Date: Sun, 13 Jul 2014 16:24:17 +0000 (UTC) From: Steve Kargl <kargl@FreeBSD.org> To: src-committers@freebsd.org, svn-src-all@freebsd.org, svn-src-head@freebsd.org Subject: svn commit: r268590 - head/lib/msun/src Message-ID: <201407131624.s6DGOHqc058487@svn.freebsd.org>
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Author: kargl Date: Sun Jul 13 16:24:16 2014 New Revision: 268590 URL: http://svnweb.freebsd.org/changeset/base/268590 Log: * Use 9 digits instead of 11 digits in efx and efx8. * 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. Modified: head/lib/msun/src/s_erff.c Modified: head/lib/msun/src/s_erff.c ============================================================================== --- head/lib/msun/src/s_erff.c Sun Jul 13 16:05:33 2014 (r268589) +++ head/lib/msun/src/s_erff.c Sun Jul 13 16:24:16 2014 (r268590) @@ -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);
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