1 use super::{exp, fabs, get_high_word, with_set_low_word}; 2 /* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunPro, a Sun Microsystems, Inc. business. test_parser()8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 /* double erf(double x) 14 * double erfc(double x) 15 * x 16 * 2 |\ 17 * erf(x) = --------- | exp(-t*t)dt 18 * sqrt(pi) \| 19 * 0 20 * 21 * erfc(x) = 1-erf(x) 22 * Note that 23 * erf(-x) = -erf(x) 24 * erfc(-x) = 2 - erfc(x) 25 * 26 * Method: 27 * 1. For |x| in [0, 0.84375] 28 * erf(x) = x + x*R(x^2) 29 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 30 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 31 * where R = P/Q where P is an odd poly of degree 8 and 32 * Q is an odd poly of degree 10. 33 * -57.90 34 * | R - (erf(x)-x)/x | <= 2 35 * 36 * 37 * Remark. The formula is derived by noting 38 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 39 * and that 40 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 41 * is close to one. The interval is chosen because the fix 42 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 43 * near 0.6174), and by some experiment, 0.84375 is chosen to 44 * guarantee the error is less than one ulp for erf. 45 * 46 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 47 * c = 0.84506291151 rounded to single (24 bits) 48 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 49 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 50 * 1+(c+P1(s)/Q1(s)) if x < 0 51 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 52 * Remark: here we use the taylor series expansion at x=1. 53 * erf(1+s) = erf(1) + s*Poly(s) 54 * = 0.845.. + P1(s)/Q1(s) 55 * That is, we use rational approximation to approximate 56 * erf(1+s) - (c = (single)0.84506291151) 57 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 58 * where 59 * P1(s) = degree 6 poly in s 60 * Q1(s) = degree 6 poly in s 61 * 62 * 3. For x in [1.25,1/0.35(~2.857143)], 63 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 64 * erf(x) = 1 - erfc(x) 65 * where 66 * R1(z) = degree 7 poly in z, (z=1/x^2) 67 * S1(z) = degree 8 poly in z 68 * 69 * 4. For x in [1/0.35,28] 70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 71 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 72 * = 2.0 - tiny (if x <= -6) 73 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 74 * erf(x) = sign(x)*(1.0 - tiny) 75 * where 76 * R2(z) = degree 6 poly in z, (z=1/x^2) 77 * S2(z) = degree 7 poly in z 78 * 79 * Note1: 80 * To compute exp(-x*x-0.5625+R/S), let s be a single 81 * precision number and s := x; then 82 * -x*x = -s*s + (s-x)*(s+x) 83 * exp(-x*x-0.5626+R/S) = 84 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 85 * Note2: 86 * Here 4 and 5 make use of the asymptotic series 87 * exp(-x*x) 88 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 89 * x*sqrt(pi) 90 * We use rational approximation to approximate 91 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 92 * Here is the error bound for R1/S1 and R2/S2 93 * |R1/S1 - f(x)| < 2**(-62.57) 94 * |R2/S2 - f(x)| < 2**(-61.52) 95 * 96 * 5. For inf > x >= 28 97 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 98 * erfc(x) = tiny*tiny (raise underflow) if x > 0 99 * = 2 - tiny if x<0 100 * 101 * 7. Special case: 102 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 103 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 104 * erfc/erf(NaN) is NaN 105 */ 106 107 const ERX: f64 = 8.45062911510467529297e-01; /* 0x3FEB0AC1, 0x60000000 */ 108 /* 109 * Coefficients for approximation to erf on [0,0.84375] 110 */ 111 const EFX8: f64 = 1.02703333676410069053e+00; /* 0x3FF06EBA, 0x8214DB69 */ 112 const PP0: f64 = 1.28379167095512558561e-01; /* 0x3FC06EBA, 0x8214DB68 */ 113 const PP1: f64 = -3.25042107247001499370e-01; /* 0xBFD4CD7D, 0x691CB913 */ 114 const PP2: f64 = -2.84817495755985104766e-02; /* 0xBF9D2A51, 0xDBD7194F */ 115 const PP3: f64 = -5.77027029648944159157e-03; /* 0xBF77A291, 0x236668E4 */ 116 const PP4: f64 = -2.37630166566501626084e-05; /* 0xBEF8EAD6, 0x120016AC */ 117 const QQ1: f64 = 3.97917223959155352819e-01; /* 0x3FD97779, 0xCDDADC09 */ 118 const QQ2: f64 = 6.50222499887672944485e-02; /* 0x3FB0A54C, 0x5536CEBA */ 119 const QQ3: f64 = 5.08130628187576562776e-03; /* 0x3F74D022, 0xC4D36B0F */ 120 const QQ4: f64 = 1.32494738004321644526e-04; /* 0x3F215DC9, 0x221C1A10 */ 121 const QQ5: f64 = -3.96022827877536812320e-06; /* 0xBED09C43, 0x42A26120 */ 122 /* 123 * Coefficients for approximation to erf in [0.84375,1.25] 124 */ 125 const PA0: f64 = -2.36211856075265944077e-03; /* 0xBF6359B8, 0xBEF77538 */ 126 const PA1: f64 = 4.14856118683748331666e-01; /* 0x3FDA8D00, 0xAD92B34D */ 127 const PA2: f64 = -3.72207876035701323847e-01; /* 0xBFD7D240, 0xFBB8C3F1 */ 128 const PA3: f64 = 3.18346619901161753674e-01; /* 0x3FD45FCA, 0x805120E4 */ 129 const PA4: f64 = -1.10894694282396677476e-01; /* 0xBFBC6398, 0x3D3E28EC */ 130 const PA5: f64 = 3.54783043256182359371e-02; /* 0x3FA22A36, 0x599795EB */ 131 const PA6: f64 = -2.16637559486879084300e-03; /* 0xBF61BF38, 0x0A96073F */ 132 const QA1: f64 = 1.06420880400844228286e-01; /* 0x3FBB3E66, 0x18EEE323 */ 133 const QA2: f64 = 5.40397917702171048937e-01; /* 0x3FE14AF0, 0x92EB6F33 */ 134 const QA3: f64 = 7.18286544141962662868e-02; /* 0x3FB2635C, 0xD99FE9A7 */ 135 const QA4: f64 = 1.26171219808761642112e-01; /* 0x3FC02660, 0xE763351F */ 136 const QA5: f64 = 1.36370839120290507362e-02; /* 0x3F8BEDC2, 0x6B51DD1C */ 137 const QA6: f64 = 1.19844998467991074170e-02; /* 0x3F888B54, 0x5735151D */ 138 /* 139 * Coefficients for approximation to erfc in [1.25,1/0.35] 140 */ 141 const RA0: f64 = -9.86494403484714822705e-03; /* 0xBF843412, 0x600D6435 */ 142 const RA1: f64 = -6.93858572707181764372e-01; /* 0xBFE63416, 0xE4BA7360 */ 143 const RA2: f64 = -1.05586262253232909814e+01; /* 0xC0251E04, 0x41B0E726 */ 144 const RA3: f64 = -6.23753324503260060396e+01; /* 0xC04F300A, 0xE4CBA38D */ 145 const RA4: f64 = -1.62396669462573470355e+02; /* 0xC0644CB1, 0x84282266 */ 146 const RA5: f64 = -1.84605092906711035994e+02; /* 0xC067135C, 0xEBCCABB2 */ 147 const RA6: f64 = -8.12874355063065934246e+01; /* 0xC0545265, 0x57E4D2F2 */ 148 const RA7: f64 = -9.81432934416914548592e+00; /* 0xC023A0EF, 0xC69AC25C */ 149 const SA1: f64 = 1.96512716674392571292e+01; /* 0x4033A6B9, 0xBD707687 */ 150 const SA2: f64 = 1.37657754143519042600e+02; /* 0x4061350C, 0x526AE721 */ 151 const SA3: f64 = 4.34565877475229228821e+02; /* 0x407B290D, 0xD58A1A71 */ 152 const SA4: f64 = 6.45387271733267880336e+02; /* 0x40842B19, 0x21EC2868 */ 153 const SA5: f64 = 4.29008140027567833386e+02; /* 0x407AD021, 0x57700314 */ 154 const SA6: f64 = 1.08635005541779435134e+02; /* 0x405B28A3, 0xEE48AE2C */ 155 const SA7: f64 = 6.57024977031928170135e+00; /* 0x401A47EF, 0x8E484A93 */ 156 const SA8: f64 = -6.04244152148580987438e-02; /* 0xBFAEEFF2, 0xEE749A62 */ 157 /* 158 * Coefficients for approximation to erfc in [1/.35,28] 159 */ 160 const RB0: f64 = -9.86494292470009928597e-03; /* 0xBF843412, 0x39E86F4A */ 161 const RB1: f64 = -7.99283237680523006574e-01; /* 0xBFE993BA, 0x70C285DE */ 162 const RB2: f64 = -1.77579549177547519889e+01; /* 0xC031C209, 0x555F995A */ 163 const RB3: f64 = -1.60636384855821916062e+02; /* 0xC064145D, 0x43C5ED98 */ 164 const RB4: f64 = -6.37566443368389627722e+02; /* 0xC083EC88, 0x1375F228 */ 165 const RB5: f64 = -1.02509513161107724954e+03; /* 0xC0900461, 0x6A2E5992 */ 166 const RB6: f64 = -4.83519191608651397019e+02; /* 0xC07E384E, 0x9BDC383F */ 167 const SB1: f64 = 3.03380607434824582924e+01; /* 0x403E568B, 0x261D5190 */ 168 const SB2: f64 = 3.25792512996573918826e+02; /* 0x40745CAE, 0x221B9F0A */ 169 const SB3: f64 = 1.53672958608443695994e+03; /* 0x409802EB, 0x189D5118 */ 170 const SB4: f64 = 3.19985821950859553908e+03; /* 0x40A8FFB7, 0x688C246A */ 171 const SB5: f64 = 2.55305040643316442583e+03; /* 0x40A3F219, 0xCEDF3BE6 */ 172 const SB6: f64 = 4.74528541206955367215e+02; /* 0x407DA874, 0xE79FE763 */ 173 const SB7: f64 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 174 175 fn erfc1(x: f64) -> f64 { 176 let s: f64; 177 let p: f64; 178 let q: f64; 179 180 s = fabs(x) - 1.0; 181 p = PA0 + s * (PA1 + s * (PA2 + s * (PA3 + s * (PA4 + s * (PA5 + s * PA6))))); 182 q = 1.0 + s * (QA1 + s * (QA2 + s * (QA3 + s * (QA4 + s * (QA5 + s * QA6))))); 183 184 1.0 - ERX - p / q 185 } 186 187 fn erfc2(ix: u32, mut x: f64) -> f64 { 188 let s: f64; 189 let r: f64; 190 let big_s: f64; 191 let z: f64; 192 193 if ix < 0x3ff40000 { 194 /* |x| < 1.25 */ 195 return erfc1(x); 196 } 197 198 x = fabs(x); 199 s = 1.0 / (x * x); 200 if ix < 0x4006db6d { 201 /* |x| < 1/.35 ~ 2.85714 */ 202 r = RA0 + s * (RA1 + s * (RA2 + s * (RA3 + s * (RA4 + s * (RA5 + s * (RA6 + s * RA7)))))); 203 big_s = 1.0 204 + s * (SA1 205 + s * (SA2 + s * (SA3 + s * (SA4 + s * (SA5 + s * (SA6 + s * (SA7 + s * SA8))))))); 206 } else { 207 /* |x| > 1/.35 */ 208 r = RB0 + s * (RB1 + s * (RB2 + s * (RB3 + s * (RB4 + s * (RB5 + s * RB6))))); 209 big_s = 210 1.0 + s * (SB1 + s * (SB2 + s * (SB3 + s * (SB4 + s * (SB5 + s * (SB6 + s * SB7)))))); 211 } 212 z = with_set_low_word(x, 0); 213 214 exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / big_s) / x 215 } 216 217 /// Error function (f64) 218 /// 219 /// Calculates an approximation to the “error function”, which estimates 220 /// the probability that an observation will fall within x standard 221 /// deviations of the mean (assuming a normal distribution). 222 pub fn erf(x: f64) -> f64 { 223 let r: f64; 224 let s: f64; 225 let z: f64; 226 let y: f64; 227 let mut ix: u32; 228 let sign: usize; 229 230 ix = get_high_word(x); 231 sign = (ix >> 31) as usize; 232 ix &= 0x7fffffff; 233 if ix >= 0x7ff00000 { 234 /* erf(nan)=nan, erf(+-inf)=+-1 */ 235 return 1.0 - 2.0 * (sign as f64) + 1.0 / x; 236 } 237 if ix < 0x3feb0000 { 238 /* |x| < 0.84375 */ 239 if ix < 0x3e300000 { 240 /* |x| < 2**-28 */ 241 /* avoid underflow */ 242 return 0.125 * (8.0 * x + EFX8 * x); 243 } 244 z = x * x; 245 r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); 246 s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); 247 y = r / s; 248 return x + x * y; 249 } 250 if ix < 0x40180000 { 251 /* 0.84375 <= |x| < 6 */ 252 y = 1.0 - erfc2(ix, x); 253 } else { 254 let x1p_1022 = f64::from_bits(0x0010000000000000); 255 y = 1.0 - x1p_1022; 256 } 257 258 if sign != 0 { 259 -y 260 } else { 261 y 262 } 263 } 264 265 /// Error function (f64) 266 /// 267 /// Calculates the complementary probability. 268 /// Is `1 - erf(x)`. Is computed directly, so that you can use it to avoid 269 /// the loss of precision that would result from subtracting 270 /// large probabilities (on large `x`) from 1. 271 pub fn erfc(x: f64) -> f64 { 272 let r: f64; 273 let s: f64; 274 let z: f64; 275 let y: f64; 276 let mut ix: u32; 277 let sign: usize; 278 279 ix = get_high_word(x); 280 sign = (ix >> 31) as usize; 281 ix &= 0x7fffffff; 282 if ix >= 0x7ff00000 { 283 /* erfc(nan)=nan, erfc(+-inf)=0,2 */ 284 return 2.0 * (sign as f64) + 1.0 / x; 285 } 286 if ix < 0x3feb0000 { 287 /* |x| < 0.84375 */ 288 if ix < 0x3c700000 { 289 /* |x| < 2**-56 */ 290 return 1.0 - x; 291 } 292 z = x * x; 293 r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4))); 294 s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5)))); 295 y = r / s; 296 if sign != 0 || ix < 0x3fd00000 { 297 /* x < 1/4 */ 298 return 1.0 - (x + x * y); 299 } 300 return 0.5 - (x - 0.5 + x * y); 301 } 302 if ix < 0x403c0000 { 303 /* 0.84375 <= |x| < 28 */ 304 if sign != 0 { 305 return 2.0 - erfc2(ix, x); 306 } else { 307 return erfc2(ix, x); 308 } 309 } 310 311 let x1p_1022 = f64::from_bits(0x0010000000000000); 312 if sign != 0 { 313 2.0 - x1p_1022 314 } else { 315 x1p_1022 * x1p_1022 316 } 317 } 318