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Diffstat (limited to 'sysdeps/ieee754/dbl-64/s_erf.c')
| -rw-r--r-- | sysdeps/ieee754/dbl-64/s_erf.c | 431 |
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diff --git a/sysdeps/ieee754/dbl-64/s_erf.c b/sysdeps/ieee754/dbl-64/s_erf.c new file mode 100644 index 0000000000..d8b6629a72 --- /dev/null +++ b/sysdeps/ieee754/dbl-64/s_erf.c @@ -0,0 +1,431 @@ +/* @(#)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. +*/ + +#if defined(LIBM_SCCS) && !defined(lint) +static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $"; +#endif + +/* 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<0 + * = 2.0 - tiny (if x <= -6) + * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else + * erf(x) = sign(x)*(1.0 - tiny) + * where + * R2(z) = degree 6 poly in z, (z=1/x^2) + * S2(z) = degree 7 poly in z + * + * Note1: + * To compute exp(-x*x-0.5625+R/S), let s be a single + * precision number and s := x; then + * -x*x = -s*s + (s-x)*(s+x) + * exp(-x*x-0.5626+R/S) = + * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); + * Note2: + * Here 4 and 5 make use of the asymptotic series + * exp(-x*x) + * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) + * x*sqrt(pi) + * We use rational approximation to approximate + * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 + * Here is the error bound for R1/S1 and R2/S2 + * |R1/S1 - f(x)| < 2**(-62.57) + * |R2/S2 - f(x)| < 2**(-61.52) + * + * 5. For inf > 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 "math.h" +#include "math_private.h" + +#ifdef __STDC__ +static const double +#else +static double +#endif +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 */ + +#ifdef __STDC__ + double __erf(double x) +#else + double __erf(x) + double x; +#endif +{ + 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; +#ifdef DO_NOT_USE_THIS + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); +#else + 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; +#endif + 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; +#ifdef DO_NOT_USE_THIS + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); +#else + 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; +#endif + 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 */ +#ifdef DO_NOT_USE_THIS + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( + ra5+s*(ra6+s*ra7)))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( + sa5+s*(sa6+s*(sa7+s*sa8))))))); +#else + 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]; +#endif + } else { /* |x| >= 1/0.35 */ +#ifdef DO_NOT_USE_THIS + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( + rb5+s*rb6))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( + sb5+s*(sb6+s*sb7)))))); +#else + 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; +#endif + } + 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; +} +weak_alias (__erf, erf) +#ifdef NO_LONG_DOUBLE +strong_alias (__erf, __erfl) +weak_alias (__erf, erfl) +#endif + +#ifdef __STDC__ + double __erfc(double x) +#else + double __erfc(x) + double x; +#endif +{ + 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; +#ifdef DO_NOT_USE_THIS + r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); + s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); +#else + 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; +#endif + 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; +#ifdef DO_NOT_USE_THIS + P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); + Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); +#else + 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; +#endif + 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*/ +#ifdef DO_NOT_USE_THIS + R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( + ra5+s*(ra6+s*ra7)))))); + S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( + sa5+s*(sa6+s*(sa7+s*sa8))))))); +#else + 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]; +#endif + } 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 */ +#ifdef DO_NOT_USE_THIS + R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( + rb5+s*rb6))))); + S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( + sb5+s*(sb6+s*sb7)))))); +#else + 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; +#endif + } + 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; + } +} +weak_alias (__erfc, erfc) +#ifdef NO_LONG_DOUBLE +strong_alias (__erfc, __erfcl) +weak_alias (__erfc, erfcl) +#endif |