svn commit: r254969 - head/lib/msun/src
Steve Kargl
kargl at FreeBSD.org
Tue Aug 27 19:46:58 UTC 2013
Author: kargl
Date: Tue Aug 27 19:46:56 2013
New Revision: 254969
URL: http://svnweb.freebsd.org/changeset/base/254969
Log:
* s_erf.c:
. Use integer literal constants instead of double literal constants.
* s_erff.c:
. Use integer literal constants instead of casting double literal
constants to float.
. Update the threshold values from those carried over from erf() to
values appropriate for float.
. New sets of polynomial coefficients for the rational approximations.
These coefficients have little, but positive, effect on the maximum
error in ULP in the four intervals, but do improve the overall
speed of execution.
. Remove redundant GET_FLOAT_WORD(ix,x) as hx already contained the
contents that is packed into ix.
. Update the mask that is used to zero-out lower-order bits in x in
the intervals [1.25, 2.857143] and [2.857143, 12]. In tests on
amd64, this change improves the maximum error in ULP from 6.27739
and 63.8095 to 3.16774 and 2.92095 on these intervals for erffc().
Reviewed by: bde
Modified:
head/lib/msun/src/s_erf.c
head/lib/msun/src/s_erff.c
Modified: head/lib/msun/src/s_erf.c
==============================================================================
--- head/lib/msun/src/s_erf.c Tue Aug 27 19:10:36 2013 (r254968)
+++ head/lib/msun/src/s_erf.c Tue Aug 27 19:46:56 2013 (r254969)
@@ -201,7 +201,7 @@ erf(double x)
if(ix < 0x3feb0000) { /* |x|<0.84375 */
if(ix < 0x3e300000) { /* |x|<2**-28 */
if (ix < 0x00800000)
- return 0.125*(8.0*x+efx8*x); /*avoid underflow */
+ return (8*x+efx8*x)/8; /* avoid spurious underflow */
return x + efx*x;
}
z = x*x;
Modified: head/lib/msun/src/s_erff.c
==============================================================================
--- head/lib/msun/src/s_erff.c Tue Aug 27 19:10:36 2013 (r254968)
+++ head/lib/msun/src/s_erff.c Tue Aug 27 19:46:56 2013 (r254969)
@@ -24,75 +24,59 @@ tiny = 1e-30,
half= 5.0000000000e-01, /* 0x3F000000 */
one = 1.0000000000e+00, /* 0x3F800000 */
two = 2.0000000000e+00, /* 0x40000000 */
- /* c = (subfloat)0.84506291151 */
-erx = 8.4506291151e-01, /* 0x3f58560b */
/*
- * Coefficients for approximation to erf on [0,0.84375]
+ * Coefficients for approximation to erf on [0,0.84375]
*/
efx = 1.2837916613e-01, /* 0x3e0375d4 */
efx8= 1.0270333290e+00, /* 0x3f8375d4 */
-pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
-pp1 = -3.2504209876e-01, /* 0xbea66beb */
-pp2 = -2.8481749818e-02, /* 0xbce9528f */
-pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
-pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
-qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
-qq2 = 6.5022252500e-02, /* 0x3d852a63 */
-qq3 = 5.0813062117e-03, /* 0x3ba68116 */
-qq4 = 1.3249473704e-04, /* 0x390aee49 */
-qq5 = -3.9602282413e-06, /* 0xb684e21a */
/*
- * Coefficients for approximation to erf in [0.84375,1.25]
+ * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
+ * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
*/
-pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
-pa1 = 4.1485610604e-01, /* 0x3ed46805 */
-pa2 = -3.7220788002e-01, /* 0xbebe9208 */
-pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
-pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
-pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
-pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
-qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
-qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
-qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
-qa4 = 1.2617121637e-01, /* 0x3e013307 */
-qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
-qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
+pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */
+pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */
+pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */
+qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */
+qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */
+qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */
/*
- * Coefficients for approximation to erfc in [1.25,1/0.35]
+ * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
+ * |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
*/
-ra0 = -9.8649440333e-03, /* 0xbc21a093 */
-ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */
-ra2 = -1.0558626175e+01, /* 0xc128f022 */
-ra3 = -6.2375331879e+01, /* 0xc2798057 */
-ra4 = -1.6239666748e+02, /* 0xc322658c */
-ra5 = -1.8460508728e+02, /* 0xc3389ae7 */
-ra6 = -8.1287437439e+01, /* 0xc2a2932b */
-ra7 = -9.8143291473e+00, /* 0xc11d077e */
-sa1 = 1.9651271820e+01, /* 0x419d35ce */
-sa2 = 1.3765776062e+02, /* 0x4309a863 */
-sa3 = 4.3456588745e+02, /* 0x43d9486f */
-sa4 = 6.4538726807e+02, /* 0x442158c9 */
-sa5 = 4.2900814819e+02, /* 0x43d6810b */
-sa6 = 1.0863500214e+02, /* 0x42d9451f */
-sa7 = 6.5702495575e+00, /* 0x40d23f7c */
-sa8 = -6.0424413532e-02, /* 0xbd777f97 */
+erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */
+pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */
+pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */
+pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */
+pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */
+qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */
+qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */
+qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */
+qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */
/*
- * Coefficients for approximation to erfc in [1/.35,28]
+ * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
*/
-rb0 = -9.8649431020e-03, /* 0xbc21a092 */
-rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */
-rb2 = -1.7757955551e+01, /* 0xc18e104b */
-rb3 = -1.6063638306e+02, /* 0xc320a2ea */
-rb4 = -6.3756646729e+02, /* 0xc41f6441 */
-rb5 = -1.0250950928e+03, /* 0xc480230b */
-rb6 = -4.8351919556e+02, /* 0xc3f1c275 */
-sb1 = 3.0338060379e+01, /* 0x41f2b459 */
-sb2 = 3.2579251099e+02, /* 0x43a2e571 */
-sb3 = 1.5367296143e+03, /* 0x44c01759 */
-sb4 = 3.1998581543e+03, /* 0x4547fdbb */
-sb5 = 2.5530502930e+03, /* 0x451f90ce */
-sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
-sb7 = -2.2440952301e+01; /* 0xc1b38712 */
+ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */
+ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */
+ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */
+ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
+sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */
+sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */
+sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */
+sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */
+/*
+ * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
+ * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
+ */
+rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */
+rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */
+rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
+rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
+rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */
+sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */
+sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */
+sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */
+sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */
float
erff(float x)
@@ -107,43 +91,37 @@ erff(float x)
}
if(ix < 0x3f580000) { /* |x|<0.84375 */
- if(ix < 0x31800000) { /* |x|<2**-28 */
- if (ix < 0x04000000)
- /*avoid underflow */
- return (float)0.125*((float)8.0*x+efx8*x);
+ if(ix < 0x38800000) { /* |x|<2**-14 */
+ if (ix < 0x04000000) /* |x|<0x1p-119 */
+ return (8*x+efx8*x)/8; /* avoid spurious underflow */
return x + efx*x;
}
z = x*x;
- r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
- s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+ r = pp0+z*(pp1+z*pp2);
+ s = one+z*(qq1+z*(qq2+z*qq3));
y = r/s;
return x + x*y;
}
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
- P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
- Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+ P = pa0+s*(pa1+s*(pa2+s*pa3));
+ Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
}
- if (ix >= 0x40c00000) { /* inf>|x|>=6 */
+ if (ix >= 0x40800000) { /* inf>|x|>=4 */
if(hx>=0) return one-tiny; else return tiny-one;
}
x = fabsf(x);
s = one/(x*x);
if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */
- R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
- ra5+s*(ra6+s*ra7))))));
- S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
- sa5+s*(sa6+s*(sa7+s*sa8)))))));
+ R=ra0+s*(ra1+s*(ra2+s*ra3));
+ S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
} else { /* |x| >= 1/0.35 */
- R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
- rb5+s*rb6)))));
- S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
- sb5+s*(sb6+s*sb7))))));
- }
- GET_FLOAT_WORD(ix,x);
- SET_FLOAT_WORD(z,ix&0xfffff000);
- r = __ieee754_expf(-z*z-(float)0.5625)*__ieee754_expf((z-x)*(z+x)+R/S);
+ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
+ S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
+ }
+ SET_FLOAT_WORD(z,hx&0xffffe000);
+ r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
if(hx>=0) return one-r/x; else return r/x-one;
}
@@ -160,11 +138,11 @@ erfcf(float x)
}
if(ix < 0x3f580000) { /* |x|<0.84375 */
- if(ix < 0x23800000) /* |x|<2**-56 */
+ if(ix < 0x33800000) /* |x|<2**-24 */
return one-x;
z = x*x;
- r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
- s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
+ r = pp0+z*(pp1+z*pp2);
+ s = one+z*(qq1+z*(qq2+z*qq3));
y = r/s;
if(hx < 0x3e800000) { /* x<1/4 */
return one-(x+x*y);
@@ -176,33 +154,27 @@ erfcf(float x)
}
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
- P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
- Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
+ P = pa0+s*(pa1+s*(pa2+s*pa3));
+ Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
if(hx>=0) {
z = one-erx; return z - P/Q;
} else {
z = erx+P/Q; return one+z;
}
}
- if (ix < 0x41e00000) { /* |x|<28 */
+ if (ix < 0x41300000) { /* |x|<11 */
x = fabsf(x);
s = one/(x*x);
if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/
- R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
- ra5+s*(ra6+s*ra7))))));
- S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
- sa5+s*(sa6+s*(sa7+s*sa8)))))));
+ R=ra0+s*(ra1+s*(ra2+s*ra3));
+ S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
} else { /* |x| >= 1/.35 ~ 2.857143 */
- if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
- R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
- rb5+s*rb6)))));
- S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
- sb5+s*(sb6+s*sb7))))));
+ if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
+ R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
+ S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
}
- GET_FLOAT_WORD(ix,x);
- SET_FLOAT_WORD(z,ix&0xfffff000);
- r = __ieee754_expf(-z*z-(float)0.5625)*
- __ieee754_expf((z-x)*(z+x)+R/S);
+ SET_FLOAT_WORD(z,hx&0xffffe000);
+ r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
if(hx>0) return r/x; else return two-r/x;
} else {
if(hx>0) return tiny*tiny; else return two-tiny;
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