From 121e3a38a1d0b324696fc1671029e4b815ea7fa4 Mon Sep 17 00:00:00 2001 From: Szabolcs Nagy Date: Mon, 7 Jan 2013 23:54:13 +0100 Subject: math: erf and erfc cleanup common part of erf and erfc was put in a separate function which saved some space and the new code is using unsigned arithmetics erfcf had a bug: for some inputs in [7.95,8] the result had more than 60ulp error: in expf(-z*z - 0.5625f) the argument must be exact but not enough lowbits of z were zeroed, -SET_FLOAT_WORD(z, ix&0xfffff000); +SET_FLOAT_WORD(z, ix&0xffffe000); fixed the issue --- src/math/erfl.c | 195 ++++++++++++++++++++++++-------------------------------- 1 file changed, 84 insertions(+), 111 deletions(-) (limited to 'src/math/erfl.c') diff --git a/src/math/erfl.c b/src/math/erfl.c index f7449859..fa30d98e 100644 --- a/src/math/erfl.c +++ b/src/math/erfl.c @@ -107,15 +107,11 @@ long double erfl(long double x) } #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 static const long double -tiny = 1e-4931L, -/* c = (float)0.84506291151 */ erx = 0.845062911510467529296875L, /* * Coefficients for approximation to erf on [0,0.84375] */ -/* 2/sqrt(pi) - 1 */ -efx = 1.2837916709551257389615890312154517168810E-1L, /* 8 * (2/sqrt(pi) - 1) */ efx8 = 1.0270333367641005911692712249723613735048E0L, pp[6] = { @@ -239,84 +235,108 @@ sc[] = { /* 1.000000000000000000000000000000000000000E0 */ }; -long double erfl(long double x) +static long double erfc1(long double x) { - long double R, S, P, Q, s, y, z, r; - int32_t ix, i; - uint32_t se, i0, i1; + long double s,P,Q; - GET_LDOUBLE_WORDS(se, i0, i1, x); - ix = se & 0x7fff; + s = fabsl(x) - 1; + P = pa[0] + s * (pa[1] + s * (pa[2] + + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7])))))); + Q = qa[0] + s * (qa[1] + s * (qa[2] + + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s)))))); + return 1 - erx - P / Q; +} - if (ix >= 0x7fff) { /* erf(nan)=nan */ - i = ((se & 0xffff) >> 15) << 1; - return (long double)(1 - i) + 1.0 / x; /* erf(+-inf)=+-1 */ - } +static long double erfc2(uint32_t ix, long double x) +{ + long double s,z,R,S; + uint32_t i0,i1; - ix = (ix << 16) | (i0 >> 16); - if (ix < 0x3ffed800) { /* |x| < 0.84375 */ - if (ix < 0x3fde8000) { /* |x| < 2**-33 */ - if (ix < 0x00080000) - return 0.125 * (8.0 * x + efx8 * x); /* avoid underflow */ - return x + efx * x; - } - z = x * x; - r = pp[0] + z * (pp[1] + - z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5])))); - s = qq[0] + z * (qq[1] + - z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z))))); - y = r / s; - return x + x * y; - } - if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */ - s = fabsl(x) - 1.0; - P = pa[0] + s * (pa[1] + s * (pa[2] + - s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7])))))); - Q = qa[0] + s * (qa[1] + s * (qa[2] + - s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s)))))); - if ((se & 0x8000) == 0) - return erx + P / Q; - return -erx - P / Q; - } - if (ix >= 0x4001d555) { /* inf > |x| >= 6.6666259765625 */ - if ((se & 0x8000) == 0) - return 1.0 - tiny; - return tiny - 1.0; + if (ix < 0x3fffa000) /* 0.84375 <= |x| < 1.25 */ + return erfc1(x); + + x = fabsl(x); + s = 1 / (x * x); + if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.857 ~ 1/.35 */ + R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] + + s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8]))))))); + S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] + + s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s)))))))); + } else { /* 2.857 <= |x| */ + R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] + + s * (rb[5] + s * (rb[6] + s * rb[7])))))); + S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] + + s * (sb[5] + s * (sb[6] + s)))))); } - x = fabsl (x); - s = 1.0 / (x * x); if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */ R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] + s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8]))))))); S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] + s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s)))))))); - } else { /* 2.857 <= |x| < 6.667 */ + } else if (ix < 0x4001d555) { /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */ R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] + s * (rb[5] + s * (rb[6] + s * rb[7])))))); S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] + s * (sb[5] + s * (sb[6] + s)))))); + } else { /* 107 > |x| >= 6.666 */ + R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] + + s * (rc[4] + s * rc[5])))); + S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] + + s * (sc[4] + s)))); } z = x; - GET_LDOUBLE_WORDS(i, i0, i1, z); + GET_LDOUBLE_WORDS(ix, i0, i1, z); i1 = 0; - SET_LDOUBLE_WORDS(z, i, i0, i1); - r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S); - if ((se & 0x8000) == 0) - return 1.0 - r / x; - return r / x - 1.0; + i0 &= 0xffffff00; + SET_LDOUBLE_WORDS(z, ix, i0, i1); + return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x; } -long double erfcl(long double x) +long double erfl(long double x) { - int32_t hx, ix; - long double R, S, P, Q, s, y, z, r; - uint32_t se, i0, i1; + long double r, s, z, y; + uint32_t i0, i1, ix; + int sign; - GET_LDOUBLE_WORDS(se, i0, i1, x); - ix = se & 0x7fff; - if (ix >= 0x7fff) { /* erfc(nan) = nan, erfc(+-inf) = 0,2 */ - return (long double)(((se & 0xffff) >> 15) << 1) + 1.0 / x; + GET_LDOUBLE_WORDS(ix, i0, i1, x); + sign = ix >> 15; + ix &= 0x7fff; + if (ix == 0x7fff) { + /* erf(nan)=nan, erf(+-inf)=+-1 */ + return 1 - 2*sign + 1/x; } + ix = (ix << 16) | (i0 >> 16); + if (ix < 0x3ffed800) { /* |x| < 0.84375 */ + if (ix < 0x3fde8000) { /* |x| < 2**-33 */ + return 0.125 * (8 * x + efx8 * x); /* avoid underflow */ + } + z = x * x; + r = pp[0] + z * (pp[1] + + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5])))); + s = qq[0] + z * (qq[1] + + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z))))); + y = r / s; + return x + x * y; + } + if (ix < 0x4001d555) /* |x| < 6.6666259765625 */ + y = 1 - erfc2(ix,x); + else + y = 1 - 0x1p-16382L; + return sign ? -y : y; +} + +long double erfcl(long double x) +{ + long double r, s, z, y; + uint32_t i0, i1, ix; + int sign; + + GET_LDOUBLE_WORDS(ix, i0, i1, x); + sign = ix>>15; + ix &= 0x7fff; + if (ix == 0x7fff) + /* erfc(nan) = nan, erfc(+-inf) = 0,2 */ + return 2*sign + 1/x; ix = (ix << 16) | (i0 >> 16); if (ix < 0x3ffed800) { /* |x| < 0.84375 */ @@ -330,57 +350,10 @@ long double erfcl(long double x) y = r / s; if (ix < 0x3ffd8000) /* x < 1/4 */ return 1.0 - (x + x * y); - r = x * y; - r += x - 0.5L; - return 0.5L - r; - } - if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */ - s = fabsl(x) - 1.0; - P = pa[0] + s * (pa[1] + s * (pa[2] + - s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7])))))); - Q = qa[0] + s * (qa[1] + s * (qa[2] + - s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s)))))); - if ((se & 0x8000) == 0) { - z = 1.0 - erx; - return z - P / Q; - } - z = erx + P / Q; - return 1.0 + z; + return 0.5 - (x - 0.5 + x * y); } - if (ix < 0x4005d600) { /* |x| < 107 */ - x = fabsl(x); - s = 1.0 / (x * x); - if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */ - R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] + - s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8]))))))); - S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] + - s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s)))))))); - } else if (ix < 0x4001d555) { /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */ - R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] + - s * (rb[5] + s * (rb[6] + s * rb[7])))))); - S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] + - s * (sb[5] + s * (sb[6] + s)))))); - } else { /* 107 > |x| >= 6.666 */ - if (se & 0x8000) - return 2.0 - tiny;/* x < -6.666 */ - R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] + - s * (rc[4] + s * rc[5])))); - S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] + - s * (sc[4] + s)))); - } - z = x; - GET_LDOUBLE_WORDS(hx, i0, i1, z); - i1 = 0; - i0 &= 0xffffff00; - SET_LDOUBLE_WORDS(z, hx, i0, i1); - r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S); - if ((se & 0x8000) == 0) - return r / x; - return 2.0 - r / x; - } - - if ((se & 0x8000) == 0) - return tiny * tiny; - return 2.0 - tiny; + if (ix < 0x4005d600) /* |x| < 107 */ + return sign ? 2 - erfc2(ix,x) : erfc2(ix,x); + return sign ? 2 - 0x1p-16382L : 0x1p-16382L*0x1p-16382L; } #endif -- cgit v1.2.1