/* @(#)s_erf.c 5.1 93/09/24 */ /* * ==================================================== * 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. * ==================================================== */ /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, for performance improvement on pipelined processors. */ /* double erf(double x) * double erfc(double x) * x * 2 |\ * erf(x) = --------- | exp(-t*t)dt * sqrt(pi) \| * 0 * * erfc(x) = 1-erf(x) * Note that * erf(-x) = -erf(x) * erfc(-x) = 2 - erfc(x) * * Method: * 1. For |x| in [0, 0.84375] * erf(x) = x + x*R(x^2) * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] * where R = P/Q where P is an odd poly of degree 8 and * Q is an odd poly of degree 10. * -57.90 * | R - (erf(x)-x)/x | <= 2 * * * Remark. The formula is derived by noting * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) * and that * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 * is close to one. The interval is chosen because the fix * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is * near 0.6174), and by some experiment, 0.84375 is chosen to * guarantee the error is less than one ulp for erf. * * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and * c = 0.84506291151 rounded to single (24 bits) * erf(x) = sign(x) * (c + P1(s)/Q1(s)) * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 * 1+(c+P1(s)/Q1(s)) if x < 0 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 * Remark: here we use the taylor series expansion at x=1. * erf(1+s) = erf(1) + s*Poly(s) * = 0.845.. + P1(s)/Q1(s) * That is, we use rational approximation to approximate * erf(1+s) - (c = (single)0.84506291151) * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] * where * P1(s) = degree 6 poly in s * Q1(s) = degree 6 poly in s * * 3. For x in [1.25,1/0.35(~2.857143)], * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) * erf(x) = 1 - erfc(x) * where * R1(z) = degree 7 poly in z, (z=1/x^2) * S1(z) = degree 8 poly in z * * 4. For x in [1/0.35,28] * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 x >= 28 * erf(x) = sign(x) *(1 - tiny) (raise inexact) * erfc(x) = tiny*tiny (raise underflow) if x > 0 * = 2 - tiny if x<0 * * 7. Special case: * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, * erfc/erf(NaN) is NaN */ #include typedef int int32_t; typedef unsigned int u_int32_t; #define __ieee754_exp __mth_i_dexp extern double __ieee754_exp(double); /* LITTLE ENDIAN IEEE format */ typedef union { double value; struct { u_int32_t lsw; u_int32_t msw; } parts; } ieee_double_shape_type; /* Get the more significant 32 bit int from a double. */ #define GET_HIGH_WORD(i, d) \ do { \ ieee_double_shape_type gh_u; \ gh_u.value = (d); \ (i) = gh_u.parts.msw; \ } while (0) /* Set the less significant 32 bits of a double from an int. */ #define SET_LOW_WORD(d, v) \ do { \ ieee_double_shape_type sl_u; \ sl_u.value = (d); \ sl_u.parts.lsw = (v); \ (d) = sl_u.value; \ } while (0) static const double tiny = 1e-300, half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ /* c = (float)0.84506291151 */ erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ /* * Coefficients for approximation to erf on [0,0.84375] */ efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ pp[] = {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */ qq[] = {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */ /* * Coefficients for approximation to erf in [0.84375,1.25] */ pa[] = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */ qa[] = {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */ /* * Coefficients for approximation to erfc in [1.25,1/0.35] */ ra[] = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */ sa[] = {0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */ /* * Coefficients for approximation to erfc in [1/.35,28] */ rb[] = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */ sb[] = {0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */ double __erf(double x) { int32_t hx, ix, i; double R, S, P, Q, s, y, z, r; GET_HIGH_WORD(hx, x); ix = hx & 0x7fffffff; if (ix >= 0x7ff00000) { /* erf(nan)=nan */ i = ((u_int32_t)hx >> 31) << 1; return (double)(1 - i) + one / x; /* erf(+-inf)=+-1 */ } if (ix < 0x3feb0000) { /* |x|<0.84375 */ double r1, r2, s1, s2, s3, z2, z4; if (ix < 0x3e300000) { /* |x|<2**-28 */ if (ix < 0x00800000) return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */ return x + efx * x; } z = x * x; r1 = pp[0] + z * pp[1]; z2 = z * z; r2 = pp[2] + z * pp[3]; z4 = z2 * z2; s1 = one + z * qq[1]; s2 = qq[2] + z * qq[3]; s3 = qq[4] + z * qq[5]; r = r1 + z2 * r2 + z4 * pp[4]; s = s1 + z2 * s2 + z4 * s3; y = r / s; return x + x * y; } if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4; s = fabs(x) - one; P1 = pa[0] + s * pa[1]; s2 = s * s; Q1 = one + s * qa[1]; s4 = s2 * s2; P2 = pa[2] + s * pa[3]; s6 = s4 * s2; Q2 = qa[2] + s * qa[3]; P3 = pa[4] + s * pa[5]; Q3 = qa[4] + s * qa[5]; P4 = s6 * pa[6]; Q4 = s6 * qa[6]; P = P1 + s2 * P2 + s4 * P3 + s6 * P4; Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4; if (hx >= 0) return erx + P / Q; else return -erx - P / Q; } if (ix >= 0x40180000) { /* inf>|x|>=6 */ if (hx >= 0) return one - tiny; else return tiny - one; } x = fabs(x); s = one / (x * x); if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */ double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8; R1 = ra[0] + s * ra[1]; s2 = s * s; S1 = one + s * sa[1]; s4 = s2 * s2; R2 = ra[2] + s * ra[3]; s6 = s4 * s2; S2 = sa[2] + s * sa[3]; s8 = s4 * s4; R3 = ra[4] + s * ra[5]; S3 = sa[4] + s * sa[5]; R4 = ra[6] + s * ra[7]; S4 = sa[6] + s * sa[7]; R = R1 + s2 * R2 + s4 * R3 + s6 * R4; S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8]; } else { /* |x| >= 1/0.35 */ double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6; R1 = rb[0] + s * rb[1]; s2 = s * s; S1 = one + s * sb[1]; s4 = s2 * s2; R2 = rb[2] + s * rb[3]; s6 = s4 * s2; S2 = sb[2] + s * sb[3]; R3 = rb[4] + s * rb[5]; S3 = sb[4] + s * sb[5]; S4 = sb[6] + s * sb[7]; R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6]; S = S1 + s2 * S2 + s4 * S3 + s6 * S4; } z = x; SET_LOW_WORD(z, 0); r = __ieee754_exp(-z * z - 0.5625) * __ieee754_exp((z - x) * (z + x) + R / S); if (hx >= 0) return one - r / x; else return r / x - one; } double __erfc(double x) { int32_t hx, ix; double R, S, P, Q, s, y, z, r; GET_HIGH_WORD(hx, x); ix = hx & 0x7fffffff; if (ix >= 0x7ff00000) { /* erfc(nan)=nan */ /* erfc(+-inf)=0,2 */ return (double)(((u_int32_t)hx >> 31) << 1) + one / x; } if (ix < 0x3feb0000) { /* |x|<0.84375 */ double r1, r2, s1, s2, s3, z2, z4; if (ix < 0x3c700000) /* |x|<2**-56 */ return one - x; z = x * x; r1 = pp[0] + z * pp[1]; z2 = z * z; r2 = pp[2] + z * pp[3]; z4 = z2 * z2; s1 = one + z * qq[1]; s2 = qq[2] + z * qq[3]; s3 = qq[4] + z * qq[5]; r = r1 + z2 * r2 + z4 * pp[4]; s = s1 + z2 * s2 + z4 * s3; y = r / s; if (hx < 0x3fd00000) { /* x<1/4 */ return one - (x + x * y); } else { r = x * y; r += (x - half); return half - r; } } if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4; s = fabs(x) - one; P1 = pa[0] + s * pa[1]; s2 = s * s; Q1 = one + s * qa[1]; s4 = s2 * s2; P2 = pa[2] + s * pa[3]; s6 = s4 * s2; Q2 = qa[2] + s * qa[3]; P3 = pa[4] + s * pa[5]; Q3 = qa[4] + s * qa[5]; P4 = s6 * pa[6]; Q4 = s6 * qa[6]; P = P1 + s2 * P2 + s4 * P3 + s6 * P4; Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4; if (hx >= 0) { z = one - erx; return z - P / Q; } else { z = erx + P / Q; return one + z; } } if (ix < 0x403c0000) { /* |x|<28 */ x = fabs(x); s = one / (x * x); if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8; R1 = ra[0] + s * ra[1]; s2 = s * s; S1 = one + s * sa[1]; s4 = s2 * s2; R2 = ra[2] + s * ra[3]; s6 = s4 * s2; S2 = sa[2] + s * sa[3]; s8 = s4 * s4; R3 = ra[4] + s * ra[5]; S3 = sa[4] + s * sa[5]; R4 = ra[6] + s * ra[7]; S4 = sa[6] + s * sa[7]; R = R1 + s2 * R2 + s4 * R3 + s6 * R4; S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8]; } else { /* |x| >= 1/.35 ~ 2.857143 */ double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6; if (hx < 0 && ix >= 0x40180000) return two - tiny; /* x < -6 */ R1 = rb[0] + s * rb[1]; s2 = s * s; S1 = one + s * sb[1]; s4 = s2 * s2; R2 = rb[2] + s * rb[3]; s6 = s4 * s2; S2 = sb[2] + s * sb[3]; R3 = rb[4] + s * rb[5]; S3 = sb[4] + s * sb[5]; S4 = sb[6] + s * sb[7]; R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6]; S = S1 + s2 * S2 + s4 * S3 + s6 * S4; } z = x; SET_LOW_WORD(z, 0); r = __ieee754_exp(-z * z - 0.5625) * __ieee754_exp((z - x) * (z + x) + R / S); if (hx > 0) return r / x; else return two - r / x; } else { if (hx > 0) return tiny * tiny; else return two - tiny; } }