s_erf.c revision 254969
1235368Sgnn/* @(#)s_erf.c 5.1 93/09/24 */ 2235368Sgnn/* 3235368Sgnn * ==================================================== 4235368Sgnn * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5235368Sgnn * 6235368Sgnn * Developed at SunPro, a Sun Microsystems, Inc. business. 7235368Sgnn * Permission to use, copy, modify, and distribute this 8235368Sgnn * software is freely granted, provided that this notice 9235368Sgnn * is preserved. 10235368Sgnn * ==================================================== 11235368Sgnn */ 12235368Sgnn 13235368Sgnn#include <sys/cdefs.h> 14235368Sgnn__FBSDID("$FreeBSD: head/lib/msun/src/s_erf.c 254969 2013-08-27 19:46:56Z kargl $"); 15235368Sgnn 16235368Sgnn/* double erf(double x) 17235368Sgnn * double erfc(double x) 18235368Sgnn * x 19235368Sgnn * 2 |\ 20235368Sgnn * erf(x) = --------- | exp(-t*t)dt 21235368Sgnn * sqrt(pi) \| 22235368Sgnn * 0 23235368Sgnn * 24235368Sgnn * erfc(x) = 1-erf(x) 25235368Sgnn * Note that 26235368Sgnn * erf(-x) = -erf(x) 27235368Sgnn * erfc(-x) = 2 - erfc(x) 28235368Sgnn * 29235368Sgnn * Method: 30235368Sgnn * 1. For |x| in [0, 0.84375] 31235368Sgnn * erf(x) = x + x*R(x^2) 32235368Sgnn * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 33235368Sgnn * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 34235368Sgnn * where R = P/Q where P is an odd poly of degree 8 and 35235368Sgnn * Q is an odd poly of degree 10. 36235368Sgnn * -57.90 37235368Sgnn * | R - (erf(x)-x)/x | <= 2 38235368Sgnn * 39235368Sgnn * 40235368Sgnn * Remark. The formula is derived by noting 41235368Sgnn * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 42235368Sgnn * and that 43235368Sgnn * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 44235368Sgnn * is close to one. The interval is chosen because the fix 45235368Sgnn * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 46235368Sgnn * near 0.6174), and by some experiment, 0.84375 is chosen to 47235368Sgnn * guarantee the error is less than one ulp for erf. 48235368Sgnn * 49235368Sgnn * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 50235368Sgnn * c = 0.84506291151 rounded to single (24 bits) 51235368Sgnn * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 52235368Sgnn * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 53235368Sgnn * 1+(c+P1(s)/Q1(s)) if x < 0 54235368Sgnn * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 55235368Sgnn * Remark: here we use the taylor series expansion at x=1. 56235368Sgnn * erf(1+s) = erf(1) + s*Poly(s) 57235368Sgnn * = 0.845.. + P1(s)/Q1(s) 58235368Sgnn * That is, we use rational approximation to approximate 59235368Sgnn * erf(1+s) - (c = (single)0.84506291151) 60235368Sgnn * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 61235368Sgnn * where 62235368Sgnn * P1(s) = degree 6 poly in s 63235368Sgnn * Q1(s) = degree 6 poly in s 64235368Sgnn * 65235368Sgnn * 3. For x in [1.25,1/0.35(~2.857143)], 66235368Sgnn * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 67235368Sgnn * erf(x) = 1 - erfc(x) 68235368Sgnn * where 69235368Sgnn * R1(z) = degree 7 poly in z, (z=1/x^2) 70235368Sgnn * S1(z) = degree 8 poly in z 71235368Sgnn * 72235368Sgnn * 4. For x in [1/0.35,28] 73235368Sgnn * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 74235368Sgnn * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 75235368Sgnn * = 2.0 - tiny (if x <= -6) 76235368Sgnn * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 77235368Sgnn * erf(x) = sign(x)*(1.0 - tiny) 78235368Sgnn * where 79235368Sgnn * R2(z) = degree 6 poly in z, (z=1/x^2) 80235368Sgnn * S2(z) = degree 7 poly in z 81235368Sgnn * 82235368Sgnn * Note1: 83235368Sgnn * To compute exp(-x*x-0.5625+R/S), let s be a single 84235368Sgnn * precision number and s := x; then 85235368Sgnn * -x*x = -s*s + (s-x)*(s+x) 86235368Sgnn * exp(-x*x-0.5626+R/S) = 87235368Sgnn * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 88235368Sgnn * Note2: 89235368Sgnn * Here 4 and 5 make use of the asymptotic series 90235368Sgnn * exp(-x*x) 91235368Sgnn * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 92235368Sgnn * x*sqrt(pi) 93235368Sgnn * We use rational approximation to approximate 94235368Sgnn * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 95235368Sgnn * Here is the error bound for R1/S1 and R2/S2 96235368Sgnn * |R1/S1 - f(x)| < 2**(-62.57) 97235368Sgnn * |R2/S2 - f(x)| < 2**(-61.52) 98235368Sgnn * 99235368Sgnn * 5. For inf > x >= 28 100235368Sgnn * erf(x) = sign(x) *(1 - tiny) (raise inexact) 101235368Sgnn * erfc(x) = tiny*tiny (raise underflow) if x > 0 102235368Sgnn * = 2 - tiny if x<0 103235368Sgnn * 104235368Sgnn * 7. Special case: 105235368Sgnn * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 106235368Sgnn * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 107235368Sgnn * erfc/erf(NaN) is NaN 108235368Sgnn */ 109235368Sgnn 110235368Sgnn 111235368Sgnn#include "math.h" 112235368Sgnn#include "math_private.h" 113235368Sgnn 114235368Sgnnstatic const double 115235368Sgnntiny = 1e-300, 116235368Sgnnhalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 117235368Sgnnone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 118235368Sgnntwo = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 119235368Sgnn /* c = (float)0.84506291151 */ 120235368Sgnnerx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 121235368Sgnn/* 122235368Sgnn * Coefficients for approximation to erf on [0,0.84375] 123235368Sgnn */ 124235368Sgnnefx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 125235368Sgnnefx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 126235368Sgnnpp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 127235368Sgnnpp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 128235368Sgnnpp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 129235368Sgnnpp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 130235368Sgnnpp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 131235368Sgnnqq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 132235368Sgnnqq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 133235368Sgnnqq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 134235368Sgnnqq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 135235368Sgnnqq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 136235368Sgnn/* 137235368Sgnn * Coefficients for approximation to erf in [0.84375,1.25] 138235368Sgnn */ 139235368Sgnnpa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 140235368Sgnnpa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 141235368Sgnnpa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 142235368Sgnnpa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 143235368Sgnnpa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 144235368Sgnnpa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 145235368Sgnnpa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 146235368Sgnnqa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 147235368Sgnnqa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 148235368Sgnnqa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 149235368Sgnnqa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 150235368Sgnnqa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 151235368Sgnnqa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 152235368Sgnn/* 153235368Sgnn * Coefficients for approximation to erfc in [1.25,1/0.35] 154235368Sgnn */ 155235368Sgnnra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 156235368Sgnnra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 157235368Sgnnra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 158235368Sgnnra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 159235368Sgnnra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 160235368Sgnnra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 161235368Sgnnra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 162235368Sgnnra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 163235368Sgnnsa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 164235368Sgnnsa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 165235368Sgnnsa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 166235368Sgnnsa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 167235368Sgnnsa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 168235368Sgnnsa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 169235368Sgnnsa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 170235368Sgnnsa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 171235368Sgnn/* 172235368Sgnn * Coefficients for approximation to erfc in [1/.35,28] 173235368Sgnn */ 174235368Sgnnrb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 175235368Sgnnrb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 176235368Sgnnrb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 177235368Sgnnrb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 178235368Sgnnrb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 179235368Sgnnrb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 180235368Sgnnrb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 181235368Sgnnsb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 182235368Sgnnsb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 183235368Sgnnsb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 184235368Sgnnsb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 185235368Sgnnsb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 186235368Sgnnsb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 187235368Sgnnsb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 188235368Sgnn 189235368Sgnndouble 190235368Sgnnerf(double x) 191235368Sgnn{ 192235368Sgnn int32_t hx,ix,i; 193235368Sgnn double R,S,P,Q,s,y,z,r; 194235368Sgnn GET_HIGH_WORD(hx,x); 195235368Sgnn ix = hx&0x7fffffff; 196235368Sgnn if(ix>=0x7ff00000) { /* erf(nan)=nan */ 197235368Sgnn i = ((u_int32_t)hx>>31)<<1; 198235368Sgnn return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 199235368Sgnn } 200235368Sgnn 201235368Sgnn if(ix < 0x3feb0000) { /* |x|<0.84375 */ 202235368Sgnn if(ix < 0x3e300000) { /* |x|<2**-28 */ 203235368Sgnn if (ix < 0x00800000) 204235368Sgnn return (8*x+efx8*x)/8; /* avoid spurious underflow */ 205235368Sgnn return x + efx*x; 206235368Sgnn } 207235368Sgnn z = x*x; 208235368Sgnn r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 209235368Sgnn s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 210235368Sgnn y = r/s; 211235368Sgnn return x + x*y; 212235368Sgnn } 213235368Sgnn if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 214235368Sgnn s = fabs(x)-one; 215235368Sgnn P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 216235368Sgnn Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 217235368Sgnn if(hx>=0) return erx + P/Q; else return -erx - P/Q; 218235368Sgnn } 219235368Sgnn if (ix >= 0x40180000) { /* inf>|x|>=6 */ 220235368Sgnn if(hx>=0) return one-tiny; else return tiny-one; 221235368Sgnn } 222235368Sgnn x = fabs(x); 223235368Sgnn s = one/(x*x); 224235368Sgnn if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 225235368Sgnn R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 226235368Sgnn ra5+s*(ra6+s*ra7)))))); 227235368Sgnn S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 228235368Sgnn sa5+s*(sa6+s*(sa7+s*sa8))))))); 229235368Sgnn } else { /* |x| >= 1/0.35 */ 230235368Sgnn R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 231235368Sgnn rb5+s*rb6))))); 232235368Sgnn S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 233235368Sgnn sb5+s*(sb6+s*sb7)))))); 234235368Sgnn } 235235368Sgnn z = x; 236235368Sgnn SET_LOW_WORD(z,0); 237235368Sgnn r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 238235368Sgnn if(hx>=0) return one-r/x; else return r/x-one; 239235368Sgnn} 240235368Sgnn 241235368Sgnndouble 242235368Sgnnerfc(double x) 243235368Sgnn{ 244235368Sgnn int32_t hx,ix; 245235368Sgnn double R,S,P,Q,s,y,z,r; 246235368Sgnn GET_HIGH_WORD(hx,x); 247235368Sgnn ix = hx&0x7fffffff; 248235368Sgnn if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 249235368Sgnn /* erfc(+-inf)=0,2 */ 250235368Sgnn return (double)(((u_int32_t)hx>>31)<<1)+one/x; 251235368Sgnn } 252235368Sgnn 253235368Sgnn if(ix < 0x3feb0000) { /* |x|<0.84375 */ 254235368Sgnn if(ix < 0x3c700000) /* |x|<2**-56 */ 255235368Sgnn return one-x; 256235368Sgnn z = x*x; 257235368Sgnn r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 258235368Sgnn s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 259235368Sgnn y = r/s; 260235368Sgnn if(hx < 0x3fd00000) { /* x<1/4 */ 261235368Sgnn return one-(x+x*y); 262235368Sgnn } else { 263235368Sgnn r = x*y; 264235368Sgnn r += (x-half); 265235368Sgnn return half - r ; 266235368Sgnn } 267235368Sgnn } 268235368Sgnn if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 269235368Sgnn s = fabs(x)-one; 270235368Sgnn P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 271235368Sgnn Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 272235368Sgnn if(hx>=0) { 273235368Sgnn z = one-erx; return z - P/Q; 274235368Sgnn } else { 275235368Sgnn z = erx+P/Q; return one+z; 276235368Sgnn } 277235368Sgnn } 278235368Sgnn if (ix < 0x403c0000) { /* |x|<28 */ 279235368Sgnn x = fabs(x); 280235368Sgnn s = one/(x*x); 281235368Sgnn if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 282235368Sgnn R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 283235368Sgnn ra5+s*(ra6+s*ra7)))))); 284235368Sgnn S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 285235368Sgnn sa5+s*(sa6+s*(sa7+s*sa8))))))); 286235368Sgnn } else { /* |x| >= 1/.35 ~ 2.857143 */ 287235368Sgnn if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 288235368Sgnn R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 289235368Sgnn rb5+s*rb6))))); 290235368Sgnn S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 291235368Sgnn sb5+s*(sb6+s*sb7)))))); 292235368Sgnn } 293235368Sgnn z = x; 294235368Sgnn SET_LOW_WORD(z,0); 295235368Sgnn r = __ieee754_exp(-z*z-0.5625)* 296235368Sgnn __ieee754_exp((z-x)*(z+x)+R/S); 297235368Sgnn if(hx>0) return r/x; else return two-r/x; 298235368Sgnn } else { 299235368Sgnn if(hx>0) return tiny*tiny; else return two-tiny; 300235368Sgnn } 301235368Sgnn} 302235368Sgnn