s_erf.c revision 268588
177701Sbrian/* @(#)s_erf.c 5.1 93/09/24 */ 285964Sbrian/* 377701Sbrian * ==================================================== 477701Sbrian * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 577701Sbrian * 677701Sbrian * Developed at SunPro, a Sun Microsystems, Inc. business. 777701Sbrian * Permission to use, copy, modify, and distribute this 877701Sbrian * software is freely granted, provided that this notice 977701Sbrian * is preserved. 1077701Sbrian * ==================================================== 1177701Sbrian */ 1277701Sbrian 1377701Sbrian#include <sys/cdefs.h> 1477701Sbrian__FBSDID("$FreeBSD: head/lib/msun/src/s_erf.c 268588 2014-07-13 15:45:45Z kargl $"); 1577701Sbrian 1677701Sbrian/* double erf(double x) 1777701Sbrian * double erfc(double x) 1877701Sbrian * x 1977701Sbrian * 2 |\ 2077701Sbrian * erf(x) = --------- | exp(-t*t)dt 2177701Sbrian * sqrt(pi) \| 2277701Sbrian * 0 2377701Sbrian * 2477701Sbrian * erfc(x) = 1-erf(x) 2577701Sbrian * Note that 2677701Sbrian * erf(-x) = -erf(x) 2784195Sdillon * erfc(-x) = 2 - erfc(x) 2884195Sdillon * 2984195Sdillon * Method: 3026026Sbrian * 1. For |x| in [0, 0.84375] 3126026Sbrian * erf(x) = x + x*R(x^2) 3226026Sbrian * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 3326026Sbrian * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 3426026Sbrian * where R = P/Q where P is an odd poly of degree 8 and 3526026Sbrian * Q is an odd poly of degree 10. 3626026Sbrian * -57.90 3726026Sbrian * | R - (erf(x)-x)/x | <= 2 3826026Sbrian * 3926026Sbrian * 4026026Sbrian * Remark. The formula is derived by noting 4126026Sbrian * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 4226026Sbrian * and that 4326026Sbrian * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 4459047Sru * is close to one. The interval is chosen because the fix 4526026Sbrian * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 4626026Sbrian * near 0.6174), and by some experiment, 0.84375 is chosen to 4726026Sbrian * guarantee the error is less than one ulp for erf. 4826026Sbrian * 4926026Sbrian * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 50131612Sdes * c = 0.84506291151 rounded to single (24 bits) 51131612Sdes * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 52131612Sdes * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 5326026Sbrian * 1+(c+P1(s)/Q1(s)) if x < 0 5426026Sbrian * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 55131612Sdes * Remark: here we use the taylor series expansion at x=1. 56131612Sdes * erf(1+s) = erf(1) + s*Poly(s) 5726026Sbrian * = 0.845.. + P1(s)/Q1(s) 5826026Sbrian * That is, we use rational approximation to approximate 59131612Sdes * erf(1+s) - (c = (single)0.84506291151) 60131612Sdes * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 61131612Sdes * where 62131612Sdes * P1(s) = degree 6 poly in s 63131612Sdes * Q1(s) = degree 6 poly in s 64131612Sdes * 65131612Sdes * 3. For x in [1.25,1/0.35(~2.857143)], 66131612Sdes * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 6726026Sbrian * erf(x) = 1 - erfc(x) 6826026Sbrian * where 69131612Sdes * R1(z) = degree 7 poly in z, (z=1/x^2) 70131612Sdes * S1(z) = degree 8 poly in z 7126026Sbrian * 7226026Sbrian * 4. For x in [1/0.35,28] 73131612Sdes * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 74131612Sdes * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 7526026Sbrian * = 2.0 - tiny (if x <= -6) 7626026Sbrian * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 77131612Sdes * erf(x) = sign(x)*(1.0 - tiny) 78131612Sdes * where 79131612Sdes * R2(z) = degree 6 poly in z, (z=1/x^2) 80131612Sdes * S2(z) = degree 7 poly in z 81131612Sdes * 82131612Sdes * Note1: 83131612Sdes * To compute exp(-x*x-0.5625+R/S), let s be a single 84131612Sdes * precision number and s := x; then 85131612Sdes * -x*x = -s*s + (s-x)*(s+x) 8626026Sbrian * exp(-x*x-0.5626+R/S) = 8726026Sbrian * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 88131612Sdes * Note2: 89131612Sdes * Here 4 and 5 make use of the asymptotic series 90131612Sdes * exp(-x*x) 91131612Sdes * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 92131612Sdes * x*sqrt(pi) 93131612Sdes * We use rational approximation to approximate 9427864Sbrian * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 9527864Sbrian * Here is the error bound for R1/S1 and R2/S2 96131612Sdes * |R1/S1 - f(x)| < 2**(-62.57) 97131612Sdes * |R2/S2 - f(x)| < 2**(-61.52) 98131612Sdes * 99131612Sdes * 5. For inf > x >= 28 10027864Sbrian * erf(x) = sign(x) *(1 - tiny) (raise inexact) 10141759Sdillon * erfc(x) = tiny*tiny (raise underflow) if x > 0 10241759Sdillon * = 2 - tiny if x<0 10341759Sdillon * 10499207Sbrian * 7. Special case: 10561861Sru * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 10661861Sru * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 10763899Sarchie * erfc/erf(NaN) is NaN 10863899Sarchie */ 10963899Sarchie 11063899Sarchie 11132377Seivind#include "math.h" 11226026Sbrian#include "math_private.h" 11326026Sbrian 114145921Sglebius/* XXX Prevent compilers from erroneously constant folding: */ 115145921Sglebiusstatic const volatile double tiny= 1e-300; 116164798Spiso 117164798Spisostatic const double 118188294Spisohalf= 0.5, 119145921Sglebiusone = 1, 12026026Sbriantwo = 2, 121162674Spiso/* c = (float)0.84506291151 */ 122145921Sglebiuserx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 123164798Spiso/* 124162674Spiso * In the domain [0, 2**-28], only the first term in the power series 125162674Spiso * expansion of erf(x) is used. The magnitude of the first neglected 126162674Spiso * terms is less than 2**-84. 127145921Sglebius */ 12826026Sbrianefx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 12926026Sbrianefx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 13026026Sbrian/* 13126026Sbrian * Coefficients for approximation to erf on [0,0.84375] 13226026Sbrian */ 13326026Sbrianpp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 13426026Sbrianpp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 13526026Sbrianpp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 136145921Sglebiuspp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 137145932Sglebiuspp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 138145921Sglebiusqq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 139162674Spisoqq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 140145921Sglebiusqq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 141162674Spisoqq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 142145932Sglebiusqq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 14326026Sbrian/* 144162674Spiso * Coefficients for approximation to erf in [0.84375,1.25] 145145921Sglebius */ 14626026Sbrianpa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 147188294Spisopa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 148188294Spisopa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 149188294Spisopa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 150188294Spisopa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 151188294Spisopa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 152188294Spisopa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 153188294Spisoqa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 154188294Spisoqa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 155188294Spisoqa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 156188294Spisoqa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 157188294Spisoqa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 158127689Sdesqa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 159127689Sdes/* 160127689Sdes * Coefficients for approximation to erfc in [1.25,1/0.35] 161131693Sdes */ 16226026Sbrianra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 163131693Sdesra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 164131693Sdesra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 165131693Sdesra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 166131693Sdesra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 167131693Sdesra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 168131693Sdesra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 169131693Sdesra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 170131693Sdessa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 171127689Sdessa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 17226026Sbriansa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 17326026Sbriansa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 17426026Sbriansa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 17526026Sbriansa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 17644307Sbriansa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 17726026Sbriansa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 17851125Sru/* 17926026Sbrian * Coefficients for approximation to erfc in [1/.35,28] 18026026Sbrian */ 18126026Sbrianrb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 18226026Sbrianrb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 18326026Sbrianrb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 18426026Sbrianrb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 185176884Spisorb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 18626026Sbrianrb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 187176884Spisorb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 18826026Sbriansb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 18926026Sbriansb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 19026026Sbriansb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 191176884Spisosb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 19226026Sbriansb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 19326026Sbriansb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 194131614Sdessb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 195127094Sdes 196176884Spisodouble 197131614Sdeserf(double x) 198176884Spiso{ 199131614Sdes int32_t hx,ix,i; 200127094Sdes double R,S,P,Q,s,y,z,r; 201127094Sdes GET_HIGH_WORD(hx,x); 202176884Spiso ix = hx&0x7fffffff; 203131614Sdes if(ix>=0x7ff00000) { /* erf(nan)=nan */ 204127094Sdes i = ((u_int32_t)hx>>31)<<1; 205127094Sdes return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 20626026Sbrian } 20726026Sbrian 20826026Sbrian if(ix < 0x3feb0000) { /* |x|<0.84375 */ 209176884Spiso if(ix < 0x3e300000) { /* |x|<2**-28 */ 21026026Sbrian if (ix < 0x00800000) 21126026Sbrian return (8*x+efx8*x)/8; /* avoid spurious underflow */ 212131614Sdes return x + efx*x; 213127094Sdes } 214176884Spiso z = x*x; 215131614Sdes r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 216176884Spiso s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 217131614Sdes y = r/s; 218127094Sdes return x + x*y; 219127094Sdes } 220176884Spiso if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 221131614Sdes s = fabs(x)-one; 222127094Sdes P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 223127094Sdes Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 22426026Sbrian if(hx>=0) return erx + P/Q; else return -erx - P/Q; 22526026Sbrian } 22626026Sbrian if (ix >= 0x40180000) { /* inf>|x|>=6 */ 22726026Sbrian if(hx>=0) return one-tiny; else return tiny-one; 22826026Sbrian } 22926026Sbrian x = fabs(x); 23099207Sbrian s = one/(x*x); 23126026Sbrian if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 23265280Sru R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 23365280Sru S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 23459726Sru s*sa8))))))); 23526026Sbrian } else { /* |x| >= 1/0.35 */ 23626026Sbrian R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 23726026Sbrian S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 23826026Sbrian } 23926026Sbrian z = x; 24026026Sbrian SET_LOW_WORD(z,0); 24126026Sbrian r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 24226026Sbrian if(hx>=0) return one-r/x; else return r/x-one; 24326026Sbrian} 24426026Sbrian 24526026Sbriandouble 24626026Sbrianerfc(double x) 24726026Sbrian{ 24826026Sbrian int32_t hx,ix; 24959356Sru double R,S,P,Q,s,y,z,r; 25026026Sbrian GET_HIGH_WORD(hx,x); 25126026Sbrian ix = hx&0x7fffffff; 25226026Sbrian if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 25326026Sbrian /* erfc(+-inf)=0,2 */ 25426026Sbrian return (double)(((u_int32_t)hx>>31)<<1)+one/x; 25526026Sbrian } 25626026Sbrian 25726026Sbrian if(ix < 0x3feb0000) { /* |x|<0.84375 */ 25859356Sru if(ix < 0x3c700000) /* |x|<2**-56 */ 25926026Sbrian return one-x; 26026026Sbrian z = x*x; 26126026Sbrian r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 26226026Sbrian s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 26326026Sbrian y = r/s; 26426026Sbrian if(hx < 0x3fd00000) { /* x<1/4 */ 26526026Sbrian return one-(x+x*y); 26626026Sbrian } else { 26726026Sbrian r = x*y; 26826026Sbrian r += (x-half); 26926026Sbrian return half - r ; 27026026Sbrian } 271127094Sdes } 272127094Sdes if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 273127094Sdes s = fabs(x)-one; 27426026Sbrian P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 275133719Sphk Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 276127094Sdes if(hx>=0) { 277131566Sphk z = one-erx; return z - P/Q; 27826026Sbrian } else { 279177098Spiso z = erx+P/Q; return one+z; 280177098Spiso } 281177098Spiso } 282177098Spiso if (ix < 0x403c0000) { /* |x|<28 */ 283177098Spiso x = fabs(x); 28459726Sru s = one/(x*x); 285127094Sdes if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 286179920Smav R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))); 28726026Sbrian S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+ 288127094Sdes s*sa8))))))); 289131566Sphk } else { /* |x| >= 1/.35 ~ 2.857143 */ 29026026Sbrian if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 29126026Sbrian R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))); 29226026Sbrian S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))); 293124621Sphk } 29426026Sbrian z = x; 295165243Spiso SET_LOW_WORD(z,0); 296165243Spiso r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 29726026Sbrian if(hx>0) return r/x; else return two-r/x; 29865280Sru } else { 29965280Sru if(hx>0) return tiny*tiny; else return two-tiny; 30026026Sbrian } 301131614Sdes} 302127094Sdes