1 // Copyright 2010 The Go Authors. All rights reserved. 2 // Use of this source code is governed by a BSD-style 3 // license that can be found in the LICENSE file. 4 5 package math 6 7 /* 8 Floating-point error function and complementary error function. 9 */ 10 11 // The original C code and the long comment below are 12 // from FreeBSD's /usr/src/lib/msun/src/s_erf.c and 13 // came with this notice. The go code is a simplified 14 // version of the original C. 15 // 16 // ==================================================== 17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 18 // 19 // Developed at SunPro, a Sun Microsystems, Inc. business. 20 // Permission to use, copy, modify, and distribute this 21 // software is freely granted, provided that this notice 22 // is preserved. 23 // ==================================================== 24 // 25 // 26 // double erf(double x) 27 // double erfc(double x) 28 // x 29 // 2 |\ 30 // erf(x) = --------- | exp(-t*t)dt 31 // sqrt(pi) \| 32 // 0 33 // 34 // erfc(x) = 1-erf(x) 35 // Note that 36 // erf(-x) = -erf(x) 37 // erfc(-x) = 2 - erfc(x) 38 // 39 // Method: 40 // 1. For |x| in [0, 0.84375] 41 // erf(x) = x + x*R(x**2) 42 // erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 43 // = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 44 // where R = P/Q where P is an odd poly of degree 8 and 45 // Q is an odd poly of degree 10. 46 // -57.90 47 // | R - (erf(x)-x)/x | <= 2 48 // 49 // 50 // Remark. The formula is derived by noting 51 // erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....) 52 // and that 53 // 2/sqrt(pi) = 1.128379167095512573896158903121545171688 54 // is close to one. The interval is chosen because the fix 55 // point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 56 // near 0.6174), and by some experiment, 0.84375 is chosen to 57 // guarantee the error is less than one ulp for erf. 58 // 59 // 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 60 // c = 0.84506291151 rounded to single (24 bits) 61 // erf(x) = sign(x) * (c + P1(s)/Q1(s)) 62 // erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 63 // 1+(c+P1(s)/Q1(s)) if x < 0 64 // |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 65 // Remark: here we use the taylor series expansion at x=1. 66 // erf(1+s) = erf(1) + s*Poly(s) 67 // = 0.845.. + P1(s)/Q1(s) 68 // That is, we use rational approximation to approximate 69 // erf(1+s) - (c = (single)0.84506291151) 70 // Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 71 // where 72 // P1(s) = degree 6 poly in s 73 // Q1(s) = degree 6 poly in s 74 // 75 // 3. For x in [1.25,1/0.35(~2.857143)], 76 // erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 77 // erf(x) = 1 - erfc(x) 78 // where 79 // R1(z) = degree 7 poly in z, (z=1/x**2) 80 // S1(z) = degree 8 poly in z 81 // 82 // 4. For x in [1/0.35,28] 83 // erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 84 // = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 85 // = 2.0 - tiny (if x <= -6) 86 // erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 87 // erf(x) = sign(x)*(1.0 - tiny) 88 // where 89 // R2(z) = degree 6 poly in z, (z=1/x**2) 90 // S2(z) = degree 7 poly in z 91 // 92 // Note1: 93 // To compute exp(-x*x-0.5625+R/S), let s be a single 94 // precision number and s := x; then 95 // -x*x = -s*s + (s-x)*(s+x) 96 // exp(-x*x-0.5626+R/S) = 97 // exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 98 // Note2: 99 // Here 4 and 5 make use of the asymptotic series 100 // exp(-x*x) 101 // erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) ) 102 // x*sqrt(pi) 103 // We use rational approximation to approximate 104 // g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625 105 // Here is the error bound for R1/S1 and R2/S2 106 // |R1/S1 - f(x)| < 2**(-62.57) 107 // |R2/S2 - f(x)| < 2**(-61.52) 108 // 109 // 5. For inf > x >= 28 110 // erf(x) = sign(x) *(1 - tiny) (raise inexact) 111 // erfc(x) = tiny*tiny (raise underflow) if x > 0 112 // = 2 - tiny if x<0 113 // 114 // 7. Special case: 115 // erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 116 // erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 117 // erfc/erf(NaN) is NaN 118 119 const ( 120 erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000 121 // Coefficients for approximation to erf in [0, 0.84375] 122 efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69 123 efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69 124 pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68 125 pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913 126 pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F 127 pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4 128 pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC 129 qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09 130 qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA 131 qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F 132 qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10 133 qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120 134 // Coefficients for approximation to erf in [0.84375, 1.25] 135 pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538 136 pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D 137 pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1 138 pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4 139 pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC 140 pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB 141 pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F 142 qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323 143 qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33 144 qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7 145 qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F 146 qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C 147 qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D 148 // Coefficients for approximation to erfc in [1.25, 1/0.35] 149 ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435 150 ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360 151 ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726 152 ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D 153 ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266 154 ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2 155 ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2 156 ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C 157 sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687 158 sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721 159 sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71 160 sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868 161 sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314 162 sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C 163 sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93 164 sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62 165 // Coefficients for approximation to erfc in [1/.35, 28] 166 rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A 167 rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE 168 rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A 169 rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98 170 rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228 171 rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992 172 rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F 173 sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190 174 sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A 175 sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118 176 sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A 177 sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6 178 sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763 179 sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62 180 ) 181 182 // Erf returns the error function of x. 183 // 184 // Special cases are: 185 // Erf(+Inf) = 1 186 // Erf(-Inf) = -1 187 // Erf(NaN) = NaN 188 func Erf(x float64) float64 { 189 const ( 190 VeryTiny = 2.848094538889218e-306 // 0x0080000000000000 191 Small = 1.0 / (1 << 28) // 2**-28 192 ) 193 // special cases 194 switch { 195 case IsNaN(x): 196 return NaN() 197 case IsInf(x, 1): 198 return 1 199 case IsInf(x, -1): 200 return -1 201 } 202 sign := false 203 if x < 0 { 204 x = -x 205 sign = true 206 } 207 if x < 0.84375 { // |x| < 0.84375 208 var temp float64 209 if x < Small { // |x| < 2**-28 210 if x < VeryTiny { 211 temp = 0.125 * (8.0*x + efx8*x) // avoid underflow 212 } else { 213 temp = x + efx*x 214 } 215 } else { 216 z := x * x 217 r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4))) 218 s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))) 219 y := r / s 220 temp = x + x*y 221 } 222 if sign { 223 return -temp 224 } 225 return temp 226 } 227 if x < 1.25 { // 0.84375 <= |x| < 1.25 228 s := x - 1 229 P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))) 230 Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))) 231 if sign { 232 return -erx - P/Q 233 } 234 return erx + P/Q 235 } 236 if x >= 6 { // inf > |x| >= 6 237 if sign { 238 return -1 239 } 240 return 1 241 } 242 s := 1 / (x * x) 243 var R, S float64 244 if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143 245 R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))) 246 S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8))))))) 247 } else { // |x| >= 1 / 0.35 ~ 2.857143 248 R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))) 249 S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))) 250 } 251 z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x 252 r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S) 253 if sign { 254 return r/x - 1 255 } 256 return 1 - r/x 257 } 258 259 // Erfc returns the complementary error function of x. 260 // 261 // Special cases are: 262 // Erfc(+Inf) = 0 263 // Erfc(-Inf) = 2 264 // Erfc(NaN) = NaN 265 func Erfc(x float64) float64 { 266 const Tiny = 1.0 / (1 << 56) // 2**-56 267 // special cases 268 switch { 269 case IsNaN(x): 270 return NaN() 271 case IsInf(x, 1): 272 return 0 273 case IsInf(x, -1): 274 return 2 275 } 276 sign := false 277 if x < 0 { 278 x = -x 279 sign = true 280 } 281 if x < 0.84375 { // |x| < 0.84375 282 var temp float64 283 if x < Tiny { // |x| < 2**-56 284 temp = x 285 } else { 286 z := x * x 287 r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4))) 288 s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))) 289 y := r / s 290 if x < 0.25 { // |x| < 1/4 291 temp = x + x*y 292 } else { 293 temp = 0.5 + (x*y + (x - 0.5)) 294 } 295 } 296 if sign { 297 return 1 + temp 298 } 299 return 1 - temp 300 } 301 if x < 1.25 { // 0.84375 <= |x| < 1.25 302 s := x - 1 303 P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))) 304 Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))) 305 if sign { 306 return 1 + erx + P/Q 307 } 308 return 1 - erx - P/Q 309 310 } 311 if x < 28 { // |x| < 28 312 s := 1 / (x * x) 313 var R, S float64 314 if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143 315 R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))) 316 S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8))))))) 317 } else { // |x| >= 1 / 0.35 ~ 2.857143 318 if sign && x > 6 { 319 return 2 // x < -6 320 } 321 R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))) 322 S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))) 323 } 324 z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x 325 r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S) 326 if sign { 327 return 2 - r/x 328 } 329 return r / x 330 } 331 if sign { 332 return 2 333 } 334 return 0 335 } 336
