1 /* @(#)s_erf.c 5.1 93/09/24 */
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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
16 #if defined(LIBM_SCCS) && !defined(lint)
17 static char rcsid
[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
20 /* double erf(double x)
21 * double erfc(double x)
24 * erf(x) = --------- | exp(-t*t)dt
31 * erfc(-x) = 2 - erfc(x)
34 * 1. For |x| in [0, 0.84375]
35 * erf(x) = x + x*R(x^2)
36 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
37 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
38 * where R = P/Q where P is an odd poly of degree 8 and
39 * Q is an odd poly of degree 10.
41 * | R - (erf(x)-x)/x | <= 2
44 * Remark. The formula is derived by noting
45 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
47 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
48 * is close to one. The interval is chosen because the fix
49 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
50 * near 0.6174), and by some experiment, 0.84375 is chosen to
51 * guarantee the error is less than one ulp for erf.
53 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
54 * c = 0.84506291151 rounded to single (24 bits)
55 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
56 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
57 * 1+(c+P1(s)/Q1(s)) if x < 0
58 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
59 * Remark: here we use the taylor series expansion at x=1.
60 * erf(1+s) = erf(1) + s*Poly(s)
61 * = 0.845.. + P1(s)/Q1(s)
62 * That is, we use rational approximation to approximate
63 * erf(1+s) - (c = (single)0.84506291151)
64 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
66 * P1(s) = degree 6 poly in s
67 * Q1(s) = degree 6 poly in s
69 * 3. For x in [1.25,1/0.35(~2.857143)],
70 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
71 * erf(x) = 1 - erfc(x)
73 * R1(z) = degree 7 poly in z, (z=1/x^2)
74 * S1(z) = degree 8 poly in z
76 * 4. For x in [1/0.35,28]
77 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
78 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
79 * = 2.0 - tiny (if x <= -6)
80 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
81 * erf(x) = sign(x)*(1.0 - tiny)
83 * R2(z) = degree 6 poly in z, (z=1/x^2)
84 * S2(z) = degree 7 poly in z
87 * To compute exp(-x*x-0.5625+R/S), let s be a single
88 * precision number and s := x; then
89 * -x*x = -s*s + (s-x)*(s+x)
90 * exp(-x*x-0.5626+R/S) =
91 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
93 * Here 4 and 5 make use of the asymptotic series
95 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
97 * We use rational approximation to approximate
98 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
99 * Here is the error bound for R1/S1 and R2/S2
100 * |R1/S1 - f(x)| < 2**(-62.57)
101 * |R2/S2 - f(x)| < 2**(-61.52)
103 * 5. For inf > x >= 28
104 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
105 * erfc(x) = tiny*tiny (raise underflow) if x > 0
109 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
110 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
111 * erfc/erf(NaN) is NaN
118 #include <math_private.h>
122 half
= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
123 one
= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
124 two
= 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
125 /* c = (float)0.84506291151 */
126 erx
= 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
128 * Coefficients for approximation to erf on [0,0.84375]
130 efx
= 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
131 pp
[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
132 -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
133 -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
134 -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
135 -2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */
136 qq
[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
137 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
138 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
139 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
140 -3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */
142 * Coefficients for approximation to erf in [0.84375,1.25]
144 pa
[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
145 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
146 -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
147 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
148 -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
149 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
150 -2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */
151 qa
[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
152 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
153 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
154 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
155 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
156 1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */
158 * Coefficients for approximation to erfc in [1.25,1/0.35]
160 ra
[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
161 -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
162 -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
163 -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
164 -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
165 -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
166 -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
167 -9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */
168 sa
[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
169 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
170 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
171 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
172 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
173 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
174 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
175 -6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */
177 * Coefficients for approximation to erfc in [1/.35,28]
179 rb
[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
180 -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
181 -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
182 -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
183 -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
184 -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
185 -4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */
186 sb
[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
187 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
188 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
189 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
190 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
191 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
192 -2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */
198 double R
, S
, P
, Q
, s
, y
, z
, r
;
199 GET_HIGH_WORD (hx
, x
);
200 ix
= hx
& 0x7fffffff;
201 if (ix
>= 0x7ff00000) /* erf(nan)=nan */
203 i
= ((u_int32_t
) hx
>> 31) << 1;
204 return (double) (1 - i
) + one
/ x
; /* erf(+-inf)=+-1 */
207 if (ix
< 0x3feb0000) /* |x|<0.84375 */
209 double r1
, r2
, s1
, s2
, s3
, z2
, z4
;
210 if (ix
< 0x3e300000) /* |x|<2**-28 */
214 /* Avoid spurious underflow. */
215 double ret
= 0.0625 * (16.0 * x
+ (16.0 * efx
) * x
);
216 if (fabs (ret
) < DBL_MIN
)
218 double force_underflow
= ret
* ret
;
219 math_force_eval (force_underflow
);
226 r1
= pp
[0] + z
* pp
[1]; z2
= z
* z
;
227 r2
= pp
[2] + z
* pp
[3]; z4
= z2
* z2
;
228 s1
= one
+ z
* qq
[1];
229 s2
= qq
[2] + z
* qq
[3];
230 s3
= qq
[4] + z
* qq
[5];
231 r
= r1
+ z2
* r2
+ z4
* pp
[4];
232 s
= s1
+ z2
* s2
+ z4
* s3
;
236 if (ix
< 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
238 double s2
, s4
, s6
, P1
, P2
, P3
, P4
, Q1
, Q2
, Q3
, Q4
;
240 P1
= pa
[0] + s
* pa
[1]; s2
= s
* s
;
241 Q1
= one
+ s
* qa
[1]; s4
= s2
* s2
;
242 P2
= pa
[2] + s
* pa
[3]; s6
= s4
* s2
;
243 Q2
= qa
[2] + s
* qa
[3];
244 P3
= pa
[4] + s
* pa
[5];
245 Q3
= qa
[4] + s
* qa
[5];
248 P
= P1
+ s2
* P2
+ s4
* P3
+ s6
* P4
;
249 Q
= Q1
+ s2
* Q2
+ s4
* Q3
+ s6
* Q4
;
255 if (ix
>= 0x40180000) /* inf>|x|>=6 */
264 if (ix
< 0x4006DB6E) /* |x| < 1/0.35 */
266 double R1
, R2
, R3
, R4
, S1
, S2
, S3
, S4
, s2
, s4
, s6
, s8
;
267 R1
= ra
[0] + s
* ra
[1]; s2
= s
* s
;
268 S1
= one
+ s
* sa
[1]; s4
= s2
* s2
;
269 R2
= ra
[2] + s
* ra
[3]; s6
= s4
* s2
;
270 S2
= sa
[2] + s
* sa
[3]; s8
= s4
* s4
;
271 R3
= ra
[4] + s
* ra
[5];
272 S3
= sa
[4] + s
* sa
[5];
273 R4
= ra
[6] + s
* ra
[7];
274 S4
= sa
[6] + s
* sa
[7];
275 R
= R1
+ s2
* R2
+ s4
* R3
+ s6
* R4
;
276 S
= S1
+ s2
* S2
+ s4
* S3
+ s6
* S4
+ s8
* sa
[8];
278 else /* |x| >= 1/0.35 */
280 double R1
, R2
, R3
, S1
, S2
, S3
, S4
, s2
, s4
, s6
;
281 R1
= rb
[0] + s
* rb
[1]; s2
= s
* s
;
282 S1
= one
+ s
* sb
[1]; s4
= s2
* s2
;
283 R2
= rb
[2] + s
* rb
[3]; s6
= s4
* s2
;
284 S2
= sb
[2] + s
* sb
[3];
285 R3
= rb
[4] + s
* rb
[5];
286 S3
= sb
[4] + s
* sb
[5];
287 S4
= sb
[6] + s
* sb
[7];
288 R
= R1
+ s2
* R2
+ s4
* R3
+ s6
* rb
[6];
289 S
= S1
+ s2
* S2
+ s4
* S3
+ s6
* S4
;
293 r
= __ieee754_exp (-z
* z
- 0.5625) *
294 __ieee754_exp ((z
- x
) * (z
+ x
) + R
/ S
);
300 weak_alias (__erf
, erf
)
301 #ifdef NO_LONG_DOUBLE
302 strong_alias (__erf
, __erfl
)
303 weak_alias (__erf
, erfl
)
310 double R
, S
, P
, Q
, s
, y
, z
, r
;
311 GET_HIGH_WORD (hx
, x
);
312 ix
= hx
& 0x7fffffff;
313 if (ix
>= 0x7ff00000) /* erfc(nan)=nan */
314 { /* erfc(+-inf)=0,2 */
315 return (double) (((u_int32_t
) hx
>> 31) << 1) + one
/ x
;
318 if (ix
< 0x3feb0000) /* |x|<0.84375 */
320 double r1
, r2
, s1
, s2
, s3
, z2
, z4
;
321 if (ix
< 0x3c700000) /* |x|<2**-56 */
324 r1
= pp
[0] + z
* pp
[1]; z2
= z
* z
;
325 r2
= pp
[2] + z
* pp
[3]; z4
= z2
* z2
;
326 s1
= one
+ z
* qq
[1];
327 s2
= qq
[2] + z
* qq
[3];
328 s3
= qq
[4] + z
* qq
[5];
329 r
= r1
+ z2
* r2
+ z4
* pp
[4];
330 s
= s1
+ z2
* s2
+ z4
* s3
;
332 if (hx
< 0x3fd00000) /* x<1/4 */
334 return one
- (x
+ x
* y
);
343 if (ix
< 0x3ff40000) /* 0.84375 <= |x| < 1.25 */
345 double s2
, s4
, s6
, P1
, P2
, P3
, P4
, Q1
, Q2
, Q3
, Q4
;
347 P1
= pa
[0] + s
* pa
[1]; s2
= s
* s
;
348 Q1
= one
+ s
* qa
[1]; s4
= s2
* s2
;
349 P2
= pa
[2] + s
* pa
[3]; s6
= s4
* s2
;
350 Q2
= qa
[2] + s
* qa
[3];
351 P3
= pa
[4] + s
* pa
[5];
352 Q3
= qa
[4] + s
* qa
[5];
355 P
= P1
+ s2
* P2
+ s4
* P3
+ s6
* P4
;
356 Q
= Q1
+ s2
* Q2
+ s4
* Q3
+ s6
* Q4
;
359 z
= one
- erx
; return z
- P
/ Q
;
363 z
= erx
+ P
/ Q
; return one
+ z
;
366 if (ix
< 0x403c0000) /* |x|<28 */
370 if (ix
< 0x4006DB6D) /* |x| < 1/.35 ~ 2.857143*/
372 double R1
, R2
, R3
, R4
, S1
, S2
, S3
, S4
, s2
, s4
, s6
, s8
;
373 R1
= ra
[0] + s
* ra
[1]; s2
= s
* s
;
374 S1
= one
+ s
* sa
[1]; s4
= s2
* s2
;
375 R2
= ra
[2] + s
* ra
[3]; s6
= s4
* s2
;
376 S2
= sa
[2] + s
* sa
[3]; s8
= s4
* s4
;
377 R3
= ra
[4] + s
* ra
[5];
378 S3
= sa
[4] + s
* sa
[5];
379 R4
= ra
[6] + s
* ra
[7];
380 S4
= sa
[6] + s
* sa
[7];
381 R
= R1
+ s2
* R2
+ s4
* R3
+ s6
* R4
;
382 S
= S1
+ s2
* S2
+ s4
* S3
+ s6
* S4
+ s8
* sa
[8];
384 else /* |x| >= 1/.35 ~ 2.857143 */
386 double R1
, R2
, R3
, S1
, S2
, S3
, S4
, s2
, s4
, s6
;
387 if (hx
< 0 && ix
>= 0x40180000)
388 return two
- tiny
; /* x < -6 */
389 R1
= rb
[0] + s
* rb
[1]; s2
= s
* s
;
390 S1
= one
+ s
* sb
[1]; s4
= s2
* s2
;
391 R2
= rb
[2] + s
* rb
[3]; s6
= s4
* s2
;
392 S2
= sb
[2] + s
* sb
[3];
393 R3
= rb
[4] + s
* rb
[5];
394 S3
= sb
[4] + s
* sb
[5];
395 S4
= sb
[6] + s
* sb
[7];
396 R
= R1
+ s2
* R2
+ s4
* R3
+ s6
* rb
[6];
397 S
= S1
+ s2
* S2
+ s4
* S3
+ s6
* S4
;
401 r
= __ieee754_exp (-z
* z
- 0.5625) *
402 __ieee754_exp ((z
- x
) * (z
+ x
) + R
/ S
);
405 #if FLT_EVAL_METHOD != 0
410 __set_errno (ERANGE
);
420 __set_errno (ERANGE
);
427 weak_alias (__erfc
, erfc
)
428 #ifdef NO_LONG_DOUBLE
429 strong_alias (__erfc
, __erfcl
)
430 weak_alias (__erfc
, erfcl
)