vendor/libm-0.1.4/src/math/erf.rs

   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.
   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 }