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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
339 lines
7.6 KiB
C
339 lines
7.6 KiB
C
/* asin.c
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*
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* Inverse circular sine
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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, asin();
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*
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* y = asin( 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 radian angle between -pi/2 and +pi/2 whose sine is x.
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*
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* A rational function of the form x + x**3 P(x**2)/Q(x**2)
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* is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is
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* transformed by the identity
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*
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* asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
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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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* DEC -1, 1 40000 2.6e-17 7.1e-18
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* IEEE -1, 1 10^6 1.9e-16 5.4e-17
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*
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*
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* ERROR MESSAGES:
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*
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* message condition value returned
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* asin domain |x| > 1 NAN
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*
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*/
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/* acos()
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*
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* Inverse circular cosine
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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, acos();
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*
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* y = acos( 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 radian angle between 0 and pi whose cosine
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* is x.
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*
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* Analytically, acos(x) = pi/2 - asin(x). However if |x| is
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* near 1, there is cancellation error in subtracting asin(x)
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* from pi/2. Hence if x < -0.5,
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*
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* acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) );
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*
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* or if x > +0.5,
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*
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* acos(x) = 2.0 * asin( sqrt((1-x)/2) ).
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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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* DEC -1, 1 50000 3.3e-17 8.2e-18
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* IEEE -1, 1 10^6 2.2e-16 6.5e-17
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*
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*
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* ERROR MESSAGES:
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*
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* message condition value returned
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* asin domain |x| > 1 NAN
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*/
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/* asin.c */
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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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/* arcsin(x) = x + x^3 P(x^2)/Q(x^2)
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0 <= x <= 0.625
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Peak relative error = 1.2e-18 */
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#if UNK
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static double P[6] = {
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4.253011369004428248960E-3,
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-6.019598008014123785661E-1,
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5.444622390564711410273E0,
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-1.626247967210700244449E1,
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1.956261983317594739197E1,
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-8.198089802484824371615E0,
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};
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static double Q[5] = {
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/* 1.000000000000000000000E0, */
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-1.474091372988853791896E1,
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7.049610280856842141659E1,
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-1.471791292232726029859E2,
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1.395105614657485689735E2,
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-4.918853881490881290097E1,
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};
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#endif
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#if DEC
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static short P[24] = {
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0036213,0056330,0057244,0053234,
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0140032,0015011,0114762,0160255,
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0040656,0035130,0136121,0067313,
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0141202,0014616,0170474,0101731,
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0041234,0100076,0151674,0111310,
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0141003,0025540,0033165,0077246,
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};
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static short Q[20] = {
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/* 0040200,0000000,0000000,0000000, */
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0141153,0155310,0055360,0072530,
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0041614,0177001,0027764,0101237,
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0142023,0026733,0064653,0133266,
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0042013,0101264,0023775,0176351,
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0141504,0140420,0050660,0036543,
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};
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#endif
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#if IBMPC
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static short P[24] = {
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0x8ad3,0x0bd4,0x6b9b,0x3f71,
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0x5c16,0x333e,0x4341,0xbfe3,
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0x2dd9,0x178a,0xc74b,0x4015,
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0x907b,0xde27,0x4331,0xc030,
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0x9259,0xda77,0x9007,0x4033,
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0xafd5,0x06ce,0x656c,0xc020,
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};
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static short Q[20] = {
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/* 0x0000,0x0000,0x0000,0x3ff0, */
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0x0eab,0x0b5e,0x7b59,0xc02d,
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0x9054,0x25fe,0x9fc0,0x4051,
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0x76d7,0x6d35,0x65bb,0xc062,
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0xbf9d,0x84ff,0x7056,0x4061,
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0x07ac,0x0a36,0x9822,0xc048,
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};
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#endif
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#if MIEEE
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static short P[24] = {
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0x3f71,0x6b9b,0x0bd4,0x8ad3,
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0xbfe3,0x4341,0x333e,0x5c16,
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0x4015,0xc74b,0x178a,0x2dd9,
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0xc030,0x4331,0xde27,0x907b,
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0x4033,0x9007,0xda77,0x9259,
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0xc020,0x656c,0x06ce,0xafd5,
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};
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static short Q[20] = {
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/* 0x3ff0,0x0000,0x0000,0x0000, */
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0xc02d,0x7b59,0x0b5e,0x0eab,
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0x4051,0x9fc0,0x25fe,0x9054,
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0xc062,0x65bb,0x6d35,0x76d7,
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0x4061,0x7056,0x84ff,0xbf9d,
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0xc048,0x9822,0x0a36,0x07ac,
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};
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#endif
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/* arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x))
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0 <= x <= 0.5
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Peak relative error = 4.2e-18 */
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#if UNK
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static double R[5] = {
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2.967721961301243206100E-3,
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-5.634242780008963776856E-1,
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6.968710824104713396794E0,
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-2.556901049652824852289E1,
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2.853665548261061424989E1,
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};
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static double S[4] = {
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/* 1.000000000000000000000E0, */
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-2.194779531642920639778E1,
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1.470656354026814941758E2,
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-3.838770957603691357202E2,
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3.424398657913078477438E2,
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};
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#endif
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#if DEC
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static short R[20] = {
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0036102,0077034,0142164,0174103,
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0140020,0036222,0147711,0044173,
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0040736,0177655,0153631,0171523,
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0141314,0106525,0060015,0055474,
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0041344,0045422,0003630,0040344,
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};
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static short S[16] = {
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/* 0040200,0000000,0000000,0000000, */
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0141257,0112425,0132772,0166136,
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0042023,0010315,0075523,0175020,
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0142277,0170104,0126203,0017563,
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0042253,0034115,0102662,0022757,
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};
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#endif
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#if IBMPC
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static short R[20] = {
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0x9f08,0x988e,0x4fc3,0x3f68,
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0x290f,0x59f9,0x0792,0xbfe2,
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0x3e6a,0xbaf3,0xdff5,0x401b,
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0xab68,0xac01,0x91aa,0xc039,
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0x081d,0x40f3,0x8962,0x403c,
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};
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static short S[16] = {
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/* 0x0000,0x0000,0x0000,0x3ff0, */
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0x5d8c,0xb6bf,0xf2a2,0xc035,
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0x7f42,0xaf6a,0x6219,0x4062,
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0x63ee,0x9590,0xfe08,0xc077,
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0x44be,0xb0b6,0x6709,0x4075,
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};
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#endif
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#if MIEEE
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static short R[20] = {
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0x3f68,0x4fc3,0x988e,0x9f08,
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0xbfe2,0x0792,0x59f9,0x290f,
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0x401b,0xdff5,0xbaf3,0x3e6a,
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0xc039,0x91aa,0xac01,0xab68,
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0x403c,0x8962,0x40f3,0x081d,
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};
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static short S[16] = {
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/* 0x3ff0,0x0000,0x0000,0x0000, */
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0xc035,0xf2a2,0xb6bf,0x5d8c,
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0x4062,0x6219,0xaf6a,0x7f42,
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0xc077,0xfe08,0x9590,0x63ee,
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0x4075,0x6709,0xb0b6,0x44be,
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};
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#endif
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/* pi/2 = PIO2 + MOREBITS. */
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#ifdef DEC
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#define MOREBITS 5.721188726109831840122E-18
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#else
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#define MOREBITS 6.123233995736765886130E-17
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#endif
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#ifdef ANSIPROT
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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 double c_sqrt ( double );
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double c_asin ( double );
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#else
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double c_sqrt(), polevl(), p1evl();
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double c_asin();
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#endif
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extern double PIO2, PIO4, NAN;
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double c_asin(x)
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double x;
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{
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double a, p, z, zz;
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short sign;
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if( x > 0 )
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{
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sign = 1;
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a = x;
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}
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else
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{
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sign = -1;
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a = -x;
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}
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if( a > 1.0 )
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{
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mtherr( "asin", DOMAIN );
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return( NAN );
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}
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if( a > 0.625 )
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{
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/* arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x)) */
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zz = 1.0 - a;
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p = zz * polevl( zz, R, 4)/p1evl( zz, S, 4);
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zz = c_sqrt(zz+zz);
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z = PIO4 - zz;
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zz = zz * p - MOREBITS;
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z = z - zz;
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z = z + PIO4;
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}
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else
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{
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if( a < 1.0e-8 )
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{
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return(x);
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}
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zz = a * a;
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z = zz * polevl( zz, P, 5)/p1evl( zz, Q, 5);
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z = a * z + a;
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}
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if( sign < 0 )
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z = -z;
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return(z);
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}
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double c_acos(x)
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double x;
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{
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if( (x < -1.0) || (x > 1.0) )
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{
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mtherr( "acos", DOMAIN );
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return( NAN );
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}
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return PIO2 - c_asin(x) + MOREBITS;
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}
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