erf.mx raw
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 //
186 // Erf(+Inf) = 1
187 // Erf(-Inf) = -1
188 // Erf(NaN) = NaN
189 func Erf(x float64) float64 {
190 if haveArchErf {
191 return archErf(x)
192 }
193 return erf(x)
194 }
195
196 func erf(x float64) float64 {
197 const (
198 VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
199 Small = 1.0 / (1 << 28) // 2**-28
200 )
201 // special cases
202 switch {
203 case IsNaN(x):
204 return NaN()
205 case IsInf(x, 1):
206 return 1
207 case IsInf(x, -1):
208 return -1
209 }
210 sign := false
211 if x < 0 {
212 x = -x
213 sign = true
214 }
215 if x < 0.84375 { // |x| < 0.84375
216 var temp float64
217 if x < Small { // |x| < 2**-28
218 if x < VeryTiny {
219 temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
220 } else {
221 temp = x + efx*x
222 }
223 } else {
224 z := x * x
225 r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
226 s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
227 y := r / s
228 temp = x + x*y
229 }
230 if sign {
231 return -temp
232 }
233 return temp
234 }
235 if x < 1.25 { // 0.84375 <= |x| < 1.25
236 s := x - 1
237 P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
238 Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
239 if sign {
240 return -erx - P/Q
241 }
242 return erx + P/Q
243 }
244 if x >= 6 { // inf > |x| >= 6
245 if sign {
246 return -1
247 }
248 return 1
249 }
250 s := 1 / (x * x)
251 var R, S float64
252 if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
253 R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
254 S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
255 } else { // |x| >= 1 / 0.35 ~ 2.857143
256 R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
257 S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
258 }
259 z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
260 r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
261 if sign {
262 return r/x - 1
263 }
264 return 1 - r/x
265 }
266
267 // Erfc returns the complementary error function of x.
268 //
269 // Special cases are:
270 //
271 // Erfc(+Inf) = 0
272 // Erfc(-Inf) = 2
273 // Erfc(NaN) = NaN
274 func Erfc(x float64) float64 {
275 if haveArchErfc {
276 return archErfc(x)
277 }
278 return erfc(x)
279 }
280
281 func erfc(x float64) float64 {
282 const Tiny = 1.0 / (1 << 56) // 2**-56
283 // special cases
284 switch {
285 case IsNaN(x):
286 return NaN()
287 case IsInf(x, 1):
288 return 0
289 case IsInf(x, -1):
290 return 2
291 }
292 sign := false
293 if x < 0 {
294 x = -x
295 sign = true
296 }
297 if x < 0.84375 { // |x| < 0.84375
298 var temp float64
299 if x < Tiny { // |x| < 2**-56
300 temp = x
301 } else {
302 z := x * x
303 r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
304 s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
305 y := r / s
306 if x < 0.25 { // |x| < 1/4
307 temp = x + x*y
308 } else {
309 temp = 0.5 + (x*y + (x - 0.5))
310 }
311 }
312 if sign {
313 return 1 + temp
314 }
315 return 1 - temp
316 }
317 if x < 1.25 { // 0.84375 <= |x| < 1.25
318 s := x - 1
319 P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
320 Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
321 if sign {
322 return 1 + erx + P/Q
323 }
324 return 1 - erx - P/Q
325
326 }
327 if x < 28 { // |x| < 28
328 s := 1 / (x * x)
329 var R, S float64
330 if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
331 R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
332 S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
333 } else { // |x| >= 1 / 0.35 ~ 2.857143
334 if sign && x > 6 {
335 return 2 // x < -6
336 }
337 R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
338 S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
339 }
340 z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
341 r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
342 if sign {
343 return 2 - r/x
344 }
345 return r / x
346 }
347 if sign {
348 return 2
349 }
350 return 0
351 }
352