/* asin.c * * Inverse circular sine * * * * SYNOPSIS: * * double x, y, asin(); * * y = asin( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose sine is x. * * A rational function of the form x + x**3 P(x**2)/Q(x**2) * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is * transformed by the identity * * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -1, 1 40000 2.6e-17 7.1e-18 * IEEE -1, 1 10^6 1.9e-16 5.4e-17 * * * ERROR MESSAGES: * * message condition value returned * asin domain |x| > 1 NAN * */ /* acos() * * Inverse circular cosine * * * * SYNOPSIS: * * double x, y, acos(); * * y = acos( x ); * * * * DESCRIPTION: * * Returns radian angle between 0 and pi whose cosine * is x. * * Analytically, acos(x) = pi/2 - asin(x). However if |x| is * near 1, there is cancellation error in subtracting asin(x) * from pi/2. Hence if x < -0.5, * * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); * * or if x > +0.5, * * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -1, 1 50000 3.3e-17 8.2e-18 * IEEE -1, 1 10^6 2.2e-16 6.5e-17 * * * ERROR MESSAGES: * * message condition value returned * asin domain |x| > 1 NAN */ /* asin.c */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier */ #include "mconf.h" /* arcsin(x) = x + x^3 P(x^2)/Q(x^2) 0 <= x <= 0.625 Peak relative error = 1.2e-18 */ #if UNK static double P[6] = { 4.253011369004428248960E-3, -6.019598008014123785661E-1, 5.444622390564711410273E0, -1.626247967210700244449E1, 1.956261983317594739197E1, -8.198089802484824371615E0, }; static double Q[5] = { /* 1.000000000000000000000E0, */ -1.474091372988853791896E1, 7.049610280856842141659E1, -1.471791292232726029859E2, 1.395105614657485689735E2, -4.918853881490881290097E1, }; #endif #if DEC static short P[24] = { 0036213,0056330,0057244,0053234, 0140032,0015011,0114762,0160255, 0040656,0035130,0136121,0067313, 0141202,0014616,0170474,0101731, 0041234,0100076,0151674,0111310, 0141003,0025540,0033165,0077246, }; static short Q[20] = { /* 0040200,0000000,0000000,0000000, */ 0141153,0155310,0055360,0072530, 0041614,0177001,0027764,0101237, 0142023,0026733,0064653,0133266, 0042013,0101264,0023775,0176351, 0141504,0140420,0050660,0036543, }; #endif #if IBMPC static short P[24] = { 0x8ad3,0x0bd4,0x6b9b,0x3f71, 0x5c16,0x333e,0x4341,0xbfe3, 0x2dd9,0x178a,0xc74b,0x4015, 0x907b,0xde27,0x4331,0xc030, 0x9259,0xda77,0x9007,0x4033, 0xafd5,0x06ce,0x656c,0xc020, }; static short Q[20] = { /* 0x0000,0x0000,0x0000,0x3ff0, */ 0x0eab,0x0b5e,0x7b59,0xc02d, 0x9054,0x25fe,0x9fc0,0x4051, 0x76d7,0x6d35,0x65bb,0xc062, 0xbf9d,0x84ff,0x7056,0x4061, 0x07ac,0x0a36,0x9822,0xc048, }; #endif #if MIEEE static short P[24] = { 0x3f71,0x6b9b,0x0bd4,0x8ad3, 0xbfe3,0x4341,0x333e,0x5c16, 0x4015,0xc74b,0x178a,0x2dd9, 0xc030,0x4331,0xde27,0x907b, 0x4033,0x9007,0xda77,0x9259, 0xc020,0x656c,0x06ce,0xafd5, }; static short Q[20] = { /* 0x3ff0,0x0000,0x0000,0x0000, */ 0xc02d,0x7b59,0x0b5e,0x0eab, 0x4051,0x9fc0,0x25fe,0x9054, 0xc062,0x65bb,0x6d35,0x76d7, 0x4061,0x7056,0x84ff,0xbf9d, 0xc048,0x9822,0x0a36,0x07ac, }; #endif /* arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x)) 0 <= x <= 0.5 Peak relative error = 4.2e-18 */ #if UNK static double R[5] = { 2.967721961301243206100E-3, -5.634242780008963776856E-1, 6.968710824104713396794E0, -2.556901049652824852289E1, 2.853665548261061424989E1, }; static double S[4] = { /* 1.000000000000000000000E0, */ -2.194779531642920639778E1, 1.470656354026814941758E2, -3.838770957603691357202E2, 3.424398657913078477438E2, }; #endif #if DEC static short R[20] = { 0036102,0077034,0142164,0174103, 0140020,0036222,0147711,0044173, 0040736,0177655,0153631,0171523, 0141314,0106525,0060015,0055474, 0041344,0045422,0003630,0040344, }; static short S[16] = { /* 0040200,0000000,0000000,0000000, */ 0141257,0112425,0132772,0166136, 0042023,0010315,0075523,0175020, 0142277,0170104,0126203,0017563, 0042253,0034115,0102662,0022757, }; #endif #if IBMPC static short R[20] = { 0x9f08,0x988e,0x4fc3,0x3f68, 0x290f,0x59f9,0x0792,0xbfe2, 0x3e6a,0xbaf3,0xdff5,0x401b, 0xab68,0xac01,0x91aa,0xc039, 0x081d,0x40f3,0x8962,0x403c, }; static short S[16] = { /* 0x0000,0x0000,0x0000,0x3ff0, */ 0x5d8c,0xb6bf,0xf2a2,0xc035, 0x7f42,0xaf6a,0x6219,0x4062, 0x63ee,0x9590,0xfe08,0xc077, 0x44be,0xb0b6,0x6709,0x4075, }; #endif #if MIEEE static short R[20] = { 0x3f68,0x4fc3,0x988e,0x9f08, 0xbfe2,0x0792,0x59f9,0x290f, 0x401b,0xdff5,0xbaf3,0x3e6a, 0xc039,0x91aa,0xac01,0xab68, 0x403c,0x8962,0x40f3,0x081d, }; static short S[16] = { /* 0x3ff0,0x0000,0x0000,0x0000, */ 0xc035,0xf2a2,0xb6bf,0x5d8c, 0x4062,0x6219,0xaf6a,0x7f42, 0xc077,0xfe08,0x9590,0x63ee, 0x4075,0x6709,0xb0b6,0x44be, }; #endif /* pi/2 = PIO2 + MOREBITS. */ #ifdef DEC #define MOREBITS 5.721188726109831840122E-18 #else #define MOREBITS 6.123233995736765886130E-17 #endif #ifdef ANSIPROT extern double polevl ( double, void *, int ); extern double p1evl ( double, void *, int ); extern double c_sqrt ( double ); double c_asin ( double ); #else double c_sqrt(), polevl(), p1evl(); double c_asin(); #endif extern double PIO2, PIO4, NAN; double c_asin(x) double x; { double a, p, z, zz; short sign; if( x > 0 ) { sign = 1; a = x; } else { sign = -1; a = -x; } if( a > 1.0 ) { mtherr( "asin", DOMAIN ); return( NAN ); } if( a > 0.625 ) { /* arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x)) */ zz = 1.0 - a; p = zz * polevl( zz, R, 4)/p1evl( zz, S, 4); zz = c_sqrt(zz+zz); z = PIO4 - zz; zz = zz * p - MOREBITS; z = z - zz; z = z + PIO4; } else { if( a < 1.0e-8 ) { return(x); } zz = a * a; z = zz * polevl( zz, P, 5)/p1evl( zz, Q, 5); z = a * z + a; } if( sign < 0 ) z = -z; return(z); } double c_acos(x) double x; { if( (x < -1.0) || (x > 1.0) ) { mtherr( "acos", DOMAIN ); return( NAN ); } return PIO2 - c_asin(x) + MOREBITS; }