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The Cephes Mathematical Library author Steve Moshier has released the Cephes Mathematical Library under the BSD license. This patch patch replaces the cephes license with the BSD license for all Cephes Mathematical Library code. See also: https://bugs.gentoo.org/687276 See also: https://lists.debian.org/debian-legal/2004/12/msg00295.html See also: https://github.com/deepmind/torch-cephes/blob/master/LICENSE.txt See also: https://github.com/nearform/node-cephes/blob/master/LICENSE # Conflicts: # src/math/readme.txt
365 lines
8 KiB
C
365 lines
8 KiB
C
/* log.c
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*
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* Natural logarithm
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, y, log();
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*
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* y = log( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the base e (2.718...) logarithm of x.
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*
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* The argument is separated into its exponent and fractional
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* parts. If the exponent is between -1 and +1, the logarithm
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* of the fraction is approximated by
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*
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* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
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*
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* Otherwise, setting z = 2(x-1)/x+1),
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*
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* log(x) = z + z**3 P(z)/Q(z).
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*
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE 0.5, 2.0 150000 1.44e-16 5.06e-17
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* IEEE +-MAXNUM 30000 1.20e-16 4.78e-17
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* DEC 0, 10 170000 1.8e-17 6.3e-18
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*
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* In the tests over the interval [+-MAXNUM], the logarithms
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* of the random arguments were uniformly distributed over
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* [0, MAXLOG].
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*
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* ERROR MESSAGES:
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*
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* log singularity: x = 0; returns -INFINITY
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* log domain: x < 0; returns NAN
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*/
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/*
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Cephes Math Library Release 2.8: June, 2000
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Copyright 1984, 1995, 2000 by Stephen L. Moshier
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions are met:
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1. Redistributions of source code must retain the above copyright notice,
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this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. Neither the name of the <ORGANIZATION> nor the names of its
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contributors may be used to endorse or promote products derived from
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this software without specific prior written permission.
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THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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POSSIBILITY OF SUCH DAMAGE.
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*/
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#include "mconf.h"
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static char fname[] = {"log"};
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/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
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* 1/sqrt(2) <= x < sqrt(2)
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*/
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#ifdef UNK
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static double P[] = {
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1.01875663804580931796E-4,
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4.97494994976747001425E-1,
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4.70579119878881725854E0,
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1.44989225341610930846E1,
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1.79368678507819816313E1,
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7.70838733755885391666E0,
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};
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static double Q[] = {
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/* 1.00000000000000000000E0, */
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1.12873587189167450590E1,
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4.52279145837532221105E1,
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8.29875266912776603211E1,
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7.11544750618563894466E1,
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2.31251620126765340583E1,
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};
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#endif
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#ifdef DEC
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static unsigned short P[] = {
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0037777,0127270,0162547,0057274,
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0041001,0054665,0164317,0005341,
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0041451,0034104,0031640,0105773,
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0041677,0011276,0123617,0160135,
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0041701,0126603,0053215,0117250,
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0041420,0115777,0135206,0030232,
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};
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static unsigned short Q[] = {
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/*0040200,0000000,0000000,0000000,*/
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0041220,0144332,0045272,0174241,
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0041742,0164566,0035720,0130431,
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0042246,0126327,0166065,0116357,
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0042372,0033420,0157525,0124560,
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0042271,0167002,0066537,0172303,
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0041730,0164777,0113711,0044407,
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};
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#endif
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#ifdef IBMPC
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static unsigned short P[] = {
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0x1bb0,0x93c3,0xb4c2,0x3f1a,
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0x52f2,0x3f56,0xd6f5,0x3fdf,
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0x6911,0xed92,0xd2ba,0x4012,
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0xeb2e,0xc63e,0xff72,0x402c,
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0xc84d,0x924b,0xefd6,0x4031,
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0xdcf8,0x7d7e,0xd563,0x401e,
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};
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static unsigned short Q[] = {
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/*0x0000,0x0000,0x0000,0x3ff0,*/
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0xef8e,0xae97,0x9320,0x4026,
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0xc033,0x4e19,0x9d2c,0x4046,
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0xbdbd,0xa326,0xbf33,0x4054,
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0xae21,0xeb5e,0xc9e2,0x4051,
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0x25b2,0x9e1f,0x200a,0x4037,
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};
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#endif
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#ifdef MIEEE
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static unsigned short P[] = {
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0x3f1a,0xb4c2,0x93c3,0x1bb0,
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0x3fdf,0xd6f5,0x3f56,0x52f2,
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0x4012,0xd2ba,0xed92,0x6911,
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0x402c,0xff72,0xc63e,0xeb2e,
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0x4031,0xefd6,0x924b,0xc84d,
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0x401e,0xd563,0x7d7e,0xdcf8,
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};
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static unsigned short Q[] = {
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/*0x3ff0,0x0000,0x0000,0x0000,*/
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0x4026,0x9320,0xae97,0xef8e,
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0x4046,0x9d2c,0x4e19,0xc033,
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0x4054,0xbf33,0xa326,0xbdbd,
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0x4051,0xc9e2,0xeb5e,0xae21,
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0x4037,0x200a,0x9e1f,0x25b2,
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};
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#endif
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/* Coefficients for log(x) = z + z**3 P(z)/Q(z),
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* where z = 2(x-1)/(x+1)
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* 1/sqrt(2) <= x < sqrt(2)
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*/
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#ifdef UNK
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static double R[3] = {
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-7.89580278884799154124E-1,
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1.63866645699558079767E1,
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-6.41409952958715622951E1,
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};
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static double S[3] = {
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/* 1.00000000000000000000E0,*/
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-3.56722798256324312549E1,
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3.12093766372244180303E2,
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-7.69691943550460008604E2,
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};
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#endif
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#ifdef DEC
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static unsigned short R[12] = {
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0140112,0020756,0161540,0072035,
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0041203,0013743,0114023,0155527,
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0141600,0044060,0104421,0050400,
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};
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static unsigned short S[12] = {
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/*0040200,0000000,0000000,0000000,*/
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0141416,0130152,0017543,0064122,
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0042234,0006000,0104527,0020155,
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0142500,0066110,0146631,0174731,
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};
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#endif
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#ifdef IBMPC
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static unsigned short R[12] = {
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0x0e84,0xdc6c,0x443d,0xbfe9,
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0x7b6b,0x7302,0x62fc,0x4030,
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0x2a20,0x1122,0x0906,0xc050,
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};
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static unsigned short S[12] = {
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/*0x0000,0x0000,0x0000,0x3ff0,*/
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0x6d0a,0x43ec,0xd60d,0xc041,
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0xe40e,0x112a,0x8180,0x4073,
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0x3f3b,0x19b3,0x0d89,0xc088,
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};
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#endif
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#ifdef MIEEE
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static unsigned short R[12] = {
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0xbfe9,0x443d,0xdc6c,0x0e84,
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0x4030,0x62fc,0x7302,0x7b6b,
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0xc050,0x0906,0x1122,0x2a20,
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};
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static unsigned short S[12] = {
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/*0x3ff0,0x0000,0x0000,0x0000,*/
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0xc041,0xd60d,0x43ec,0x6d0a,
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0x4073,0x8180,0x112a,0xe40e,
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0xc088,0x0d89,0x19b3,0x3f3b,
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};
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#endif
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#ifdef ANSIPROT
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extern double frexp ( double, int * );
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extern double ldexp ( double, int );
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extern double polevl ( double, void *, int );
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extern double p1evl ( double, void *, int );
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extern int isnan ( double );
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extern int isfinite ( double );
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#else
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double frexp(), ldexp(), polevl(), p1evl();
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int isnan(), isfinite();
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#endif
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#define SQRTH 0.70710678118654752440
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extern double INFINITY, NAN;
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double c_log(x)
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double x;
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{
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int e;
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#ifdef DEC
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short *q;
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#endif
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double y, z;
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#ifdef NANS
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if( isnan(x) )
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return(x);
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#endif
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#ifdef INFINITIES
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if( x == INFINITY )
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return(x);
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#endif
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/* Test for domain */
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if( x <= 0.0 )
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{
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if( x == 0.0 )
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{
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mtherr( fname, SING );
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return( -INFINITY );
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}
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else
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{
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mtherr( fname, DOMAIN );
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return( NAN );
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}
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}
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/* separate mantissa from exponent */
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#ifdef DEC
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q = (short *)&x;
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e = *q; /* short containing exponent */
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e = ((e >> 7) & 0377) - 0200; /* the exponent */
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*q &= 0177; /* strip exponent from x */
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*q |= 040000; /* x now between 0.5 and 1 */
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#endif
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/* Note, frexp is used so that denormal numbers
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* will be handled properly.
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*/
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#ifdef IBMPC
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x = frexp( x, &e );
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/*
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q = (short *)&x;
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q += 3;
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e = *q;
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e = ((e >> 4) & 0x0fff) - 0x3fe;
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*q &= 0x0f;
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*q |= 0x3fe0;
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*/
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#endif
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/* Equivalent C language standard library function: */
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#ifdef UNK
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x = frexp( x, &e );
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#endif
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#ifdef MIEEE
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x = frexp( x, &e );
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#endif
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/* logarithm using log(x) = z + z**3 P(z)/Q(z),
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* where z = 2(x-1)/x+1)
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*/
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if( (e > 2) || (e < -2) )
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{
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if( x < SQRTH )
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{ /* 2( 2x-1 )/( 2x+1 ) */
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e -= 1;
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z = x - 0.5;
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y = 0.5 * z + 0.5;
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}
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else
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{ /* 2 (x-1)/(x+1) */
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z = x - 0.5;
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z -= 0.5;
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y = 0.5 * x + 0.5;
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}
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x = z / y;
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/* rational form */
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z = x*x;
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z = x * ( z * polevl( z, R, 2 ) / p1evl( z, S, 3 ) );
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y = e;
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z = z - y * 2.121944400546905827679e-4;
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z = z + x;
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z = z + e * 0.693359375;
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goto ldone;
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}
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/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
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if( x < SQRTH )
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{
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e -= 1;
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x = ldexp( x, 1 ) - 1.0; /* 2x - 1 */
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}
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else
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{
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x = x - 1.0;
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}
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/* rational form */
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z = x*x;
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#if DEC
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y = x * ( z * polevl( x, P, 5 ) / p1evl( x, Q, 6 ) );
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#else
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y = x * ( z * polevl( x, P, 5 ) / p1evl( x, Q, 5 ) );
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#endif
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if( e )
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y = y - e * 2.121944400546905827679e-4;
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y = y - ldexp( z, -1 ); /* y - 0.5 * z */
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z = x + y;
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if( e )
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z = z + e * 0.693359375;
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ldone:
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return( z );
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}
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