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JeffBezansonJeffBezanson
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new polynomials for erff, by Steve Kargl
these are both faster and more accurate see http://svnweb.freebsd.org/base/head/lib/msun/src/s_erff.c?view=log
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‎src/s_erff.c‎

Lines changed: 72 additions & 99 deletions
Original file line numberDiff line numberDiff line change
@@ -24,75 +24,60 @@ tiny = 1e-30,
2424
half= 5.0000000000e-01, /* 0x3F000000 */
2525
one = 1.0000000000e+00, /* 0x3F800000 */
2626
two = 2.0000000000e+00, /* 0x40000000 */
27-
/* c = (subfloat)0.84506291151 */
28-
erx = 8.4506291151e-01, /* 0x3f58560b */
2927
/*
30-
* Coefficients for approximation to erf on [0,0.84375]
28+
* Coefficients for approximation to erf on [0,0.84375]
3129
*/
3230
efx = 1.2837916613e-01, /* 0x3e0375d4 */
3331
efx8= 1.0270333290e+00, /* 0x3f8375d4 */
34-
pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
35-
pp1 = -3.2504209876e-01, /* 0xbea66beb */
36-
pp2 = -2.8481749818e-02, /* 0xbce9528f */
37-
pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
38-
pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
39-
qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
40-
qq2 = 6.5022252500e-02, /* 0x3d852a63 */
41-
qq3 = 5.0813062117e-03, /* 0x3ba68116 */
42-
qq4 = 1.3249473704e-04, /* 0x390aee49 */
43-
qq5 = -3.9602282413e-06, /* 0xb684e21a */
4432
/*
45-
* Coefficients for approximation to erf in [0.84375,1.25]
33+
* Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
34+
* |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
4635
*/
47-
pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
48-
pa1 = 4.1485610604e-01, /* 0x3ed46805 */
49-
pa2 = -3.7220788002e-01, /* 0xbebe9208 */
50-
pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
51-
pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
52-
pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
53-
pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
54-
qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
55-
qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
56-
qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
57-
qa4 = 1.2617121637e-01, /* 0x3e013307 */
58-
qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
59-
qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
36+
pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */
37+
pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */
38+
pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */
39+
qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */
40+
qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */
41+
qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */
6042
/*
61-
* Coefficients for approximation to erfc in [1.25,1/0.35]
43+
* Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
44+
* |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
6245
*/
63-
ra0 = -9.8649440333e-03, /* 0xbc21a093 */
64-
ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */
65-
ra2 = -1.0558626175e+01, /* 0xc128f022 */
66-
ra3 = -6.2375331879e+01, /* 0xc2798057 */
67-
ra4 = -1.6239666748e+02, /* 0xc322658c */
68-
ra5 = -1.8460508728e+02, /* 0xc3389ae7 */
69-
ra6 = -8.1287437439e+01, /* 0xc2a2932b */
70-
ra7 = -9.8143291473e+00, /* 0xc11d077e */
71-
sa1 = 1.9651271820e+01, /* 0x419d35ce */
72-
sa2 = 1.3765776062e+02, /* 0x4309a863 */
73-
sa3 = 4.3456588745e+02, /* 0x43d9486f */
74-
sa4 = 6.4538726807e+02, /* 0x442158c9 */
75-
sa5 = 4.2900814819e+02, /* 0x43d6810b */
76-
sa6 = 1.0863500214e+02, /* 0x42d9451f */
77-
sa7 = 6.5702495575e+00, /* 0x40d23f7c */
78-
sa8 = -6.0424413532e-02, /* 0xbd777f97 */
46+
erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */
47+
pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */
48+
pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */
49+
pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */
50+
pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */
51+
qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */
52+
qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */
53+
qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */
54+
qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */
7955
/*
80-
* Coefficients for approximation to erfc in [1/.35,28]
56+
* Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
57+
* |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
8158
*/
82-
rb0 = -9.8649431020e-03, /* 0xbc21a092 */
83-
rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */
84-
rb2 = -1.7757955551e+01, /* 0xc18e104b */
85-
rb3 = -1.6063638306e+02, /* 0xc320a2ea */
86-
rb4 = -6.3756646729e+02, /* 0xc41f6441 */
87-
rb5 = -1.0250950928e+03, /* 0xc480230b */
88-
rb6 = -4.8351919556e+02, /* 0xc3f1c275 */
89-
sb1 = 3.0338060379e+01, /* 0x41f2b459 */
90-
sb2 = 3.2579251099e+02, /* 0x43a2e571 */
91-
sb3 = 1.5367296143e+03, /* 0x44c01759 */
92-
sb4 = 3.1998581543e+03, /* 0x4547fdbb */
93-
sb5 = 2.5530502930e+03, /* 0x451f90ce */
94-
sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
95-
sb7 = -2.2440952301e+01; /* 0xc1b38712 */
59+
ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */
60+
ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */
61+
ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */
62+
ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
63+
sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */
64+
sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */
65+
sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */
66+
sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */
67+
/*
68+
* Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
69+
* |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
70+
*/
71+
rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */
72+
rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */
73+
rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
74+
rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
75+
rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */
76+
sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */
77+
sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */
78+
sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */
79+
sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */
80+
9681

9782
DLLEXPORT float
9883
erff(float x)
@@ -107,43 +92,37 @@ erff(float x)
10792
}
10893

10994
if(ix < 0x3f580000) { /* |x|<0.84375 */
110-
if(ix < 0x31800000) { /* |x|<2**-28 */
111-
if (ix < 0x04000000)
112-
/*avoid underflow */
113-
return (float)0.125*((float)8.0*x+efx8*x);
95+
if(ix < 0x38800000) { /* |x|<2**-14 */
96+
if (ix < 0x04000000) /* |x|<0x1p-119 */
97+
return (8*x+efx8*x)/8; /* avoid spurious underflow */
11498
return x + efx*x;
11599
}
116100
z = x*x;
117-
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
118-
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
101+
r = pp0+z*(pp1+z*pp2);
102+
s = one+z*(qq1+z*(qq2+z*qq3));
119103
y = r/s;
120104
return x + x*y;
121105
}
122106
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
123107
s = fabsf(x)-one;
124-
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
125-
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
108+
P = pa0+s*(pa1+s*(pa2+s*pa3));
109+
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
126110
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
127111
}
128-
if (ix >= 0x40c00000) { /* inf>|x|>=6 */
112+
if (ix >= 0x40800000) { /* inf>|x|>=4 */
129113
if(hx>=0) return one-tiny; else return tiny-one;
130114
}
131115
x = fabsf(x);
132116
s = one/(x*x);
133117
if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */
134-
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
135-
ra5+s*(ra6+s*ra7))))));
136-
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
137-
sa5+s*(sa6+s*(sa7+s*sa8)))))));
118+
R=ra0+s*(ra1+s*(ra2+s*ra3));
119+
S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
138120
} else { /* |x| >= 1/0.35 */
139-
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
140-
rb5+s*rb6)))));
141-
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
142-
sb5+s*(sb6+s*sb7))))));
143-
}
144-
GET_FLOAT_WORD(ix,x);
145-
SET_FLOAT_WORD(z,ix&0xfffff000);
146-
r = __ieee754_expf(-z*z-(float)0.5625)*__ieee754_expf((z-x)*(z+x)+R/S);
121+
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
122+
S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
123+
}
124+
SET_FLOAT_WORD(z,hx&0xffffe000);
125+
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
147126
if(hx>=0) return one-r/x; else return r/x-one;
148127
}
149128

@@ -160,11 +139,11 @@ erfcf(float x)
160139
}
161140

162141
if(ix < 0x3f580000) { /* |x|<0.84375 */
163-
if(ix < 0x23800000) /* |x|<2**-56 */
142+
if(ix < 0x33800000) /* |x|<2**-56 */
164143
return one-x;
165144
z = x*x;
166-
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
167-
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
145+
r = pp0+z*(pp1+z*pp2);
146+
s = one+z*(qq1+z*(qq2+z*qq3));
168147
y = r/s;
169148
if(hx < 0x3e800000) { /* x<1/4 */
170149
return one-(x+x*y);
@@ -176,33 +155,27 @@ erfcf(float x)
176155
}
177156
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
178157
s = fabsf(x)-one;
179-
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
180-
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
158+
P = pa0+s*(pa1+s*(pa2+s*pa3));
159+
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
181160
if(hx>=0) {
182161
z = one-erx; return z - P/Q;
183162
} else {
184163
z = erx+P/Q; return one+z;
185164
}
186165
}
187-
if (ix < 0x41e00000) { /* |x|<28 */
166+
if (ix < 0x41300000) { /* |x|<28 */
188167
x = fabsf(x);
189168
s = one/(x*x);
190169
if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/
191-
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
192-
ra5+s*(ra6+s*ra7))))));
193-
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
194-
sa5+s*(sa6+s*(sa7+s*sa8)))))));
170+
R=ra0+s*(ra1+s*(ra2+s*ra3));
171+
S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
195172
} else { /* |x| >= 1/.35 ~ 2.857143 */
196-
if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
197-
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
198-
rb5+s*rb6)))));
199-
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
200-
sb5+s*(sb6+s*sb7))))));
173+
if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
174+
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
175+
S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
201176
}
202-
GET_FLOAT_WORD(ix,x);
203-
SET_FLOAT_WORD(z,ix&0xfffff000);
204-
r = __ieee754_expf(-z*z-(float)0.5625)*
205-
__ieee754_expf((z-x)*(z+x)+R/S);
177+
SET_FLOAT_WORD(z,hx&0xffffe000);
178+
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
206179
if(hx>0) return r/x; else return two-r/x;
207180
} else {
208181
if(hx>0) return tiny*tiny; else return two-tiny;

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