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.
8 Floating-point error function and complementary error function.
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.
16 // ====================================================
17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
23 // ====================================================
26 // double erf(double x)
27 // double erfc(double x)
30 // erf(x) = --------- | exp(-t*t)dt
37 // erfc(-x) = 2 - erfc(x)
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.
47 // | R - (erf(x)-x)/x | <= 2
50 // Remark. The formula is derived by noting
51 // erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
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.
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]
72 // P1(s) = degree 6 poly in s
73 // Q1(s) = degree 6 poly in s
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)
79 // R1(z) = degree 7 poly in z, (z=1/x**2)
80 // S1(z) = degree 8 poly in z
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)
89 // R2(z) = degree 6 poly in z, (z=1/x**2)
90 // S2(z) = degree 7 poly in z
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);
99 // Here 4 and 5 make use of the asymptotic series
101 // erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
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)
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
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
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
182 // Erf returns the error function of x.
184 // Special cases are:
188 func Erf(x
float64) float64 {
193 func libc_erf(float64) float64
195 func erf(x
float64) float64 {
197 VeryTiny
= 2.848094538889218e-306 // 0x0080000000000000
198 Small
= 1.0 / (1 << 28) // 2**-28
214 if x
< 0.84375 { // |x| < 0.84375
216 if x
< Small
{ // |x| < 2**-28
218 temp
= 0.125 * (8.0*x
+ efx8
*x
) // avoid underflow
224 r
:= pp0
+ z
*(pp1
+z
*(pp2
+z
*(pp3
+z
*pp4
)))
225 s
:= 1 + z
*(qq1
+z
*(qq2
+z
*(qq3
+z
*(qq4
+z
*qq5
))))
234 if x
< 1.25 { // 0.84375 <= |x| < 1.25
236 P
:= pa0
+ s
*(pa1
+s
*(pa2
+s
*(pa3
+s
*(pa4
+s
*(pa5
+s
*pa6
)))))
237 Q
:= 1 + s
*(qa1
+s
*(qa2
+s
*(qa3
+s
*(qa4
+s
*(qa5
+s
*qa6
)))))
243 if x
>= 6 { // inf > |x| >= 6
251 if x
< 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
252 R
= ra0
+ s
*(ra1
+s
*(ra2
+s
*(ra3
+s
*(ra4
+s
*(ra5
+s
*(ra6
+s
*ra7
))))))
253 S
= 1 + s
*(sa1
+s
*(sa2
+s
*(sa3
+s
*(sa4
+s
*(sa5
+s
*(sa6
+s
*(sa7
+s
*sa8
)))))))
254 } else { // |x| >= 1 / 0.35 ~ 2.857143
255 R
= rb0
+ s
*(rb1
+s
*(rb2
+s
*(rb3
+s
*(rb4
+s
*(rb5
+s
*rb6
)))))
256 S
= 1 + s
*(sb1
+s
*(sb2
+s
*(sb3
+s
*(sb4
+s
*(sb5
+s
*(sb6
+s
*sb7
))))))
258 z
:= Float64frombits(Float64bits(x
) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
259 r
:= Exp(-z
*z
-0.5625) * Exp((z
-x
)*(z
+x
)+R
/S
)
266 // Erfc returns the complementary error function of x.
268 // Special cases are:
272 func Erfc(x
float64) float64 {
277 func libc_erfc(float64) float64
279 func erfc(x
float64) float64 {
280 const Tiny
= 1.0 / (1 << 56) // 2**-56
295 if x
< 0.84375 { // |x| < 0.84375
297 if x
< Tiny
{ // |x| < 2**-56
301 r
:= pp0
+ z
*(pp1
+z
*(pp2
+z
*(pp3
+z
*pp4
)))
302 s
:= 1 + z
*(qq1
+z
*(qq2
+z
*(qq3
+z
*(qq4
+z
*qq5
))))
304 if x
< 0.25 { // |x| < 1/4
307 temp
= 0.5 + (x
*y
+ (x
- 0.5))
315 if x
< 1.25 { // 0.84375 <= |x| < 1.25
317 P
:= pa0
+ s
*(pa1
+s
*(pa2
+s
*(pa3
+s
*(pa4
+s
*(pa5
+s
*pa6
)))))
318 Q
:= 1 + s
*(qa1
+s
*(qa2
+s
*(qa3
+s
*(qa4
+s
*(qa5
+s
*qa6
)))))
325 if x
< 28 { // |x| < 28
328 if x
< 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
329 R
= ra0
+ s
*(ra1
+s
*(ra2
+s
*(ra3
+s
*(ra4
+s
*(ra5
+s
*(ra6
+s
*ra7
))))))
330 S
= 1 + s
*(sa1
+s
*(sa2
+s
*(sa3
+s
*(sa4
+s
*(sa5
+s
*(sa6
+s
*(sa7
+s
*sa8
)))))))
331 } else { // |x| >= 1 / 0.35 ~ 2.857143
335 R
= rb0
+ s
*(rb1
+s
*(rb2
+s
*(rb3
+s
*(rb4
+s
*(rb5
+s
*rb6
)))))
336 S
= 1 + s
*(sb1
+s
*(sb2
+s
*(sb3
+s
*(sb4
+s
*(sb5
+s
*(sb6
+s
*sb7
))))))
338 z
:= Float64frombits(Float64bits(x
) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
339 r
:= Exp(-z
*z
-0.5625) * Exp((z
-x
)*(z
+x
)+R
/S
)