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1/* @(#)s_erf.c 5.1 93/09/24 */
2/*
3 * ====================================================
5 *
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
9 * is preserved.
10 * ====================================================
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11 */
12
967141b1 13
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14/* double erf(double x)
15 * double erfc(double x)
16 * x
17 * 2 |\
18 * erf(x) = --------- | exp(-t*t)dt
19 * sqrt(pi) \|
20 * 0
21 *
22 * erfc(x) = 1-erf(x)
23 * Note that
24 * erf(-x) = -erf(x)
25 * erfc(-x) = 2 - erfc(x)
26 *
27 * Method:
28 * 1. For |x| in [0, 0.84375]
29 * erf(x) = x + x*R(x^2)
30 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
31 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
32 * where R = P/Q where P is an odd poly of degree 8 and
33 * Q is an odd poly of degree 10.
34 * -57.90
35 * | R - (erf(x)-x)/x | <= 2
36 *
37 *
38 * Remark. The formula is derived by noting
39 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
40 * and that
41 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
42 * is close to one. The interval is chosen because the fix
43 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
44 * near 0.6174), and by some experiment, 0.84375 is chosen to
45 * guarantee the error is less than one ulp for erf.
46 *
47 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
48 * c = 0.84506291151 rounded to single (24 bits)
49 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
50 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
51 * 1+(c+P1(s)/Q1(s)) if x < 0
52 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
53 * Remark: here we use the taylor series expansion at x=1.
54 * erf(1+s) = erf(1) + s*Poly(s)
55 * = 0.845.. + P1(s)/Q1(s)
56 * That is, we use rational approximation to approximate
57 * erf(1+s) - (c = (single)0.84506291151)
58 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
59 * where
60 * P1(s) = degree 6 poly in s
61 * Q1(s) = degree 6 poly in s
62 *
63 * 3. For x in [1.25,1/0.35(~2.857143)],
64 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
65 * erf(x) = 1 - erfc(x)
66 * where
67 * R1(z) = degree 7 poly in z, (z=1/x^2)
68 * S1(z) = degree 8 poly in z
69 *
70 * 4. For x in [1/0.35,28]
71 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
72 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
73 * = 2.0 - tiny (if x <= -6)
74 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
75 * erf(x) = sign(x)*(1.0 - tiny)
76 * where
77 * R2(z) = degree 6 poly in z, (z=1/x^2)
78 * S2(z) = degree 7 poly in z
79 *
80 * Note1:
81 * To compute exp(-x*x-0.5625+R/S), let s be a single
82 * precision number and s := x; then
83 * -x*x = -s*s + (s-x)*(s+x)
84 * exp(-x*x-0.5626+R/S) =
85 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
86 * Note2:
87 * Here 4 and 5 make use of the asymptotic series
88 * exp(-x*x)
89 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
90 * x*sqrt(pi)
91 * We use rational approximation to approximate
92 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
93 * Here is the error bound for R1/S1 and R2/S2
94 * |R1/S1 - f(x)| < 2**(-62.57)
95 * |R2/S2 - f(x)| < 2**(-61.52)
96 *
97 * 5. For inf > x >= 28
98 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
99 * erfc(x) = tiny*tiny (raise underflow) if x > 0
100 * = 2 - tiny if x<0
101 *
102 * 7. Special case:
103 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
104 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
105 * erfc/erf(NaN) is NaN
106 */
107
108
6ff43c94 109#include "math.h"
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110#include "math_private.h"
111
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112/* XXX Prevent compilers from erroneously constant folding: */
113static const volatile double tiny= 1e-300;
114
b34b60bc 115static const double
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116half= 0.5,
117one = 1,
118two = 2,
119/* c = (float)0.84506291151 */
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120erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
121/*
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122 * In the domain [0, 2**-28], only the first term in the power series
123 * expansion of erf(x) is used. The magnitude of the first neglected
124 * terms is less than 2**-84.
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125 */
126efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
127efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
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128/*
129 * Coefficients for approximation to erf on [0,0.84375]
130 */
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131pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
132pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
133pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
134pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
135pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
136qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
137qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
138qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
139qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
140qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
141/*
2fedfd5c 142 * Coefficients for approximation to erf in [0.84375,1.25]
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143 */
144pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
145pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
146pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
147pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
148pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
149pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
150pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
151qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
152qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
153qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
154qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
155qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
156qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
157/*
2fedfd5c 158 * Coefficients for approximation to erfc in [1.25,1/0.35]
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159 */
160ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
161ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
162ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
163ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
164ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
165ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
166ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
167ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
168sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
169sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
170sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
171sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
172sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
173sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
174sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
175sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
176/*
2fedfd5c 177 * Coefficients for approximation to erfc in [1/.35,28]
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178 */
179rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
180rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
181rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
182rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
183rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
184rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
185rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
186sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
187sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
188sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
189sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
190sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
191sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
192sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
193
194double
195erf(double x)
196{
197 int32_t hx,ix,i;
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 */
202 i = ((u_int32_t)hx>>31)<<1;
203 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
204 }
205
206 if(ix < 0x3feb0000) { /* |x|<0.84375 */
207 if(ix < 0x3e300000) { /* |x|<2**-28 */
208 if (ix < 0x00800000)
967141b1 209 return (8*x+efx8*x)/8; /* avoid spurious underflow */
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210 return x + efx*x;
211 }
212 z = x*x;
213 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
214 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
215 y = r/s;
216 return x + x*y;
217 }
218 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
219 s = fabs(x)-one;
220 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
221 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
222 if(hx>=0) return erx + P/Q; else return -erx - P/Q;
223 }
224 if (ix >= 0x40180000) { /* inf>|x|>=6 */
225 if(hx>=0) return one-tiny; else return tiny-one;
226 }
227 x = fabs(x);
228 s = one/(x*x);
229 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
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230 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
231 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
232 s*sa8)))))));
b34b60bc 233 } else { /* |x| >= 1/0.35 */
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234 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
235 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
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236 }
237 z = x;
238 SET_LOW_WORD(z,0);
6ff43c94 239 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
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240 if(hx>=0) return one-r/x; else return r/x-one;
241}
242
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243#if (LDBL_MANT_DIG == 53)
244__weak_reference(erf, erfl);
245#endif
246
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247double
248erfc(double x)
249{
250 int32_t hx,ix;
251 double R,S,P,Q,s,y,z,r;
252 GET_HIGH_WORD(hx,x);
253 ix = hx&0x7fffffff;
254 if(ix>=0x7ff00000) { /* erfc(nan)=nan */
255 /* erfc(+-inf)=0,2 */
256 return (double)(((u_int32_t)hx>>31)<<1)+one/x;
257 }
258
259 if(ix < 0x3feb0000) { /* |x|<0.84375 */
260 if(ix < 0x3c700000) /* |x|<2**-56 */
261 return one-x;
262 z = x*x;
263 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
264 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
265 y = r/s;
266 if(hx < 0x3fd00000) { /* x<1/4 */
267 return one-(x+x*y);
268 } else {
269 r = x*y;
270 r += (x-half);
271 return half - r ;
272 }
273 }
274 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
275 s = fabs(x)-one;
276 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
277 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
278 if(hx>=0) {
279 z = one-erx; return z - P/Q;
280 } else {
281 z = erx+P/Q; return one+z;
282 }
283 }
284 if (ix < 0x403c0000) { /* |x|<28 */
285 x = fabs(x);
286 s = one/(x*x);
287 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
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288 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
289 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+
290 s*sa8)))))));
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291 } else { /* |x| >= 1/.35 ~ 2.857143 */
292 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
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293 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
294 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
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295 }
296 z = x;
297 SET_LOW_WORD(z,0);
2fedfd5c 298 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
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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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304
305#if (LDBL_MANT_DIG == 53)
306__weak_reference(erfc, erfcl);
307#endif