325 lines
6.3 KiB
C
325 lines
6.3 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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*/
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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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const 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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const 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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const 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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const 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 sqrt ( double );
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double asin ( double );
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#else
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double sqrt(), polevl(), p1evl();
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double asin();
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#endif
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extern double PIO2, PIO4, NAN;
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double 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 = 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 acos(x)
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double x;
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{
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double z;
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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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if( x > 0.5 )
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{
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return( 2.0 * asin( sqrt(0.5 - 0.5*x) ) );
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}
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z = PIO4 - asin(x);
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z = z + MOREBITS;
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z = z + PIO4;
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return( z );
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}
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