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Diffstat (limited to 'sysdeps/libm-ieee754/s_erf.c')
| -rw-r--r-- | sysdeps/libm-ieee754/s_erf.c | 431 |
1 files changed, 0 insertions, 431 deletions
diff --git a/sysdeps/libm-ieee754/s_erf.c b/sysdeps/libm-ieee754/s_erf.c deleted file mode 100644 index d8b6629a72..0000000000 --- a/sysdeps/libm-ieee754/s_erf.c +++ /dev/null @@ -1,431 +0,0 @@ -/* @(#)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 |