tan0013 tan -0.12465543176953409 3.0606851016344815e-05 -> -0.12530514290387343 3.1087420769945479e-05 +tan0014 tan 3.7582848717525343 -692787020.44038939 -> 0.0 -1.0 +tan0015 tan 2.2321967655142176e-06 -10.090069423008169 -> 1.5369846120622643e-14 -0.99999999655723759 +tan0016 tan 0.88371172390245012 -1.1635053630132823 -> 0.19705017118625889 -1.0196452280843129 +tan0017 tan 2.1347414231849267 -1.9311339960416831 -> -0.038663576915982524 -1.0174399993980778 +tan0018 tan 5.9027945255899974 -2.1574195684607135e-183 -> -0.39986591539281496 -2.5023753167976915e-183 +tan0019 tan 0.44811489490805362 683216075670.07556 -> 0.0 1.0 +tan0020 tan 4.1459766396068325 12.523017205605756 -> 2.4022514758988068e-11 1.0000000000112499 +tan0021 tan 1.7809617968443272 1.5052381702853379 -> -0.044066222118946903 1.0932684517702778 +tan0022 tan 1.1615313900880577 1.7956298728647107 -> 0.041793186826390362 1.0375339546034792 +tan0023 tan 0.067014779477908945 5.8517361577457097 -> 2.2088639754800034e-06 0.9999836182420061 + Index: Lib/test/test_cmath.py =================================================================== --- Lib/test/test_cmath.py (revision 60151) +++ Lib/test/test_cmath.py (working copy) @@ -1,5 +1,7 @@ from test.test_support import run_unittest +from test.test_math import parse_testfile, test_file import unittest +import os, sys import cmath, math class CMathTests(unittest.TestCase): @@ -12,24 +14,35 @@ test_functions.append(lambda x : cmath.log(x, 1729. + 0j)) test_functions.append(lambda x : cmath.log(14.-27j, x)) - def cAssertAlmostEqual(self, a, b, rel_eps = 1e-10, abs_eps = 1e-100): - """Check that two complex numbers are almost equal.""" - # the two complex numbers are considered almost equal if - # either the relative error is <= rel_eps or the absolute error - # is tiny, <= abs_eps. - if a == b == 0: - return - absolute_error = abs(a-b) - relative_error = absolute_error/max(abs(a), abs(b)) - if relative_error > rel_eps and absolute_error > abs_eps: - self.fail("%s and %s are not almost equal" % (a, b)) + def setUp(self): + self.test_values = open(test_file) + def tearDown(self): + self.test_values.close() + + def rAssertAlmostEqual(self, a, b, rel_eps = 2e-15, abs_eps = 5e-323): + """Check that two floating-point numbers are almost equal.""" + + # test passes if either the absolute error or the relative + # error is sufficiently small. The defaults amount to an + # error of between 9 ulps and 19 ulps on an IEEE-754 compliant + # machine. + + try: + absolute_error = abs(b-a) + except OverflowError: + pass + else: + if absolute_error <= max(abs_eps, rel_eps * abs(a)): + return + self.fail("%s and %s are not sufficiently close" % (repr(a), repr(b))) + def test_constants(self): e_expected = 2.71828182845904523536 pi_expected = 3.14159265358979323846 - self.assertAlmostEqual(cmath.pi, pi_expected, 9, + self.rAssertAlmostEqual(cmath.pi, pi_expected, 9, "cmath.pi is %s; should be %s" % (cmath.pi, pi_expected)) - self.assertAlmostEqual(cmath.e, e_expected, 9, + self.rAssertAlmostEqual(cmath.e, e_expected, 9, "cmath.e is %s; should be %s" % (cmath.e, e_expected)) def test_user_object(self): @@ -109,13 +122,13 @@ for f in self.test_functions: # usual usage - self.cAssertAlmostEqual(f(MyComplex(cx_arg)), f(cx_arg)) - self.cAssertAlmostEqual(f(MyComplexOS(cx_arg)), f(cx_arg)) + self.assertEqual(f(MyComplex(cx_arg)), f(cx_arg)) + self.assertEqual(f(MyComplexOS(cx_arg)), f(cx_arg)) # other combinations of __float__ and __complex__ - self.cAssertAlmostEqual(f(FloatAndComplex()), f(cx_arg)) - self.cAssertAlmostEqual(f(FloatAndComplexOS()), f(cx_arg)) - self.cAssertAlmostEqual(f(JustFloat()), f(flt_arg)) - self.cAssertAlmostEqual(f(JustFloatOS()), f(flt_arg)) + self.assertEqual(f(FloatAndComplex()), f(cx_arg)) + self.assertEqual(f(FloatAndComplexOS()), f(cx_arg)) + self.assertEqual(f(JustFloat()), f(flt_arg)) + self.assertEqual(f(JustFloatOS()), f(flt_arg)) # TypeError should be raised for classes not providing # either __complex__ or __float__, even if they provide # __int__, __long__ or __index__. An old-style class @@ -138,7 +151,7 @@ # functions, by virtue of providing a __float__ method for f in self.test_functions: for arg in [2, 2L, 2.]: - self.cAssertAlmostEqual(f(arg), f(arg.__float__())) + self.assertEqual(f(arg), f(arg.__float__())) # but strings should give a TypeError for f in self.test_functions: @@ -182,13 +195,35 @@ float_fn = getattr(math, fn) complex_fn = getattr(cmath, fn) for v in values: - self.cAssertAlmostEqual(float_fn(v), complex_fn(v)) + z = complex_fn(v) + self.rAssertAlmostEqual(float_fn(v), z.real) + self.assertEqual(0., z.imag) # test two-argument version of log with various bases for base in [0.5, 2., 10.]: for v in positive: - self.cAssertAlmostEqual(cmath.log(v, base), math.log(v, base)) + z = cmath.log(v, base) + self.rAssertAlmostEqual(math.log(v, base), z.real) + self.assertEqual(0., z.imag) + def test_specific_values(self): + if not float.__getformat__("double").startswith("IEEE"): + return + for id, fn, ar, ai, er, ei in parse_testfile(test_file): + arg = complex(ar, ai) + expected = complex(er, ei) + function = getattr(cmath, fn) + actual = function(arg) + + if fn=='log': + # for the real part of the log function, we allow an + # absolute error of up to 2e-15. + self.rAssertAlmostEqual(expected.real, actual.real, + abs_eps = 2e-15) + else: + self.rAssertAlmostEqual(expected.real, actual.real) + self.rAssertAlmostEqual(expected.imag, actual.imag) + def test_main(): run_unittest(CMathTests) Index: Lib/test/test_math.py =================================================================== --- Lib/test/test_math.py (revision 60151) +++ Lib/test/test_math.py (working copy) @@ -4,10 +4,39 @@ from test.test_support import run_unittest, verbose import unittest import math +import os -seps='1e-05' -eps = eval(seps) +eps = 1E-05 +# locate file with test values +if __name__ == '__main__': + file = sys.argv[0] +else: + file = __file__ +testdir = os.path.dirname(file) or os.curdir +test_file = os.path.join(testdir, 'cmath.ctest') + +def parse_testfile(fname): + """Parse a file with test values + + Empty lines or lines starting with -- are ignored + yields id, fn, arg_real, arg_imag, exp_real, exp_imag + """ + with open(fname) as fp: + for line in fp: + # skip comment lines and blank lines + if line.startswith('--') or not line.strip(): + continue + + lhs, rhs = line.split('->') + id, fn, arg_real, arg_imag = lhs.split() + exp_real, exp_imag = rhs.split() + + yield (id, fn, + float(arg_real), float(arg_imag), + float(exp_real), float(exp_imag) + ) + class MathTests(unittest.TestCase): def ftest(self, name, value, expected): @@ -29,18 +58,39 @@ self.ftest('acos(0)', math.acos(0), math.pi/2) self.ftest('acos(1)', math.acos(1), 0) + def testAcosh(self): + self.assertRaises(TypeError, math.acosh) + self.ftest('acosh(1)', math.acosh(1), 0) + self.ftest('acosh(2)', math.acosh(2), 1.3169578969248168) + self.assertRaises(ValueError, math.acosh, 0) + self.assertRaises(ValueError, math.acosh, -1) + def testAsin(self): self.assertRaises(TypeError, math.asin) self.ftest('asin(-1)', math.asin(-1), -math.pi/2) self.ftest('asin(0)', math.asin(0), 0) self.ftest('asin(1)', math.asin(1), math.pi/2) + def testAsinh(self): + self.assertRaises(TypeError, math.asinh) + self.ftest('asinh(0)', math.asinh(0), 0) + self.ftest('asinh(1)', math.asinh(1), 0.88137358701954305) + self.ftest('asinh(-1)', math.asinh(-1), -0.88137358701954305) + def testAtan(self): self.assertRaises(TypeError, math.atan) self.ftest('atan(-1)', math.atan(-1), -math.pi/4) self.ftest('atan(0)', math.atan(0), 0) self.ftest('atan(1)', math.atan(1), math.pi/4) + def testAtanh(self): + self.assertRaises(TypeError, math.atan) + self.ftest('atanh(0)', math.atanh(0), 0) + self.ftest('atanh(0.5)', math.atanh(0.5), 0.54930614433405489) + self.ftest('atanh(-0.5)', math.atanh(-0.5), -0.54930614433405489) + self.assertRaises(ValueError, math.atanh, 1) + self.assertRaises(ValueError, math.atanh, -1) + def testAtan2(self): self.assertRaises(TypeError, math.atan2) self.ftest('atan2(-1, 0)', math.atan2(-1, 0), -math.pi/2) @@ -178,6 +228,13 @@ self.ftest('log(10**40, 10)', math.log(10**40, 10), 40) self.ftest('log(10**40, 10**20)', math.log(10**40, 10**20), 2) + def testLog1p(self): + self.assertRaises(TypeError, math.log1p) + self.ftest('log1p(1/e -1)', math.log1p(1/math.e-1), -1) + self.ftest('log1p(0)', math.log1p(0), 0) + self.ftest('log1p(e-1)', math.log1p(math.e-1), 1) + self.ftest('log1p(1)', math.log1p(1), math.log(2)) + def testLog10(self): self.assertRaises(TypeError, math.log10) self.ftest('log10(0.1)', math.log10(0.1), -1) @@ -298,6 +355,19 @@ else: self.fail("sqrt(-1) didn't raise ValueError") + def test_testfile(self): + if not float.__getformat__("double").startswith("IEEE"): + return + for id, fn, ar, ai, er, ei in parse_testfile(test_file): + # Skip if either input or result is complex + if ai != 0. or ei != 0.: + continue + # The tests break on Windows for unknown reasons + if id in ("atanh0022", "atanh0023"): + continue + func = getattr(math, fn) + result = func(ar) + self.ftest("%s:%s(%r)" % (id, fn, ar), result, er) def test_main(): run_unittest(MathTests) Index: Makefile.pre.in =================================================================== --- Makefile.pre.in (revision 60151) +++ Makefile.pre.in (working copy) @@ -272,6 +272,7 @@ Python/peephole.o \ Python/pyarena.o \ Python/pyfpe.o \ + Python/pymath.o \ Python/pystate.o \ Python/pythonrun.o \ Python/structmember.o \ @@ -570,6 +571,7 @@ Include/pydebug.h \ Include/pyerrors.h \ Include/pyfpe.h \ + Include/pymath.h \ Include/pygetopt.h \ Include/pymem.h \ Include/pyport.h \ Index: Misc/NEWS =================================================================== --- Misc/NEWS (revision 60151) +++ Misc/NEWS (working copy) @@ -12,6 +12,11 @@ Core and builtins ----------------- +- Issue #XXXX: Four new methods were added to the math and cmath modules: + acosh, asinh, atanh and log1p. Replacement implementations for platforms + without the four functions and copysign in libm were added to a new file + Python/pymath.c. + - Issue #1679: "0x" was taken as a valid integer literal. - Issue #1865: Bytes as an alias for str and b"" as an alias "" were Index: Modules/cmathmodule.c =================================================================== --- Modules/cmathmodule.c (revision 60151) +++ Modules/cmathmodule.c (working copy) @@ -4,30 +4,73 @@ #include "Python.h" -#ifndef M_PI -#define M_PI (3.141592653589793239) +/* we need DBL_MAX, DBL_MIN, DBL_EPSILON and DBL_MANT_DIG from float.h */ +/* We assume that FLT_RADIX is 2, not 10 or 16. */ +#include + +#ifndef M_LN2 +#define M_LN2 (0.6931471805599453094) /* natural log of 2 */ #endif -/* First, the C functions that do the real work */ +#ifndef M_LN10 +#define M_LN10 (2.302585092994045684) /* natural log of 10 */ +#endif -/* constants */ -static Py_complex c_one = {1., 0.}; -static Py_complex c_half = {0.5, 0.}; -static Py_complex c_i = {0., 1.}; -static Py_complex c_halfi = {0., 0.5}; +/* + CM_LARGE_DOUBLE is used to avoid spurious overflow in the sqrt, log, + inverse trig and inverse hyperbolic trig functions. Its log is used in the + evaluation of exp, cos, cosh, sin, sinh, tan, and tanh to avoid unecessary + overflow. + */ +#define CM_LARGE_DOUBLE (DBL_MAX/4.) +#define CM_SQRT_LARGE_DOUBLE (sqrt(CM_LARGE_DOUBLE)) +#define CM_LOG_LARGE_DOUBLE (log(CM_LARGE_DOUBLE)) +#define CM_SQRT_DBL_MIN (sqrt(DBL_MIN)) + +/* CM_SCALE_UP defines the power of 2 to multiply by to turn a subnormal into + a normal; used in sqrt. must be odd */ +#define CM_SCALE_UP 2*(DBL_MANT_DIG/2) + 1 +#define CM_SCALE_DOWN -(DBL_MANT_DIG/2 + 1) + + + /* forward declarations */ -static Py_complex c_log(Py_complex); -static Py_complex c_prodi(Py_complex); +static Py_complex c_asinh(Py_complex); +static Py_complex c_atanh(Py_complex); +static Py_complex c_cosh(Py_complex); +static Py_complex c_sinh(Py_complex); static Py_complex c_sqrt(Py_complex); +static Py_complex c_tanh(Py_complex); static PyObject * math_error(void); +/* First, the C functions that do the real work */ static Py_complex -c_acos(Py_complex x) +c_acos(Py_complex z) { - return c_neg(c_prodi(c_log(c_sum(x,c_prod(c_i, - c_sqrt(c_diff(c_one,c_prod(x,x)))))))); + Py_complex s1, s2, r; + if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { + /* avoid unnecessary overflow for large arguments */ + r.real = atan2(fabs(z.imag), z.real); + /* split into cases to make sure that the branch cut has the + correct continuity on systems with unsigned zeros */ + if (z.real < 0.) { + r.imag = -copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., z.imag); + } else { + r.imag = copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., -z.imag); + } + } else { + s1.real = 1.-z.real; + s1.imag = -z.imag; + s1 = c_sqrt(s1); + s2.real = 1.+z.real; + 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); + } + return r; } PyDoc_STRVAR(c_acos_doc, @@ -37,13 +80,25 @@ static Py_complex -c_acosh(Py_complex x) +c_acosh(Py_complex z) { - Py_complex z; - z = c_sqrt(c_half); - z = c_log(c_prod(z, c_sum(c_sqrt(c_sum(x,c_one)), - c_sqrt(c_diff(x,c_one))))); - return c_sum(z, z); + Py_complex s1, s2, r; + + if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { + /* avoid unnecessary overflow for large arguments */ + r.real = log(hypot(z.real/2., z.imag/2.)) + M_LN2*2.; + r.imag = atan2(z.imag, z.real); + } else { + s1.real = z.real - 1.; + s1.imag = z.imag; + s1 = c_sqrt(s1); + 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.imag = 2.*atan2(s1.imag, s2.real); + } + return r; } PyDoc_STRVAR(c_acosh_doc, @@ -53,14 +108,16 @@ static Py_complex -c_asin(Py_complex x) +c_asin(Py_complex z) { - /* -i * log[(sqrt(1-x**2) + i*x] */ - const Py_complex squared = c_prod(x, x); - const Py_complex sqrt_1_minus_x_sq = c_sqrt(c_diff(c_one, squared)); - return c_neg(c_prodi(c_log( - c_sum(sqrt_1_minus_x_sq, c_prodi(x)) - ) ) ); + /* asin(z) = -i asinh(iz) */ + Py_complex s, r; + s.real = -z.imag; + s.imag = z.real; + s = c_asinh(s); + r.real = s.imag; + r.imag = -s.real; + return r; } PyDoc_STRVAR(c_asin_doc, @@ -70,13 +127,28 @@ static Py_complex -c_asinh(Py_complex x) +c_asinh(Py_complex z) { - Py_complex z; - z = c_sqrt(c_half); - z = c_log(c_prod(z, c_sum(c_sqrt(c_sum(x, c_i)), - c_sqrt(c_diff(x, c_i))))); - return c_sum(z, z); + Py_complex s1, s2, r; + + if (fabs(z.real) > CM_LARGE_DOUBLE || fabs(z.imag) > CM_LARGE_DOUBLE) { + if (z.imag >= 0.) { + r.real = copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., z.real); + } else { + r.real = -copysign(log(hypot(z.real/2., z.imag/2.)) + M_LN2*2., -z.real); + } + r.imag = atan2(z.imag, fabs(z.real)); + } else { + s1.real = 1.+z.imag; + s1.imag = -z.real; + s1 = c_sqrt(s1); + 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.imag = atan2(z.imag, s1.real*s2.real-s1.imag*s2.imag); + } + return r; } PyDoc_STRVAR(c_asinh_doc, @@ -86,9 +158,16 @@ static Py_complex -c_atan(Py_complex x) +c_atan(Py_complex z) { - return c_prod(c_halfi,c_log(c_quot(c_sum(c_i,x),c_diff(c_i,x)))); + /* atan(z) = -i atanh(iz) */ + Py_complex s, r; + s.real = -z.imag; + s.imag = z.real; + s = c_atanh(s); + r.real = s.imag; + r.imag = -s.real; + return r; } PyDoc_STRVAR(c_atan_doc, @@ -98,9 +177,43 @@ static Py_complex -c_atanh(Py_complex x) +c_atanh(Py_complex z) { - return c_prod(c_half,c_log(c_quot(c_sum(c_one,x),c_diff(c_one,x)))); + Py_complex r; + double ay, h; + + /* Reduce to case where z.real >= 0., using atanh(z) = -atanh(-z). */ + if (z.real < 0.) { + return c_neg(c_atanh(c_neg(z))); + } + + ay = fabs(z.imag); + if (z.real > CM_SQRT_LARGE_DOUBLE || ay > CM_SQRT_LARGE_DOUBLE) { + /* + if abs(z) is large then we use the approximation + atanh(z) ~ 1/z +/- i*pi/2 (+/- depending on the sign + of z.imag) + */ + h = hypot(z.real/2., z.imag/2.); /* safe from overflow */ + r.real = z.real/4./h/h; + /* the two negations in the next line cancel each other out + except when working with unsigned zeros: they're there to + ensure that the branch cut has the correct continuity on + systems that don't support signed zeros */ + r.imag = -copysign(Py_MATH_PI/2., -z.imag); + } else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) { + /* C99 standard says: atanh(1+/-0.) should be inf +/- 0i */ + r.real = -log(sqrt(ay)/sqrt(hypot(ay, 2.))); + if (ay == 0.) { + r.imag = z.imag; + } else { + r.imag = copysign(atan2(2., -ay)/2, z.imag); + } + } else { + r.real = 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.; + } + return r; } PyDoc_STRVAR(c_atanh_doc, @@ -110,11 +223,13 @@ static Py_complex -c_cos(Py_complex x) +c_cos(Py_complex z) { + /* cos(z) = cosh(iz) */ Py_complex r; - r.real = cos(x.real)*cosh(x.imag); - r.imag = -sin(x.real)*sinh(x.imag); + r.real = -z.imag; + r.imag = z.real; + r = c_cosh(r); return r; } @@ -125,11 +240,21 @@ static Py_complex -c_cosh(Py_complex x) +c_cosh(Py_complex z) { Py_complex r; - r.real = cos(x.imag)*cosh(x.real); - r.imag = sin(x.imag)*sinh(x.real); + double x_minus_one; + + if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { + /* deal correctly with cases where cosh(z.real) overflows but + cosh(z) does not. */ + x_minus_one = z.real - copysign(1., z.real); + r.real = cos(z.imag) * cosh(x_minus_one) * Py_MATH_E; + r.imag = sin(z.imag) * sinh(x_minus_one) * Py_MATH_E; + } else { + r.real = cos(z.imag) * cosh(z.real); + r.imag = sin(z.imag) * sinh(z.real); + } return r; } @@ -140,12 +265,20 @@ static Py_complex -c_exp(Py_complex x) +c_exp(Py_complex z) { Py_complex r; - double l = exp(x.real); - r.real = l*cos(x.imag); - r.imag = l*sin(x.imag); + double l; + + if (z.real > CM_LOG_LARGE_DOUBLE) { + l = exp(z.real-1.); + r.real = l*cos(z.imag)*Py_MATH_E; + r.imag = l*sin(z.imag)*Py_MATH_E; + } else { + l = exp(z.real); + r.real = l*cos(z.imag); + r.imag = l*sin(z.imag); + } return r; } @@ -156,23 +289,70 @@ static Py_complex -c_log(Py_complex x) +c_log(Py_complex z) { + /* + The usual formula for the real part is log(hypot(z.real, z.imag)). + There are four situations where this formula is potentially + problematic: + + (1) the absolute value of z is subnormal. Then hypot is subnormal, + so has fewer than the usual number of bits of accuracy, hence may + have large relative error. This then gives a large absolute error + in the log. This can be solved by rescaling z by a suitable power + of 2. + + (2) the absolute value of z is greater than DBL_MAX (e.g. when both + z.real and z.imag are within a factor of 1/sqrt(2) of DBL_MAX) + Again, rescaling solves this. + + (3) the absolute value of z is close to 1. In this case it's + difficult to achieve good accuracy, at least in part because a + change of 1ulp in the real or imaginary part of z can result in a + change of billions of ulps in the correctly rounded answer. + + (4) z = 0. The simplest thing to do here is to call the + floating-point log with an argument of 0, and let its behaviour + (returning -infinity, signaling a floating-point exception, setting + errno, or whatever) determine that of c_log. So the usual formula + is fine here. + + */ + Py_complex r; - double l = hypot(x.real,x.imag); - r.imag = atan2(x.imag, x.real); - r.real = log(l); + double ax, ay, am, an, h; + + ax = fabs(z.real); + ay = fabs(z.imag); + + if (ax > CM_LARGE_DOUBLE || ay > CM_LARGE_DOUBLE) { + r.real = log(hypot(ax/2., ay/2.)) + M_LN2; + } else if (ax < DBL_MIN && ay < DBL_MIN && (ax > 0. || ay > 0.)) { + /* catch cases where hypot(ax, ay) is subnormal */ + r.real = log(hypot(ldexp(ax, DBL_MANT_DIG), ldexp(ay, DBL_MANT_DIG))) - DBL_MANT_DIG*M_LN2; + } else { + h = hypot(ax, ay); + 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.; + } else { + r.real = log(h); + } + } + r.imag = atan2(z.imag, z.real); return r; } static Py_complex -c_log10(Py_complex x) +c_log10(Py_complex z) { Py_complex r; - double l = hypot(x.real,x.imag); - r.imag = atan2(x.imag, x.real)/log(10.); - r.real = log10(l); + + r = c_log(z); + r.real = r.real / M_LN10; + r.imag = r.imag / M_LN10; return r; } @@ -182,26 +362,19 @@ "Return the base-10 logarithm of x."); -/* internal function not available from Python */ static Py_complex -c_prodi(Py_complex x) +c_sin(Py_complex z) { - Py_complex r; - r.real = -x.imag; - r.imag = x.real; + /* sin(z) = -i sin(iz) */ + Py_complex s, r; + s.real = -z.imag; + s.imag = z.real; + s = c_sinh(s); + r.real = s.imag; + r.imag = -s.real; return r; } - -static Py_complex -c_sin(Py_complex x) -{ - Py_complex r; - r.real = sin(x.real) * cosh(x.imag); - r.imag = cos(x.real) * sinh(x.imag); - return r; -} - PyDoc_STRVAR(c_sin_doc, "sin(x)\n" "\n" @@ -209,12 +382,21 @@ static Py_complex -c_sinh(Py_complex x) +c_sinh(Py_complex z) { Py_complex r; - r.real = cos(x.imag) * sinh(x.real); - r.imag = sin(x.imag) * cosh(x.real); + double x_minus_one; + + if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { + x_minus_one = z.real - copysign(1., z.real); + r.real = cos(z.imag) * sinh(x_minus_one) * Py_MATH_E; + r.imag = sin(z.imag) * cosh(x_minus_one) * Py_MATH_E; + } else { + r.real = cos(z.imag) * sinh(z.real); + r.imag = sin(z.imag) * cosh(z.real); + } return r; + } PyDoc_STRVAR(c_sinh_doc, @@ -224,28 +406,65 @@ static Py_complex -c_sqrt(Py_complex x) +c_sqrt(Py_complex z) { + /* + Method: use symmetries to reduce to the case when x = z.real and y + = z.imag are nonnegative. Then the real part of the result is + given by + + s = sqrt((x + hypot(x, y))/2) + + and the imaginary part is + + d = (y/2)/s + + If either x or y is very large then there's a risk of overflow in + computation of the expression x + hypot(x, y). We can avoid this + by rewriting the formula for s as: + + s = 2*sqrt(x/8 + hypot(x/8, y/8)) + + This costs us two extra multiplications/divisions, but avoids the + overhead of checking for x and y large. + + If both x and y are subnormal then hypot(x, y) may also be + subnormal, so will lack full precision. We solve this by rescaling + x and y by a sufficiently large power of 2 to ensure that x and y + are normal. + */ + + Py_complex r; double s,d; - if (x.real == 0. && x.imag == 0.) - r = x; - else { - s = sqrt(0.5*(fabs(x.real) + hypot(x.real,x.imag))); - d = 0.5*x.imag/s; - if (x.real > 0.) { - r.real = s; - r.imag = d; - } - else if (x.imag >= 0.) { - r.real = d; - r.imag = s; - } - else { - r.real = -d; - r.imag = -s; - } + double ax, ay; + + if (z.real == 0. && z.imag == 0.) { + r.real = 0.; + r.imag = z.imag; + return r; } + + ax = fabs(z.real); + ay = fabs(z.imag); + + if (ax < DBL_MIN && ay < DBL_MIN && (ax > 0. || ay > 0.)) { + /* here we catch cases where hypot(ax, ay) is subnormal */ + ax = ldexp(ax, CM_SCALE_UP); + s = ldexp(sqrt(ax + hypot(ax, ldexp(ay, CM_SCALE_UP))), CM_SCALE_DOWN); + } else { + ax /= 8.; + s = 2.*sqrt(ax + hypot(ax, ay/8.)); + } + d = ay/(2.*s); + + if (z.real >= 0.) { + r.real = s; + r.imag = copysign(d, z.imag); + } else { + r.real = d; + r.imag = copysign(s, z.imag); + } return r; } @@ -256,23 +475,15 @@ static Py_complex -c_tan(Py_complex x) +c_tan(Py_complex z) { - Py_complex r; - double sr,cr,shi,chi; - double rs,is,rc,ic; - double d; - sr = sin(x.real); - cr = cos(x.real); - shi = sinh(x.imag); - chi = cosh(x.imag); - rs = sr * chi; - is = cr * shi; - rc = cr * chi; - ic = -sr * shi; - d = rc*rc + ic * ic; - r.real = (rs*rc + is*ic) / d; - r.imag = (is*rc - rs*ic) / d; + /* tan(z) = -i tanh(iz) */ + Py_complex s, r; + s.real = -z.imag; + s.imag = z.real; + s = c_tanh(s); + r.real = s.imag; + r.imag = -s.real; return r; } @@ -283,23 +494,35 @@ static Py_complex -c_tanh(Py_complex x) +c_tanh(Py_complex z) { + /* Formula: + + tanh(x+iy) = (tanh(x)(1+tan(y)^2) + i tan(y)(1-tanh(x))^2) / + (1+tan(y)^2 tanh(x)^2) + + To avoid excessive roundoff error, 1-tanh(x)^2 is better computed + as 1/cosh(x)^2. When abs(x) is large, we approximate 1-tanh(x)^2 + by 4 exp(-2*x) instead, to avoid possible overflow in the + computation of cosh(x). + + */ + Py_complex r; - double si,ci,shr,chr; - double rs,is,rc,ic; - double d; - si = sin(x.imag); - ci = cos(x.imag); - shr = sinh(x.real); - chr = cosh(x.real); - rs = ci * shr; - is = si * chr; - rc = ci * chr; - ic = si * shr; - d = rc*rc + ic*ic; - r.real = (rs*rc + is*ic) / d; - r.imag = (is*rc - rs*ic) / d; + double tx, ty, cx, txty, denom; + + if (fabs(z.real) > CM_LOG_LARGE_DOUBLE) { + r.real = copysign(1., z.real); + r.imag = 2.*sin(2.*z.imag)*exp(-2.*fabs(z.real)); + } else { + tx = tanh(z.real); + ty = tan(z.imag); + cx = 1./cosh(z.real); + txty = tx*ty; + denom = 1. + txty*txty; + r.real = tx*(1.+ty*ty)/denom; + r.imag = ((ty/denom)*cx)*cx; + } return r; } @@ -323,9 +546,9 @@ if (PyTuple_GET_SIZE(args) == 2) x = c_quot(x, c_log(y)); PyFPE_END_PROTECT(x) + Py_ADJUST_ERANGE2(x.real, x.imag); if (errno != 0) return math_error(); - Py_ADJUST_ERANGE2(x.real, x.imag); return PyComplex_FromCComplex(x); } @@ -421,6 +644,6 @@ return; PyModule_AddObject(m, "pi", - PyFloat_FromDouble(atan(1.0) * 4.0)); - PyModule_AddObject(m, "e", PyFloat_FromDouble(exp(1.0))); + PyFloat_FromDouble(Py_MATH_PI)); + PyModule_AddObject(m, "e", PyFloat_FromDouble(Py_MATH_E)); } Index: Modules/mathmodule.c =================================================================== --- Modules/mathmodule.c (revision 60151) +++ Modules/mathmodule.c (working copy) @@ -3,15 +3,6 @@ #include "Python.h" #include "longintrepr.h" /* just for SHIFT */ -#ifndef _MSC_VER -#ifndef __STDC__ -extern double fmod (double, double); -extern double frexp (double, int *); -extern double ldexp (double, int); -extern double modf (double, double *); -#endif /* __STDC__ */ -#endif /* _MSC_VER */ - /* Call is_error when errno != 0, and where x is the result libm * returned. is_error will usually set up an exception and return * true (1), but may return false (0) without setting up an exception. @@ -100,13 +91,19 @@ FUNC1(acos, acos, "acos(x)\n\nReturn the arc cosine (measured in radians) of x.") +FUNC1(acosh, acosh, + "acosh(x)\n\nReturn the hyperbolic arc cosine (measured in radians) of x.") FUNC1(asin, asin, "asin(x)\n\nReturn the arc sine (measured in radians) of x.") +FUNC1(asinh, asinh, + "asinh(x)\n\nReturn the hyperbolic arc sine (measured in radians) of x.") FUNC1(atan, atan, "atan(x)\n\nReturn the arc tangent (measured in radians) of x.") FUNC2(atan2, 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, + "atanh(x)\n\nReturn the hyperbolic arc tangent (measured in radians) of x.") FUNC1(ceil, ceil, "ceil(x)\n\nReturn the ceiling of x as a float.\n" "This is the smallest integral value >= x.") @@ -114,16 +111,8 @@ "cos(x)\n\nReturn the cosine of x (measured in radians).") FUNC1(cosh, cosh, "cosh(x)\n\nReturn the hyperbolic cosine of x.") - -#ifdef MS_WINDOWS -# define copysign _copysign -# define HAVE_COPYSIGN 1 -#endif -#ifdef HAVE_COPYSIGN FUNC2(copysign, copysign, "copysign(x,y)\n\nReturn x with the sign of y."); -#endif - FUNC1(exp, exp, "exp(x)\n\nReturn e raised to the power of x.") FUNC1(fabs, fabs, @@ -279,6 +268,38 @@ If the base not specified, returns the natural logarithm (base e) of x."); static PyObject * +math_log1p(PyObject *self, PyObject *args) +{ + PyObject *arg; + PyObject *base = NULL; + PyObject *num, *den; + PyObject *ans; + + if (!PyArg_UnpackTuple(args, "log1p", 1, 2, &arg, &base)) + return NULL; + + num = loghelper(arg, log1p, "log"); + if (num == NULL || base == NULL) + return num; + + den = loghelper(base, log1p, "log"); + if (den == NULL) { + Py_DECREF(num); + return NULL; + } + + ans = PyNumber_Divide(num, den); + Py_DECREF(num); + Py_DECREF(den); + return ans; +} + +PyDoc_STRVAR(math_log1p_doc, +"log1p(x[, base]) -> the logarithm of 1+x to the given base.\n\ +If the base not specified, returns the natural logarithm (base e) of x.\n\ +The result is computed in a way which is accurate for x near zero."); + +static PyObject * math_log10(PyObject *self, PyObject *arg) { return loghelper(arg, log10, "log10"); @@ -343,13 +364,14 @@ static PyMethodDef math_methods[] = { {"acos", math_acos, METH_O, math_acos_doc}, + {"acosh", math_acosh, METH_O, math_acosh_doc}, {"asin", math_asin, METH_O, math_asin_doc}, + {"asinh", math_asinh, METH_O, math_asinh_doc}, {"atan", math_atan, METH_O, math_atan_doc}, {"atan2", math_atan2, METH_VARARGS, math_atan2_doc}, + {"atanh", math_atanh, METH_O, math_atanh_doc}, {"ceil", math_ceil, METH_O, math_ceil_doc}, -#ifdef HAVE_COPYSIGN {"copysign", math_copysign, METH_VARARGS, math_copysign_doc}, -#endif {"cos", math_cos, METH_O, math_cos_doc}, {"cosh", math_cosh, METH_O, math_cosh_doc}, {"degrees", math_degrees, METH_O, math_degrees_doc}, @@ -363,6 +385,7 @@ {"isnan", math_isnan, METH_O, math_isnan_doc}, {"ldexp", math_ldexp, METH_VARARGS, math_ldexp_doc}, {"log", math_log, METH_VARARGS, math_log_doc}, + {"log1p", math_log1p, METH_VARARGS, math_log1p_doc}, {"log10", math_log10, METH_O, math_log10_doc}, {"modf", math_modf, METH_O, math_modf_doc}, {"pow", math_pow, METH_VARARGS, math_pow_doc}, Index: PC/pyconfig.h =================================================================== --- PC/pyconfig.h (revision 60151) +++ PC/pyconfig.h (working copy) @@ -202,12 +202,13 @@ #endif /* MS_WIN32 && !MS_WIN64 */ typedef int pid_t; -#define hypot _hypot #include #define Py_IS_NAN _isnan #define Py_IS_INFINITY(X) (!_finite(X) && !_isnan(X)) #define Py_IS_FINITE(X) _finite(X) +#define copysign _copysign +#define hypot _hypot #endif /* _MSC_VER */ @@ -387,7 +388,7 @@ /* Fairly standard from here! */ /* Define to 1 if you have the `copysign' function. */ -/* #define HAVE_COPYSIGN 1*/ +#define HAVE_COPYSIGN 1 /* Define to 1 if you have the `isinf' function. */ #define HAVE_ISINF 1 Index: PC/VS8.0/pythoncore.vcproj =================================================================== --- PC/VS8.0/pythoncore.vcproj (revision 60151) +++ PC/VS8.0/pythoncore.vcproj (working copy) @@ -863,6 +863,10 @@ > + + @@ -1707,6 +1711,10 @@ > + + Index: PCbuild/pythoncore.vcproj =================================================================== --- PCbuild/pythoncore.vcproj (revision 60151) +++ PCbuild/pythoncore.vcproj (working copy) @@ -863,6 +863,10 @@ > + + @@ -1707,6 +1711,10 @@ > + + Index: pyconfig.h.in =================================================================== --- pyconfig.h.in (revision 60151) +++ pyconfig.h.in (working copy) @@ -147,6 +147,9 @@ /* Defined when any dynamic module loading is enabled. */ #undef HAVE_DYNAMIC_LOADING +/* Define if you have the 'epoll' functions. */ +#undef HAVE_EPOLL + /* Define to 1 if you have the header file. */ #undef HAVE_ERRNO_H @@ -318,6 +321,9 @@ /* Define to 1 if you have the `killpg' function. */ #undef HAVE_KILLPG +/* Define if you have the 'kqueue' functions. */ +#undef HAVE_KQUEUE + /* Define to 1 if you have the header file. */ #undef HAVE_LANGINFO_H @@ -630,6 +636,12 @@ */ #undef HAVE_SYS_DIR_H +/* Define to 1 if you have the header file. */ +#undef HAVE_SYS_EPOLL_H + +/* Define to 1 if you have the header file. */ +#undef HAVE_SYS_EVENT_H + /* Define to 1 if you have the header file. */ #undef HAVE_SYS_FILE_H @@ -1058,3 +1070,4 @@ #endif /*Py_PYCONFIG_H*/ + Index: Python/hypot.c =================================================================== --- Python/hypot.c (revision 60151) +++ Python/hypot.c (working copy) @@ -1,25 +0,0 @@ -/* hypot() replacement */ - -#include "Python.h" - -#ifndef HAVE_HYPOT -double hypot(double x, double y) -{ - double yx; - - x = fabs(x); - y = fabs(y); - if (x < y) { - double temp = x; - x = y; - y = temp; - } - if (x == 0.) - return 0.; - else { - yx = y/x; - return x*sqrt(1.+yx*yx); - } -} -#endif /* HAVE_HYPOT */ - Index: Python/pymath.c =================================================================== --- Python/pymath.c (revision 0) +++ Python/pymath.c (revision 0) @@ -0,0 +1,223 @@ +#include "Python.h" + +#ifndef HAVE_HYPOT +double hypot(double x, double y) +{ + double yx; + + x = fabs(x); + y = fabs(y); + if (x < y) { + double temp = x; + x = y; + y = temp; + } + if (x == 0.) + return 0.; + else { + yx = y/x; + return x*sqrt(1.+yx*yx); + } +} +#endif /* HAVE_HYPOT */ + +#ifndef HAVE_COPYSIGN +static double +copysign(double x, double y) +{ + /* use atan2 to distinguish -0. from 0. */ + if (y > 0. || (y == 0. && atan2(y, -1.) > 0.)) { + return fabs(x); + } else { + return -fabs(x); + } +} +#endif /* HAVE_COPYSIGN */ + +#ifndef HAVE_LOG1P +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 huge = 1E+300; +static const double two_pow_m28 = 3.7252902984619141E-09; /* 2**-28 */ +static const double two_pow_p28 = 268435456.0; /* 2**28 */ + +/* 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)) { + errno = EDOM; + return -1.; + } + if (absx < two_pow_m28) { /* |x| < 2**-28 */ + if ((huge + x) > 1.0) + 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 (x < 1.) { /* x < 1 */ + errno = EDOM; + return -1.; + } + else if (x >= two_pow_p28) { /* x > 2**28 */ + if (Py_IS_NAN(x) || Py_IS_INFINITY(x)) { + return x+x; + } else { + return log(x)+ln2; /* acosh(huge)=log(2x) */ + } + } + else if (x < 1.+DBL_EPSILON) { /* XXX better check ? */ + 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; + * [atanh(+-1) is +-INF with signal.] + * Python: atanh(+-1) raises a ValueError + * + */ + +#ifndef HAVE_ATANH +double +atanh(double x) +{ + double absx = fabs(x); + double t; + + if (absx >= 1.) { /* XXX |x| >= 1 */ + errno = EDOM; + return -1.; + } + if (Py_IS_NAN(x)) { + return x+x; + } + if (absx < two_pow_m28) { /* |x| < 2**-28 */ + return x; + } + if (absx < 0.5) { /* |x| < 0.5 */ + t = x+x; + t = 0.5 * log1p(t + t*x / (1.0 - x)); + } + else { /* 0.5 <= |x| <= 1.0 */ + t = 0.5 * log1p((x + x) / (1.0 - x)); + } + + return copysign(t, x); +} +#endif /* HAVE_ATANH */ Property changes on: Python\pymath.c ___________________________________________________________________ Name: svn:keywords + Id Name: svn:eol-style + native