libm/math/erf.rs
1use 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
107const ERX: f64 = 8.45062911510467529297e-01; /* 0x3FEB0AC1, 0x60000000 */
108/*
109 * Coefficients for approximation to erf on [0,0.84375]
110 */
111const EFX8: f64 = 1.02703333676410069053e+00; /* 0x3FF06EBA, 0x8214DB69 */
112const PP0: f64 = 1.28379167095512558561e-01; /* 0x3FC06EBA, 0x8214DB68 */
113const PP1: f64 = -3.25042107247001499370e-01; /* 0xBFD4CD7D, 0x691CB913 */
114const PP2: f64 = -2.84817495755985104766e-02; /* 0xBF9D2A51, 0xDBD7194F */
115const PP3: f64 = -5.77027029648944159157e-03; /* 0xBF77A291, 0x236668E4 */
116const PP4: f64 = -2.37630166566501626084e-05; /* 0xBEF8EAD6, 0x120016AC */
117const QQ1: f64 = 3.97917223959155352819e-01; /* 0x3FD97779, 0xCDDADC09 */
118const QQ2: f64 = 6.50222499887672944485e-02; /* 0x3FB0A54C, 0x5536CEBA */
119const QQ3: f64 = 5.08130628187576562776e-03; /* 0x3F74D022, 0xC4D36B0F */
120const QQ4: f64 = 1.32494738004321644526e-04; /* 0x3F215DC9, 0x221C1A10 */
121const QQ5: f64 = -3.96022827877536812320e-06; /* 0xBED09C43, 0x42A26120 */
122/*
123 * Coefficients for approximation to erf in [0.84375,1.25]
124 */
125const PA0: f64 = -2.36211856075265944077e-03; /* 0xBF6359B8, 0xBEF77538 */
126const PA1: f64 = 4.14856118683748331666e-01; /* 0x3FDA8D00, 0xAD92B34D */
127const PA2: f64 = -3.72207876035701323847e-01; /* 0xBFD7D240, 0xFBB8C3F1 */
128const PA3: f64 = 3.18346619901161753674e-01; /* 0x3FD45FCA, 0x805120E4 */
129const PA4: f64 = -1.10894694282396677476e-01; /* 0xBFBC6398, 0x3D3E28EC */
130const PA5: f64 = 3.54783043256182359371e-02; /* 0x3FA22A36, 0x599795EB */
131const PA6: f64 = -2.16637559486879084300e-03; /* 0xBF61BF38, 0x0A96073F */
132const QA1: f64 = 1.06420880400844228286e-01; /* 0x3FBB3E66, 0x18EEE323 */
133const QA2: f64 = 5.40397917702171048937e-01; /* 0x3FE14AF0, 0x92EB6F33 */
134const QA3: f64 = 7.18286544141962662868e-02; /* 0x3FB2635C, 0xD99FE9A7 */
135const QA4: f64 = 1.26171219808761642112e-01; /* 0x3FC02660, 0xE763351F */
136const QA5: f64 = 1.36370839120290507362e-02; /* 0x3F8BEDC2, 0x6B51DD1C */
137const QA6: f64 = 1.19844998467991074170e-02; /* 0x3F888B54, 0x5735151D */
138/*
139 * Coefficients for approximation to erfc in [1.25,1/0.35]
140 */
141const RA0: f64 = -9.86494403484714822705e-03; /* 0xBF843412, 0x600D6435 */
142const RA1: f64 = -6.93858572707181764372e-01; /* 0xBFE63416, 0xE4BA7360 */
143const RA2: f64 = -1.05586262253232909814e+01; /* 0xC0251E04, 0x41B0E726 */
144const RA3: f64 = -6.23753324503260060396e+01; /* 0xC04F300A, 0xE4CBA38D */
145const RA4: f64 = -1.62396669462573470355e+02; /* 0xC0644CB1, 0x84282266 */
146const RA5: f64 = -1.84605092906711035994e+02; /* 0xC067135C, 0xEBCCABB2 */
147const RA6: f64 = -8.12874355063065934246e+01; /* 0xC0545265, 0x57E4D2F2 */
148const RA7: f64 = -9.81432934416914548592e+00; /* 0xC023A0EF, 0xC69AC25C */
149const SA1: f64 = 1.96512716674392571292e+01; /* 0x4033A6B9, 0xBD707687 */
150const SA2: f64 = 1.37657754143519042600e+02; /* 0x4061350C, 0x526AE721 */
151const SA3: f64 = 4.34565877475229228821e+02; /* 0x407B290D, 0xD58A1A71 */
152const SA4: f64 = 6.45387271733267880336e+02; /* 0x40842B19, 0x21EC2868 */
153const SA5: f64 = 4.29008140027567833386e+02; /* 0x407AD021, 0x57700314 */
154const SA6: f64 = 1.08635005541779435134e+02; /* 0x405B28A3, 0xEE48AE2C */
155const SA7: f64 = 6.57024977031928170135e+00; /* 0x401A47EF, 0x8E484A93 */
156const SA8: f64 = -6.04244152148580987438e-02; /* 0xBFAEEFF2, 0xEE749A62 */
157/*
158 * Coefficients for approximation to erfc in [1/.35,28]
159 */
160const RB0: f64 = -9.86494292470009928597e-03; /* 0xBF843412, 0x39E86F4A */
161const RB1: f64 = -7.99283237680523006574e-01; /* 0xBFE993BA, 0x70C285DE */
162const RB2: f64 = -1.77579549177547519889e+01; /* 0xC031C209, 0x555F995A */
163const RB3: f64 = -1.60636384855821916062e+02; /* 0xC064145D, 0x43C5ED98 */
164const RB4: f64 = -6.37566443368389627722e+02; /* 0xC083EC88, 0x1375F228 */
165const RB5: f64 = -1.02509513161107724954e+03; /* 0xC0900461, 0x6A2E5992 */
166const RB6: f64 = -4.83519191608651397019e+02; /* 0xC07E384E, 0x9BDC383F */
167const SB1: f64 = 3.03380607434824582924e+01; /* 0x403E568B, 0x261D5190 */
168const SB2: f64 = 3.25792512996573918826e+02; /* 0x40745CAE, 0x221B9F0A */
169const SB3: f64 = 1.53672958608443695994e+03; /* 0x409802EB, 0x189D5118 */
170const SB4: f64 = 3.19985821950859553908e+03; /* 0x40A8FFB7, 0x688C246A */
171const SB5: f64 = 2.55305040643316442583e+03; /* 0x40A3F219, 0xCEDF3BE6 */
172const SB6: f64 = 4.74528541206955367215e+02; /* 0x407DA874, 0xE79FE763 */
173const SB7: f64 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
174
175fn 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
187fn 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#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
223pub fn erf(x: f64) -> f64 {
224 let r: f64;
225 let s: f64;
226 let z: f64;
227 let y: f64;
228 let mut ix: u32;
229 let sign: usize;
230
231 ix = get_high_word(x);
232 sign = (ix >> 31) as usize;
233 ix &= 0x7fffffff;
234 if ix >= 0x7ff00000 {
235 /* erf(nan)=nan, erf(+-inf)=+-1 */
236 return 1.0 - 2.0 * (sign as f64) + 1.0 / x;
237 }
238 if ix < 0x3feb0000 {
239 /* |x| < 0.84375 */
240 if ix < 0x3e300000 {
241 /* |x| < 2**-28 */
242 /* avoid underflow */
243 return 0.125 * (8.0 * x + EFX8 * x);
244 }
245 z = x * x;
246 r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4)));
247 s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5))));
248 y = r / s;
249 return x + x * y;
250 }
251 if ix < 0x40180000 {
252 /* 0.84375 <= |x| < 6 */
253 y = 1.0 - erfc2(ix, x);
254 } else {
255 let x1p_1022 = f64::from_bits(0x0010000000000000);
256 y = 1.0 - x1p_1022;
257 }
258
259 if sign != 0 { -y } else { y }
260}
261
262/// Complementary error function (f64)
263///
264/// Calculates the complementary probability.
265/// Is `1 - erf(x)`. Is computed directly, so that you can use it to avoid
266/// the loss of precision that would result from subtracting
267/// large probabilities (on large `x`) from 1.
268pub fn erfc(x: f64) -> f64 {
269 let r: f64;
270 let s: f64;
271 let z: f64;
272 let y: f64;
273 let mut ix: u32;
274 let sign: usize;
275
276 ix = get_high_word(x);
277 sign = (ix >> 31) as usize;
278 ix &= 0x7fffffff;
279 if ix >= 0x7ff00000 {
280 /* erfc(nan)=nan, erfc(+-inf)=0,2 */
281 return 2.0 * (sign as f64) + 1.0 / x;
282 }
283 if ix < 0x3feb0000 {
284 /* |x| < 0.84375 */
285 if ix < 0x3c700000 {
286 /* |x| < 2**-56 */
287 return 1.0 - x;
288 }
289 z = x * x;
290 r = PP0 + z * (PP1 + z * (PP2 + z * (PP3 + z * PP4)));
291 s = 1.0 + z * (QQ1 + z * (QQ2 + z * (QQ3 + z * (QQ4 + z * QQ5))));
292 y = r / s;
293 if sign != 0 || ix < 0x3fd00000 {
294 /* x < 1/4 */
295 return 1.0 - (x + x * y);
296 }
297 return 0.5 - (x - 0.5 + x * y);
298 }
299 if ix < 0x403c0000 {
300 /* 0.84375 <= |x| < 28 */
301 if sign != 0 {
302 return 2.0 - erfc2(ix, x);
303 } else {
304 return erfc2(ix, x);
305 }
306 }
307
308 let x1p_1022 = f64::from_bits(0x0010000000000000);
309 if sign != 0 { 2.0 - x1p_1022 } else { x1p_1022 * x1p_1022 }
310}