Annotation of src/lib/libm/src/s_erf.c, Revision 1.11
1.1 jtc 1: /* @(#)s_erf.c 5.1 93/09/24 */
2: /*
3: * ====================================================
4: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5: *
6: * Developed at SunPro, a Sun Microsystems, Inc. business.
7: * Permission to use, copy, modify, and distribute this
1.10 simonb 8: * software is freely granted, provided that this notice
1.1 jtc 9: * is preserved.
10: * ====================================================
11: */
1.3 jtc 12:
1.9 lukem 13: #include <sys/cdefs.h>
1.7 jtc 14: #if defined(LIBM_SCCS) && !defined(lint)
1.11 ! wiz 15: __RCSID("$NetBSD: s_erf.c,v 1.10 1999/07/02 15:37:42 simonb Exp $");
1.3 jtc 16: #endif
1.1 jtc 17:
18: /* double erf(double x)
19: * double erfc(double x)
20: * x
21: * 2 |\
22: * erf(x) = --------- | exp(-t*t)dt
1.10 simonb 23: * sqrt(pi) \|
1.1 jtc 24: * 0
25: *
26: * erfc(x) = 1-erf(x)
1.10 simonb 27: * Note that
1.1 jtc 28: * erf(-x) = -erf(x)
29: * erfc(-x) = 2 - erfc(x)
30: *
31: * Method:
32: * 1. For |x| in [0, 0.84375]
33: * erf(x) = x + x*R(x^2)
34: * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
35: * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
36: * where R = P/Q where P is an odd poly of degree 8 and
37: * Q is an odd poly of degree 10.
38: * -57.90
39: * | R - (erf(x)-x)/x | <= 2
1.10 simonb 40: *
1.1 jtc 41: *
42: * Remark. The formula is derived by noting
43: * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
44: * and that
45: * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
46: * is close to one. The interval is chosen because the fix
47: * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
48: * near 0.6174), and by some experiment, 0.84375 is chosen to
49: * guarantee the error is less than one ulp for erf.
50: *
51: * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
52: * c = 0.84506291151 rounded to single (24 bits)
53: * erf(x) = sign(x) * (c + P1(s)/Q1(s))
54: * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
55: * 1+(c+P1(s)/Q1(s)) if x < 0
56: * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
57: * Remark: here we use the taylor series expansion at x=1.
58: * erf(1+s) = erf(1) + s*Poly(s)
59: * = 0.845.. + P1(s)/Q1(s)
60: * That is, we use rational approximation to approximate
61: * erf(1+s) - (c = (single)0.84506291151)
62: * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
1.10 simonb 63: * where
1.1 jtc 64: * P1(s) = degree 6 poly in s
65: * Q1(s) = degree 6 poly in s
66: *
1.10 simonb 67: * 3. For x in [1.25,1/0.35(~2.857143)],
1.1 jtc 68: * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
69: * erf(x) = 1 - erfc(x)
1.10 simonb 70: * where
1.1 jtc 71: * R1(z) = degree 7 poly in z, (z=1/x^2)
72: * S1(z) = degree 8 poly in z
73: *
74: * 4. For x in [1/0.35,28]
75: * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
76: * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
77: * = 2.0 - tiny (if x <= -6)
78: * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
79: * erf(x) = sign(x)*(1.0 - tiny)
80: * where
81: * R2(z) = degree 6 poly in z, (z=1/x^2)
82: * S2(z) = degree 7 poly in z
83: *
84: * Note1:
85: * To compute exp(-x*x-0.5625+R/S), let s be a single
86: * precision number and s := x; then
87: * -x*x = -s*s + (s-x)*(s+x)
1.10 simonb 88: * exp(-x*x-0.5626+R/S) =
1.1 jtc 89: * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
90: * Note2:
91: * Here 4 and 5 make use of the asymptotic series
92: * exp(-x*x)
93: * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
94: * x*sqrt(pi)
95: * We use rational approximation to approximate
96: * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
97: * Here is the error bound for R1/S1 and R2/S2
98: * |R1/S1 - f(x)| < 2**(-62.57)
99: * |R2/S2 - f(x)| < 2**(-61.52)
100: *
101: * 5. For inf > x >= 28
102: * erf(x) = sign(x) *(1 - tiny) (raise inexact)
103: * erfc(x) = tiny*tiny (raise underflow) if x > 0
104: * = 2 - tiny if x<0
105: *
106: * 7. Special case:
107: * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
1.10 simonb 108: * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
1.1 jtc 109: * erfc/erf(NaN) is NaN
110: */
111:
112:
1.5 jtc 113: #include "math.h"
114: #include "math_private.h"
1.1 jtc 115:
116: static const double
117: tiny = 1e-300,
118: half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
119: one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
120: two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
121: /* c = (float)0.84506291151 */
122: erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
123: /*
124: * Coefficients for approximation to erf on [0,0.84375]
125: */
126: efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
127: efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
128: pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
129: pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
130: pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
131: pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
132: pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
133: qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
134: qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
135: qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
136: qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
137: qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
138: /*
1.10 simonb 139: * Coefficients for approximation to erf in [0.84375,1.25]
1.1 jtc 140: */
141: pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
142: pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
143: pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
144: pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
145: pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
146: pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
147: pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
148: qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
149: qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
150: qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
151: qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
152: qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
153: qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
154: /*
155: * Coefficients for approximation to erfc in [1.25,1/0.35]
156: */
157: ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
158: ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
159: ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
160: ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
161: ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
162: ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
163: ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
164: ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
165: sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
166: sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
167: sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
168: sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
169: sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
170: sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
171: sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
172: sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
173: /*
174: * Coefficients for approximation to erfc in [1/.35,28]
175: */
176: rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
177: rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
178: rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
179: rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
180: rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
181: rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
182: rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
183: sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
184: sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
185: sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
186: sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
187: sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
188: sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
189: sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
190:
1.11 ! wiz 191: double
! 192: erf(double x)
1.1 jtc 193: {
1.6 jtc 194: int32_t hx,ix,i;
1.1 jtc 195: double R,S,P,Q,s,y,z,r;
1.5 jtc 196: GET_HIGH_WORD(hx,x);
1.1 jtc 197: ix = hx&0x7fffffff;
198: if(ix>=0x7ff00000) { /* erf(nan)=nan */
1.6 jtc 199: i = ((u_int32_t)hx>>31)<<1;
1.1 jtc 200: return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
201: }
202:
203: if(ix < 0x3feb0000) { /* |x|<0.84375 */
204: if(ix < 0x3e300000) { /* |x|<2**-28 */
1.10 simonb 205: if (ix < 0x00800000)
1.1 jtc 206: return 0.125*(8.0*x+efx8*x); /*avoid underflow */
207: return x + efx*x;
208: }
209: z = x*x;
210: r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
211: s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
212: y = r/s;
213: return x + x*y;
214: }
215: if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
216: s = fabs(x)-one;
217: P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
218: Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
219: if(hx>=0) return erx + P/Q; else return -erx - P/Q;
220: }
221: if (ix >= 0x40180000) { /* inf>|x|>=6 */
222: if(hx>=0) return one-tiny; else return tiny-one;
223: }
224: x = fabs(x);
225: s = one/(x*x);
226: if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
227: R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
228: ra5+s*(ra6+s*ra7))))));
229: S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
230: sa5+s*(sa6+s*(sa7+s*sa8)))))));
231: } else { /* |x| >= 1/0.35 */
232: R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
233: rb5+s*rb6)))));
234: S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
235: sb5+s*(sb6+s*sb7))))));
236: }
1.10 simonb 237: z = x;
1.5 jtc 238: SET_LOW_WORD(z,0);
1.1 jtc 239: r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
240: if(hx>=0) return one-r/x; else return r/x-one;
241: }
242:
1.11 ! wiz 243: double
! 244: erfc(double x)
1.1 jtc 245: {
1.6 jtc 246: int32_t hx,ix;
1.1 jtc 247: double R,S,P,Q,s,y,z,r;
1.5 jtc 248: GET_HIGH_WORD(hx,x);
1.1 jtc 249: ix = hx&0x7fffffff;
250: if(ix>=0x7ff00000) { /* erfc(nan)=nan */
251: /* erfc(+-inf)=0,2 */
1.6 jtc 252: return (double)(((u_int32_t)hx>>31)<<1)+one/x;
1.1 jtc 253: }
254:
255: if(ix < 0x3feb0000) { /* |x|<0.84375 */
256: if(ix < 0x3c700000) /* |x|<2**-56 */
257: return one-x;
258: z = x*x;
259: r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
260: s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
261: y = r/s;
262: if(hx < 0x3fd00000) { /* x<1/4 */
263: return one-(x+x*y);
264: } else {
265: r = x*y;
266: r += (x-half);
267: return half - r ;
268: }
269: }
270: if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
271: s = fabs(x)-one;
272: P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
273: Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
274: if(hx>=0) {
1.10 simonb 275: z = one-erx; return z - P/Q;
1.1 jtc 276: } else {
277: z = erx+P/Q; return one+z;
278: }
279: }
280: if (ix < 0x403c0000) { /* |x|<28 */
281: x = fabs(x);
282: s = one/(x*x);
283: if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
284: R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
285: ra5+s*(ra6+s*ra7))))));
286: S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
287: sa5+s*(sa6+s*(sa7+s*sa8)))))));
288: } else { /* |x| >= 1/.35 ~ 2.857143 */
289: if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
290: R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
291: rb5+s*rb6)))));
292: S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
293: sb5+s*(sb6+s*sb7))))));
294: }
295: z = x;
1.5 jtc 296: SET_LOW_WORD(z,0);
1.1 jtc 297: r = __ieee754_exp(-z*z-0.5625)*
298: __ieee754_exp((z-x)*(z+x)+R/S);
299: if(hx>0) return r/x; else return two-r/x;
300: } else {
301: if(hx>0) return tiny*tiny; else return two-tiny;
302: }
303: }
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