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* Use 9 digits instead of 11 digits in efx and efx8.

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* Update the domain and range of comments for the polynomial
  approximations, including using the the correct variable names
  (e.g., pp(x) instead of p(x)).

* Use hex values of the form 0x3e0375d4 instead of 0x1.06eba8p-3,
  which was obtained from printf("%.6a").

* In the domain [0.84375, 1.25], qa(x) can be reduced from a 4th
  order polynomial to 3rd order.

* In the domain [1.25,1/0.35], sa(x) can be reduced from a 4th
  order polynomial to 3rd order.

* In the domain [1/0.35, 11], the 4th order polynomials rb(x) and
  sb(x) can be reduced to 2nd and 3rd order, respectively.
This commit is contained in:
Steve Kargl 2014-07-13 16:24:16 +00:00
parent c9cb48ff4d
commit 7e1c60ae94
1 changed files with 49 additions and 54 deletions
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103
lib/msun/src/s_erff.c
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@ -32,55 +32,50 @@ erx = 8.42697144e-01, /* 0x3f57bb00 */
* expansion of erf(x) is used. The magnitude of the first neglected
* terms is less than 2**-42.
*/
efx = 1.2837916613e-01, /* 0x3e0375d4 */
efx8= 1.0270333290e+00, /* 0x3f8375d4 */
efx = 1.28379166e-01, /* 0x3e0375d4 */
efx8= 1.02703333e+00, /* 0x3f8375d4 */
/*
* Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
* |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
* Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]:
* |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-31
*/
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 */
pp0 = 1.28379166e-01, /* 0x3e0375d4 */
pp1 = -3.36030394e-01, /* 0xbeac0c2d */
pp2 = -1.86261395e-03, /* 0xbaf422f4 */
qq1 = 3.12324315e-01, /* 0x3e9fe8f9 */
qq2 = 2.16070414e-02, /* 0x3cb10140 */
qq3 = -1.98859372e-03, /* 0xbb025311 */
/*
* Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
* |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
* Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]:
* |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31
*/
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 */
pa0 = 3.65041046e-06, /* 0x3674f993 */
pa1 = 4.15109307e-01, /* 0x3ed48935 */
pa2 = -2.09395722e-01, /* 0xbe566bd5 */
pa3 = 8.67677554e-02, /* 0x3db1b34b */
qa1 = 4.95560974e-01, /* 0x3efdba2b */
qa2 = 3.71248513e-01, /* 0x3ebe1449 */
qa3 = 3.92478965e-02, /* 0x3d20c267 */
/*
* 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
* Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]:
* |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28
*/
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 */
ra0 = -9.88156721e-03, /* 0xbc21e64c */
ra1 = -5.43658376e-01, /* 0xbf0b2d32 */
ra2 = -1.66828310e+00, /* 0xbfd58a4d */
ra3 = -6.91554189e-01, /* 0xbf3109b2 */
sa1 = 4.48581553e+00, /* 0x408f8bcd */
sa2 = 4.10799170e+00, /* 0x408374ab */
sa3 = 5.53855181e-01, /* 0x3f0dc974 */
/*
* 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
* Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]:
* |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30
*/
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 */
rb0 = -9.86496918e-03, /* 0xbc21a0ae */
rb1 = -5.48049808e-01, /* 0xbf0c4cfe */
rb2 = -1.84115684e+00, /* 0xbfebab07 */
sb1 = 4.87132740e+00, /* 0x409be1ea */
sb2 = 3.04982710e+00, /* 0x4043305e */
sb3 = -7.61900663e-01; /* 0xbf430bec */
float
erff(float x)
@ -109,7 +104,7 @@ erff(float x)
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
P = pa0+s*(pa1+s*(pa2+s*pa3));
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
Q = one+s*(qa1+s*(qa2+s*qa3));
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
}
if (ix >= 0x40800000) { /* inf>|x|>=4 */
@ -117,12 +112,12 @@ erff(float x)
}
x = fabsf(x);
s = one/(x*x);
if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */
if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/0.35 */
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=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
S=one+s*(sa1+s*(sa2+s*sa3));
} else { /* |x| >= 2.85715 ~ 1/0.35 */
R=rb0+s*(rb1+s*rb2);
S=one+s*(sb1+s*(sb2+s*sb3));
}
SET_FLOAT_WORD(z,hx&0xffffe000);
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
@ -159,7 +154,7 @@ erfcf(float x)
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-one;
P = pa0+s*(pa1+s*(pa2+s*pa3));
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
Q = one+s*(qa1+s*(qa2+s*qa3));
if(hx>=0) {
z = one-erx; return z - P/Q;
} else {
@ -169,13 +164,13 @@ erfcf(float x)
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=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
} else { /* |x| >= 1/.35 ~ 2.857143 */
if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/.35 */
R=ra0+s*(ra1+s*(ra2+s*ra3));
S=one+s*(sa1+s*(sa2+s*sa3));
} else { /* |x| >= 2.85715 ~ 1/.35 */
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)));
R=rb0+s*(rb1+s*rb2);
S=one+s*(sb1+s*(sb2+s*sb3));
}
SET_FLOAT_WORD(z,hx&0xffffe000);
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);