12116Sjkh/* @(#)s_erf.c 5.1 93/09/24 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 62116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 72116Sjkh * Permission to use, copy, modify, and distribute this 88870Srgrimes * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 13176451Sdas#include <sys/cdefs.h> 14176451Sdas__FBSDID("$FreeBSD$"); 152116Sjkh 162116Sjkh/* double erf(double x) 172116Sjkh * double erfc(double x) 182116Sjkh * x 192116Sjkh * 2 |\ 202116Sjkh * erf(x) = --------- | exp(-t*t)dt 218870Srgrimes * sqrt(pi) \| 222116Sjkh * 0 232116Sjkh * 242116Sjkh * erfc(x) = 1-erf(x) 258870Srgrimes * Note that 262116Sjkh * erf(-x) = -erf(x) 272116Sjkh * erfc(-x) = 2 - erfc(x) 282116Sjkh * 292116Sjkh * Method: 302116Sjkh * 1. For |x| in [0, 0.84375] 312116Sjkh * erf(x) = x + x*R(x^2) 322116Sjkh * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 332116Sjkh * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 342116Sjkh * where R = P/Q where P is an odd poly of degree 8 and 352116Sjkh * Q is an odd poly of degree 10. 362116Sjkh * -57.90 372116Sjkh * | R - (erf(x)-x)/x | <= 2 382116Sjkh * 398870Srgrimes * 402116Sjkh * Remark. The formula is derived by noting 412116Sjkh * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 422116Sjkh * and that 432116Sjkh * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 442116Sjkh * is close to one. The interval is chosen because the fix 452116Sjkh * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 462116Sjkh * near 0.6174), and by some experiment, 0.84375 is chosen to 472116Sjkh * guarantee the error is less than one ulp for erf. 482116Sjkh * 492116Sjkh * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 502116Sjkh * c = 0.84506291151 rounded to single (24 bits) 512116Sjkh * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 522116Sjkh * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 532116Sjkh * 1+(c+P1(s)/Q1(s)) if x < 0 542116Sjkh * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 552116Sjkh * Remark: here we use the taylor series expansion at x=1. 562116Sjkh * erf(1+s) = erf(1) + s*Poly(s) 572116Sjkh * = 0.845.. + P1(s)/Q1(s) 582116Sjkh * That is, we use rational approximation to approximate 592116Sjkh * erf(1+s) - (c = (single)0.84506291151) 602116Sjkh * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 618870Srgrimes * where 622116Sjkh * P1(s) = degree 6 poly in s 632116Sjkh * Q1(s) = degree 6 poly in s 642116Sjkh * 658870Srgrimes * 3. For x in [1.25,1/0.35(~2.857143)], 662116Sjkh * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 672116Sjkh * erf(x) = 1 - erfc(x) 688870Srgrimes * where 692116Sjkh * R1(z) = degree 7 poly in z, (z=1/x^2) 702116Sjkh * S1(z) = degree 8 poly in z 712116Sjkh * 722116Sjkh * 4. For x in [1/0.35,28] 732116Sjkh * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 742116Sjkh * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 752116Sjkh * = 2.0 - tiny (if x <= -6) 762116Sjkh * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 772116Sjkh * erf(x) = sign(x)*(1.0 - tiny) 782116Sjkh * where 792116Sjkh * R2(z) = degree 6 poly in z, (z=1/x^2) 802116Sjkh * S2(z) = degree 7 poly in z 812116Sjkh * 822116Sjkh * Note1: 832116Sjkh * To compute exp(-x*x-0.5625+R/S), let s be a single 842116Sjkh * precision number and s := x; then 852116Sjkh * -x*x = -s*s + (s-x)*(s+x) 868870Srgrimes * exp(-x*x-0.5626+R/S) = 872116Sjkh * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 882116Sjkh * Note2: 892116Sjkh * Here 4 and 5 make use of the asymptotic series 902116Sjkh * exp(-x*x) 912116Sjkh * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 922116Sjkh * x*sqrt(pi) 932116Sjkh * We use rational approximation to approximate 942116Sjkh * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 952116Sjkh * Here is the error bound for R1/S1 and R2/S2 962116Sjkh * |R1/S1 - f(x)| < 2**(-62.57) 972116Sjkh * |R2/S2 - f(x)| < 2**(-61.52) 982116Sjkh * 992116Sjkh * 5. For inf > x >= 28 1002116Sjkh * erf(x) = sign(x) *(1 - tiny) (raise inexact) 1012116Sjkh * erfc(x) = tiny*tiny (raise underflow) if x > 0 1022116Sjkh * = 2 - tiny if x<0 1032116Sjkh * 1042116Sjkh * 7. Special case: 1052116Sjkh * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 1068870Srgrimes * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 1072116Sjkh * erfc/erf(NaN) is NaN 1082116Sjkh */ 1092116Sjkh 1102116Sjkh 1112116Sjkh#include "math.h" 1122116Sjkh#include "math_private.h" 1132116Sjkh 1142116Sjkhstatic const double 1152116Sjkhtiny = 1e-300, 1162116Sjkhhalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 1172116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 1182116Sjkhtwo = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 1192116Sjkh /* c = (float)0.84506291151 */ 1202116Sjkherx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 1212116Sjkh/* 1222116Sjkh * Coefficients for approximation to erf on [0,0.84375] 1232116Sjkh */ 1242116Sjkhefx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 1252116Sjkhefx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 1262116Sjkhpp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 1272116Sjkhpp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 1282116Sjkhpp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 1292116Sjkhpp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 1302116Sjkhpp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 1312116Sjkhqq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 1322116Sjkhqq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 1332116Sjkhqq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 1342116Sjkhqq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 1352116Sjkhqq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 1362116Sjkh/* 1378870Srgrimes * Coefficients for approximation to erf in [0.84375,1.25] 1382116Sjkh */ 1392116Sjkhpa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 1402116Sjkhpa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 1412116Sjkhpa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 1422116Sjkhpa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 1432116Sjkhpa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 1442116Sjkhpa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 1452116Sjkhpa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 1462116Sjkhqa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 1472116Sjkhqa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 1482116Sjkhqa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 1492116Sjkhqa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 1502116Sjkhqa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 1512116Sjkhqa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 1522116Sjkh/* 1532116Sjkh * Coefficients for approximation to erfc in [1.25,1/0.35] 1542116Sjkh */ 1552116Sjkhra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 1562116Sjkhra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 1572116Sjkhra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 1582116Sjkhra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 1592116Sjkhra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 1602116Sjkhra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 1612116Sjkhra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 1622116Sjkhra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 1632116Sjkhsa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 1642116Sjkhsa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 1652116Sjkhsa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 1662116Sjkhsa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 1672116Sjkhsa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 1682116Sjkhsa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 1692116Sjkhsa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 1702116Sjkhsa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 1712116Sjkh/* 1722116Sjkh * Coefficients for approximation to erfc in [1/.35,28] 1732116Sjkh */ 1742116Sjkhrb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 1752116Sjkhrb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 1762116Sjkhrb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 1772116Sjkhrb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 1782116Sjkhrb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 1792116Sjkhrb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 1802116Sjkhrb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 1812116Sjkhsb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 1822116Sjkhsb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 1832116Sjkhsb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 1842116Sjkhsb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 1852116Sjkhsb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 1862116Sjkhsb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 1872116Sjkhsb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 1882116Sjkh 18997413Salfreddouble 19097413Salfrederf(double x) 1912116Sjkh{ 1922116Sjkh int32_t hx,ix,i; 1932116Sjkh double R,S,P,Q,s,y,z,r; 1942116Sjkh GET_HIGH_WORD(hx,x); 1952116Sjkh ix = hx&0x7fffffff; 1962116Sjkh if(ix>=0x7ff00000) { /* erf(nan)=nan */ 1972116Sjkh i = ((u_int32_t)hx>>31)<<1; 1982116Sjkh return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 1992116Sjkh } 2002116Sjkh 2012116Sjkh if(ix < 0x3feb0000) { /* |x|<0.84375 */ 2022116Sjkh if(ix < 0x3e300000) { /* |x|<2**-28 */ 2038870Srgrimes if (ix < 0x00800000) 204254994Skargl return (8*x+efx8*x)/8; /* avoid spurious underflow */ 2052116Sjkh return x + efx*x; 2062116Sjkh } 2072116Sjkh z = x*x; 2082116Sjkh r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 2092116Sjkh s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 2102116Sjkh y = r/s; 2112116Sjkh return x + x*y; 2122116Sjkh } 2132116Sjkh if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 2142116Sjkh s = fabs(x)-one; 2152116Sjkh P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 2162116Sjkh Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 2172116Sjkh if(hx>=0) return erx + P/Q; else return -erx - P/Q; 2182116Sjkh } 2192116Sjkh if (ix >= 0x40180000) { /* inf>|x|>=6 */ 2202116Sjkh if(hx>=0) return one-tiny; else return tiny-one; 2212116Sjkh } 2222116Sjkh x = fabs(x); 2232116Sjkh s = one/(x*x); 2242116Sjkh if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 2252116Sjkh R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 2262116Sjkh ra5+s*(ra6+s*ra7)))))); 2272116Sjkh S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 2282116Sjkh sa5+s*(sa6+s*(sa7+s*sa8))))))); 2292116Sjkh } else { /* |x| >= 1/0.35 */ 2302116Sjkh R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 2312116Sjkh rb5+s*rb6))))); 2322116Sjkh S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 2332116Sjkh sb5+s*(sb6+s*sb7)))))); 2342116Sjkh } 2358870Srgrimes z = x; 2362116Sjkh SET_LOW_WORD(z,0); 2372116Sjkh r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 2382116Sjkh if(hx>=0) return one-r/x; else return r/x-one; 2392116Sjkh} 2402116Sjkh 24197413Salfreddouble 24297413Salfrederfc(double x) 2432116Sjkh{ 2442116Sjkh int32_t hx,ix; 2452116Sjkh double R,S,P,Q,s,y,z,r; 2462116Sjkh GET_HIGH_WORD(hx,x); 2472116Sjkh ix = hx&0x7fffffff; 2482116Sjkh if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 2492116Sjkh /* erfc(+-inf)=0,2 */ 2502116Sjkh return (double)(((u_int32_t)hx>>31)<<1)+one/x; 2512116Sjkh } 2522116Sjkh 2532116Sjkh if(ix < 0x3feb0000) { /* |x|<0.84375 */ 2542116Sjkh if(ix < 0x3c700000) /* |x|<2**-56 */ 2552116Sjkh return one-x; 2562116Sjkh z = x*x; 2572116Sjkh r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 2582116Sjkh s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 2592116Sjkh y = r/s; 2602116Sjkh if(hx < 0x3fd00000) { /* x<1/4 */ 2612116Sjkh return one-(x+x*y); 2622116Sjkh } else { 2632116Sjkh r = x*y; 2642116Sjkh r += (x-half); 2652116Sjkh return half - r ; 2662116Sjkh } 2672116Sjkh } 2682116Sjkh if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 2692116Sjkh s = fabs(x)-one; 2702116Sjkh P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 2712116Sjkh Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 2722116Sjkh if(hx>=0) { 2738870Srgrimes z = one-erx; return z - P/Q; 2742116Sjkh } else { 2752116Sjkh z = erx+P/Q; return one+z; 2762116Sjkh } 2772116Sjkh } 2782116Sjkh if (ix < 0x403c0000) { /* |x|<28 */ 2792116Sjkh x = fabs(x); 2802116Sjkh s = one/(x*x); 2812116Sjkh if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 2822116Sjkh R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 2832116Sjkh ra5+s*(ra6+s*ra7)))))); 2842116Sjkh S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 2852116Sjkh sa5+s*(sa6+s*(sa7+s*sa8))))))); 2862116Sjkh } else { /* |x| >= 1/.35 ~ 2.857143 */ 2872116Sjkh if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 2882116Sjkh R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 2892116Sjkh rb5+s*rb6))))); 2902116Sjkh S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 2912116Sjkh sb5+s*(sb6+s*sb7)))))); 2922116Sjkh } 2932116Sjkh z = x; 2942116Sjkh SET_LOW_WORD(z,0); 2952116Sjkh r = __ieee754_exp(-z*z-0.5625)* 2962116Sjkh __ieee754_exp((z-x)*(z+x)+R/S); 2972116Sjkh if(hx>0) return r/x; else return two-r/x; 2982116Sjkh } else { 2992116Sjkh if(hx>0) return tiny*tiny; else return two-tiny; 3002116Sjkh } 3012116Sjkh} 302