1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
12 /* double erf(double x)
13 * double erfc(double x)
16 * erf(x) = --------- | exp(-t*t)dt
23 * erfc(-x) = 2 - erfc(x)
26 * 1. For |x| in [0, 0.84375]
27 * erf(x) = x + x*R(x^2)
28 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
29 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
30 * where R = P/Q where P is an odd poly of degree 8 and
31 * Q is an odd poly of degree 10.
33 * | R - (erf(x)-x)/x | <= 2
36 * Remark. The formula is derived by noting
37 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
39 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
40 * is close to one. The interval is chosen because the fix
41 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
42 * near 0.6174), and by some experiment, 0.84375 is chosen to
43 * guarantee the error is less than one ulp for erf.
45 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
46 * c = 0.84506291151 rounded to single (24 bits)
47 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
48 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
49 * 1+(c+P1(s)/Q1(s)) if x < 0
50 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
51 * Remark: here we use the taylor series expansion at x=1.
52 * erf(1+s) = erf(1) + s*Poly(s)
53 * = 0.845.. + P1(s)/Q1(s)
54 * That is, we use rational approximation to approximate
55 * erf(1+s) - (c = (single)0.84506291151)
56 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
58 * P1(s) = degree 6 poly in s
59 * Q1(s) = degree 6 poly in s
61 * 3. For x in [1.25,1/0.35(~2.857143)],
62 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
63 * erf(x) = 1 - erfc(x)
65 * R1(z) = degree 7 poly in z, (z=1/x^2)
66 * S1(z) = degree 8 poly in z
68 * 4. For x in [1/0.35,28]
69 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
70 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
71 * = 2.0 - tiny (if x <= -6)
72 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
73 * erf(x) = sign(x)*(1.0 - tiny)
75 * R2(z) = degree 6 poly in z, (z=1/x^2)
76 * S2(z) = degree 7 poly in z
79 * To compute exp(-x*x-0.5625+R/S), let s be a single
80 * precision number and s := x; then
81 * -x*x = -s*s + (s-x)*(s+x)
82 * exp(-x*x-0.5626+R/S) =
83 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
85 * Here 4 and 5 make use of the asymptotic series
87 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
89 * We use rational approximation to approximate
90 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
91 * Here is the error bound for R1/S1 and R2/S2
92 * |R1/S1 - f(x)| < 2**(-62.57)
93 * |R2/S2 - f(x)| < 2**(-61.52)
95 * 5. For inf > x >= 28
96 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
97 * erfc(x) = tiny*tiny (raise underflow) if x > 0
101 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
102 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
103 * erfc/erf(NaN) is NaN
109 erx
= 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
111 * Coefficients for approximation to erf on [0,0.84375]
113 efx8
= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
114 pp0
= 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
115 pp1
= -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
116 pp2
= -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
117 pp3
= -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
118 pp4
= -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
119 qq1
= 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
120 qq2
= 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
121 qq3
= 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
122 qq4
= 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
123 qq5
= -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
125 * Coefficients for approximation to erf in [0.84375,1.25]
127 pa0
= -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
128 pa1
= 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
129 pa2
= -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
130 pa3
= 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
131 pa4
= -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
132 pa5
= 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
133 pa6
= -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
134 qa1
= 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
135 qa2
= 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
136 qa3
= 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
137 qa4
= 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
138 qa5
= 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
139 qa6
= 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
141 * Coefficients for approximation to erfc in [1.25,1/0.35]
143 ra0
= -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
144 ra1
= -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
145 ra2
= -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
146 ra3
= -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
147 ra4
= -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
148 ra5
= -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
149 ra6
= -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
150 ra7
= -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
151 sa1
= 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
152 sa2
= 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
153 sa3
= 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
154 sa4
= 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
155 sa5
= 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
156 sa6
= 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
157 sa7
= 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
158 sa8
= -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
160 * Coefficients for approximation to erfc in [1/.35,28]
162 rb0
= -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
163 rb1
= -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
164 rb2
= -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
165 rb3
= -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
166 rb4
= -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
167 rb5
= -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
168 rb6
= -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
169 sb1
= 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
170 sb2
= 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
171 sb3
= 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
172 sb4
= 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
173 sb5
= 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
174 sb6
= 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
175 sb7
= -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
177 static double erfc1(double x
)
182 P
= pa0
+s
*(pa1
+s
*(pa2
+s
*(pa3
+s
*(pa4
+s
*(pa5
+s
*pa6
)))));
183 Q
= 1+s
*(qa1
+s
*(qa2
+s
*(qa3
+s
*(qa4
+s
*(qa5
+s
*qa6
)))));
184 return 1 - erx
- P
/Q
;
187 static double erfc2(uint32_t ix
, double x
)
192 if (ix
< 0x3ff40000) /* |x| < 1.25 */
197 if (ix
< 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */
198 R
= ra0
+s
*(ra1
+s
*(ra2
+s
*(ra3
+s
*(ra4
+s
*(
199 ra5
+s
*(ra6
+s
*ra7
))))));
200 S
= 1.0+s
*(sa1
+s
*(sa2
+s
*(sa3
+s
*(sa4
+s
*(
201 sa5
+s
*(sa6
+s
*(sa7
+s
*sa8
)))))));
202 } else { /* |x| > 1/.35 */
203 R
= rb0
+s
*(rb1
+s
*(rb2
+s
*(rb3
+s
*(rb4
+s
*(
205 S
= 1.0+s
*(sb1
+s
*(sb2
+s
*(sb3
+s
*(sb4
+s
*(
206 sb5
+s
*(sb6
+s
*sb7
))))));
210 return exp(-z
*z
-0.5625)*exp((z
-x
)*(z
+x
)+R
/S
)/x
;
219 GET_HIGH_WORD(ix
, x
);
222 if (ix
>= 0x7ff00000) {
223 /* erf(nan)=nan, erf(+-inf)=+-1 */
224 return 1-2*sign
+ 1/x
;
226 if (ix
< 0x3feb0000) { /* |x| < 0.84375 */
227 if (ix
< 0x3e300000) { /* |x| < 2**-28 */
228 /* avoid underflow */
229 return 0.125*(8*x
+ efx8
*x
);
232 r
= pp0
+z
*(pp1
+z
*(pp2
+z
*(pp3
+z
*pp4
)));
233 s
= 1.0+z
*(qq1
+z
*(qq2
+z
*(qq3
+z
*(qq4
+z
*qq5
))));
237 if (ix
< 0x40180000) /* 0.84375 <= |x| < 6 */
241 return sign
? -y
: y
;
244 double erfc(double x
)
250 GET_HIGH_WORD(ix
, x
);
253 if (ix
>= 0x7ff00000) {
254 /* erfc(nan)=nan, erfc(+-inf)=0,2 */
257 if (ix
< 0x3feb0000) { /* |x| < 0.84375 */
258 if (ix
< 0x3c700000) /* |x| < 2**-56 */
261 r
= pp0
+z
*(pp1
+z
*(pp2
+z
*(pp3
+z
*pp4
)));
262 s
= 1.0+z
*(qq1
+z
*(qq2
+z
*(qq3
+z
*(qq4
+z
*qq5
))));
264 if (sign
|| ix
< 0x3fd00000) { /* x < 1/4 */
265 return 1.0 - (x
+x
*y
);
267 return 0.5 - (x
- 0.5 + x
*y
);
269 if (ix
< 0x403c0000) { /* 0.84375 <= |x| < 28 */
270 return sign
? 2 - erfc2(ix
,x
) : erfc2(ix
,x
);
272 return sign
? 2 - 0x1p
-1022 : 0x1p
-1022*0x1p
-1022;