GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/s_erf.c Lines: 75 80 93.8 %
Date: 2017-11-13 Branches: 36 42 85.7 %

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/* @(#)s_erf.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5
 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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13
/* double erf(double x)
14
 * double erfc(double x)
15
 *			     x
16
 *		      2      |\
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 *     erf(x)  =  ---------  | exp(-t*t)dt
18
 *	 	   sqrt(pi) \|
19
 *			     0
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 *
21
 *     erfc(x) =  1-erf(x)
22
 *  Note that
23
 *		erf(-x) = -erf(x)
24
 *		erfc(-x) = 2 - erfc(x)
25
 *
26
 * Method:
27
 *	1. For |x| in [0, 0.84375]
28
 *	    erf(x)  = x + x*R(x^2)
29
 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
30
 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
31
 *	   where R = P/Q where P is an odd poly of degree 8 and
32
 *	   Q is an odd poly of degree 10.
33
 *						 -57.90
34
 *			| R - (erf(x)-x)/x | <= 2
35
 *
36
 *
37
 *	   Remark. The formula is derived by noting
38
 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
39
 *	   and that
40
 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
41
 *	   is close to one. The interval is chosen because the fix
42
 *	   point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
43
 *	   near 0.6174), and by some experiment, 0.84375 is chosen to
44
 * 	   guarantee the error is less than one ulp for erf.
45
 *
46
 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
47
 *         c = 0.84506291151 rounded to single (24 bits)
48
 *         	erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
49
 *         	erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
50
 *			  1+(c+P1(s)/Q1(s))    if x < 0
51
 *         	|P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
52
 *	   Remark: here we use the taylor series expansion at x=1.
53
 *		erf(1+s) = erf(1) + s*Poly(s)
54
 *			 = 0.845.. + P1(s)/Q1(s)
55
 *	   That is, we use rational approximation to approximate
56
 *			erf(1+s) - (c = (single)0.84506291151)
57
 *	   Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
58
 *	   where
59
 *		P1(s) = degree 6 poly in s
60
 *		Q1(s) = degree 6 poly in s
61
 *
62
 *      3. For x in [1.25,1/0.35(~2.857143)],
63
 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
64
 *         	erf(x)  = 1 - erfc(x)
65
 *	   where
66
 *		R1(z) = degree 7 poly in z, (z=1/x^2)
67
 *		S1(z) = degree 8 poly in z
68
 *
69
 *      4. For x in [1/0.35,28]
70
 *         	erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
71
 *			= 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
72
 *			= 2.0 - tiny		(if x <= -6)
73
 *         	erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
74
 *         	erf(x)  = sign(x)*(1.0 - tiny)
75
 *	   where
76
 *		R2(z) = degree 6 poly in z, (z=1/x^2)
77
 *		S2(z) = degree 7 poly in z
78
 *
79
 *      Note1:
80
 *	   To compute exp(-x*x-0.5625+R/S), let s be a single
81
 *	   precision number and s := x; then
82
 *		-x*x = -s*s + (s-x)*(s+x)
83
 *	        exp(-x*x-0.5626+R/S) =
84
 *			exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
85
 *      Note2:
86
 *	   Here 4 and 5 make use of the asymptotic series
87
 *			  exp(-x*x)
88
 *		erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
89
 *			  x*sqrt(pi)
90
 *	   We use rational approximation to approximate
91
 *      	g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
92
 *	   Here is the error bound for R1/S1 and R2/S2
93
 *      	|R1/S1 - f(x)|  < 2**(-62.57)
94
 *      	|R2/S2 - f(x)|  < 2**(-61.52)
95
 *
96
 *      5. For inf > x >= 28
97
 *         	erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
98
 *         	erfc(x) = tiny*tiny (raise underflow) if x > 0
99
 *			= 2 - tiny if x<0
100
 *
101
 *      7. Special case:
102
 *         	erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
103
 *         	erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
104
 *	   	erfc/erf(NaN) is NaN
105
 */
106
107
#include <float.h>
108
#include <math.h>
109
110
#include "math_private.h"
111
112
static const double
113
tiny	    = 1e-300,
114
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
115
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
116
two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
117
	/* c = (float)0.84506291151 */
118
erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
119
/*
120
 * Coefficients for approximation to  erf on [0,0.84375]
121
 */
122
efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
123
efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
124
pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
125
pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
126
pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
127
pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
128
pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
129
qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
130
qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
131
qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
132
qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
133
qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
134
/*
135
 * Coefficients for approximation to  erf  in [0.84375,1.25]
136
 */
137
pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
138
pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
139
pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
140
pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
141
pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
142
pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
143
pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
144
qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
145
qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
146
qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
147
qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
148
qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
149
qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
150
/*
151
 * Coefficients for approximation to  erfc in [1.25,1/0.35]
152
 */
153
ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
154
ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
155
ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
156
ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
157
ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
158
ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
159
ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
160
ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
161
sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
162
sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
163
sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
164
sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
165
sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
166
sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
167
sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
168
sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
169
/*
170
 * Coefficients for approximation to  erfc in [1/.35,28]
171
 */
172
rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
173
rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
174
rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
175
rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
176
rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
177
rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
178
rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
179
sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
180
sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
181
sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
182
sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
183
sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
184
sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
185
sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
186
187
double
188
erf(double x)
189
{
190
	int32_t hx,ix,i;
191
	double R,S,P,Q,s,y,z,r;
192
15324
	GET_HIGH_WORD(hx,x);
193
7662
	ix = hx&0x7fffffff;
194
7662
	if(ix>=0x7ff00000) {		/* erf(nan)=nan */
195
	    i = ((u_int32_t)hx>>31)<<1;
196
	    return (double)(1-i)+one/x;	/* erf(+-inf)=+-1 */
197
	}
198
199
7662
	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
200
1260
	    if(ix < 0x3e300000) { 	/* |x|<2**-28 */
201
18
	        if (ix < 0x00800000)
202
18
		    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
203
		return x + efx*x;
204
	    }
205
1242
	    z = x*x;
206
1242
	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
207
1242
	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
208
1242
	    y = r/s;
209
1242
	    return x + x*y;
210
	}
211
6402
	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
212
738
	    s = fabs(x)-one;
213
738
	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
214
738
	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
215
1476
	    if(hx>=0) return erx + P/Q; else return -erx - P/Q;
216
	}
217
5664
	if (ix >= 0x40180000) {		/* inf>|x|>=6 */
218
732
	    if(hx>=0) return one-tiny; else return tiny-one;
219
	}
220
5298
	x = fabs(x);
221
5298
 	s = one/(x*x);
222
5298
	if(ix< 0x4006DB6E) {	/* |x| < 1/0.35 */
223
1680
	    R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
224
1680
				ra5+s*(ra6+s*ra7))))));
225
1680
	    S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
226
1680
				sa5+s*(sa6+s*(sa7+s*sa8)))))));
227
1680
	} else {	/* |x| >= 1/0.35 */
228
3618
	    R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
229
3618
				rb5+s*rb6)))));
230
3618
	    S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
231
3618
				sb5+s*(sb6+s*sb7))))));
232
	}
233
	z  = x;
234
5298
	SET_LOW_WORD(z,0);
235
5298
	r  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
236
10596
	if(hx>=0) return one-r/x; else return  r/x-one;
237
7662
}
238
DEF_STD(erf);
239
LDBL_MAYBE_CLONE(erf);
240
241
double
242
erfc(double x)
243
{
244
	int32_t hx,ix;
245
	double R,S,P,Q,s,y,z,r;
246
15816
	GET_HIGH_WORD(hx,x);
247
7908
	ix = hx&0x7fffffff;
248
7908
	if(ix>=0x7ff00000) {			/* erfc(nan)=nan */
249
						/* erfc(+-inf)=0,2 */
250
	    return (double)(((u_int32_t)hx>>31)<<1)+one/x;
251
	}
252
253
7908
	if(ix < 0x3feb0000) {		/* |x|<0.84375 */
254
564
	    if(ix < 0x3c700000)  	/* |x|<2**-56 */
255
18
		return one-x;
256
546
	    z = x*x;
257
546
	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
258
546
	    s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
259
546
	    y = r/s;
260
546
	    if(hx < 0x3fd00000) {  	/* x<1/4 */
261
372
		return one-(x+x*y);
262
	    } else {
263
174
		r = x*y;
264
174
		r += (x-half);
265
174
	        return half - r ;
266
	    }
267
	}
268
7344
	if(ix < 0x3ff40000) {		/* 0.84375 <= |x| < 1.25 */
269
654
	    s = fabs(x)-one;
270
654
	    P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
271
654
	    Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
272
654
	    if(hx>=0) {
273
426
	        z  = one-erx; return z - P/Q;
274
	    } else {
275
228
		z = erx+P/Q; return one+z;
276
	    }
277
	}
278
6690
	if (ix < 0x403c0000) {		/* |x|<28 */
279
6690
	    x = fabs(x);
280
6690
 	    s = one/(x*x);
281
6690
	    if(ix< 0x4006DB6D) {	/* |x| < 1/.35 ~ 2.857143*/
282
1476
	        R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
283
1476
				ra5+s*(ra6+s*ra7))))));
284
1476
	        S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
285
1476
				sa5+s*(sa6+s*(sa7+s*sa8)))))));
286
1476
	    } else {			/* |x| >= 1/.35 ~ 2.857143 */
287
5226
		if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
288
5202
	        R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
289
5202
				rb5+s*rb6)))));
290
5202
	        S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
291
5202
				sb5+s*(sb6+s*sb7))))));
292
	    }
293
	    z  = x;
294
6678
	    SET_LOW_WORD(z,0);
295
6678
	    r  =  exp(-z*z-0.5625) * exp((z-x)*(z+x)+R/S);
296
13356
	    if(hx>0) return r/x; else return two-r/x;
297
	} else {
298
	    if(hx>0) return tiny*tiny; else return two-tiny;
299
	}
300
7908
}
301
DEF_STD(erfc);
302
LDBL_MAYBE_CLONE(erfc);