Date: Mon, 3 Jun 2013 19:51:32 +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: r251343 - in head/lib/msun: . ld128 ld80 man src Message-ID: <201306031951.r53JpWjS051618@svn.freebsd.org>
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Author: kargl Date: Mon Jun 3 19:51:32 2013 New Revision: 251343 URL: http://svnweb.freebsd.org/changeset/base/251343 Log: ld80 and ld128 implementations of expm1l(). This code started life as a fairly faithful implementation of the algorithm found in PTP Tang, "Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, 211-222 (1992). Over the last 18-24 months, the code has under gone significant optimization and testing. Reviewed by: bde Obtained from: bde (most of the optimizations) Modified: head/lib/msun/Symbol.map head/lib/msun/ld128/s_expl.c head/lib/msun/ld80/s_expl.c head/lib/msun/man/exp.3 head/lib/msun/src/math.h head/lib/msun/src/s_expm1.c Modified: head/lib/msun/Symbol.map ============================================================================== --- head/lib/msun/Symbol.map Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/Symbol.map Mon Jun 3 19:51:32 2013 (r251343) @@ -262,6 +262,7 @@ FBSD_1.3 { ctanh; ctanhf; expl; + expm1l; log10l; log1pl; log2l; Modified: head/lib/msun/ld128/s_expl.c ============================================================================== --- head/lib/msun/ld128/s_expl.c Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/ld128/s_expl.c Mon Jun 3 19:51:32 2013 (r251343) @@ -298,3 +298,198 @@ expl(long double x) RETURNI(t * twopkp10000 * twom10000); } } + +/* + * Our T1 and T2 are chosen to be approximately the points where method + * A and method B have the same accuracy. Tang's T1 and T2 are the + * points where method A's accuracy changes by a full bit. For Tang, + * this drop in accuracy makes method A immediately less accurate than + * method B, but our larger INTERVALS makes method A 2 bits more + * accurate so it remains the most accurate method significantly + * closer to the origin despite losing the full bit in our extended + * range for it. + * + * Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2]. + * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear + * in both subintervals, so set T3 = 2**-5, which places the condition + * into the [T1, T3] interval. + */ +static const double +T1 = -0.1659, /* ~-30.625/128 * log(2) */ +T2 = 0.1659, /* ~30.625/128 * log(2) */ +T3 = 0.03125; + +/* + * Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03 + */ +static const long double +C3 = 1.66666666666666666666666666666666667e-1L, +C4 = 4.16666666666666666666666666666666645e-2L, +C5 = 8.33333333333333333333333333333371638e-3L, +C6 = 1.38888888888888888888888888891188658e-3L, +C7 = 1.98412698412698412698412697235950394e-4L, +C8 = 2.48015873015873015873015112487849040e-5L, +C9 = 2.75573192239858906525606685484412005e-6L, +C10 = 2.75573192239858906612966093057020362e-7L, +C11 = 2.50521083854417203619031960151253944e-8L, +C12 = 2.08767569878679576457272282566520649e-9L, +C13 = 1.60590438367252471783548748824255707e-10L; + +static const double +C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */ +C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */ +C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */ +C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */ +C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */ + +/* + * Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44 + */ +static const long double +D3 = 1.66666666666666666666666666666682245e-1L, +D4 = 4.16666666666666666666666666634228324e-2L, +D5 = 8.33333333333333333333333364022244481e-3L, +D6 = 1.38888888888888888888887138722762072e-3L, +D7 = 1.98412698412698412699085805424661471e-4L, +D8 = 2.48015873015873015687993712101479612e-5L, +D9 = 2.75573192239858944101036288338208042e-6L, +D10 = 2.75573192239853161148064676533754048e-7L, +D11 = 2.50521083855084570046480450935267433e-8L, +D12 = 2.08767569819738524488686318024854942e-9L, +D13 = 1.60590442297008495301927448122499313e-10L; + +static const double +D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */ +D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */ +D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */ +D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */ + +long double +expm1l(long double x) +{ + union IEEEl2bits u, v; + long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi; + long double x_lo, x2; + double dr, dx, fn, r2; + int k, n, n2; + uint16_t hx, ix; + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf or -NaN */ + return (-1 / x - 1); + return (x + x); /* x is +Inf or +NaN */ + } + if (x > o_threshold) + return (huge * huge); + /* + * expm1l() never underflows, but it must avoid + * unrepresentable large negative exponents. We used a + * much smaller threshold for large |x| above than in + * expl() so as to handle not so large negative exponents + * in the same way as large ones here. + */ + if (hx & 0x8000) /* x <= -128 */ + return (tiny - 1); /* good for x < -114ln2 - eps */ + } + + ENTERI(); + + if (T1 < x && x < T2) { + x2 = x * x; + dx = x; + + if (x < T3) { + if (ix < BIAS - 113) { /* |x| < 0x1p-113 */ + /* x (rounded) with inexact if x != 0: */ + RETURNI(x == 0 ? x : + (0x1p200 * x + fabsl(x)) * 0x1p-200); + } + q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 + + x * (C7 + x * (C8 + x * (C9 + x * (C10 + + x * (C11 + x * (C12 + x * (C13 + + dx * (C14 + dx * (C15 + dx * (C16 + + dx * (C17 + dx * C18)))))))))))))); + } else { + q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 + + x * (D7 + x * (D8 + x * (D9 + x * (D10 + + x * (D11 + x * (D12 + x * (D13 + + dx * (D14 + dx * (D15 + dx * (D16 + + dx * D17))))))))))))); + } + + x_hi = (float)x; + x_lo = x - x_hi; + hx2_hi = x_hi * x_hi / 2; + hx2_lo = x_lo * (x + x_hi) / 2; + if (ix >= BIAS - 7) + RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); + else + RETURNI(hx2_lo + q + hx2_hi + x); + } + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + /* Use a specialized rint() to get fn. Assume round-to-nearest. */ + fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52; +#if defined(HAVE_EFFICIENT_IRINT) + n = irint(fn); +#else + n = (int)fn; +#endif + n2 = (unsigned)n % INTERVALS; + k = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Prepare scale factor. */ + v.e = 1; + v.xbits.expsign = BIAS + k; + twopk = v.e; + + /* + * Evaluate lower terms of + * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). + */ + dr = r; + q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 + + dr * (A7 + dr * (A8 + dr * (A9 + dr * A10)))))))); + + t = tbl[n2].lo + tbl[n2].hi; + + if (k == 0) { + t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + + (tbl[n2].hi - 1); + RETURNI(t); + } + if (k == -1) { + t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + + (tbl[n2].hi - 2); + RETURNI(t / 2); + } + if (k < -7) { + t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + RETURNI(t * twopk - 1); + } + if (k > 2 * LDBL_MANT_DIG - 1) { + t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L - 1); + RETURNI(t * twopk - 1); + } + + v.xbits.expsign = BIAS - k; + twomk = v.e; + + if (k > LDBL_MANT_DIG - 1) + t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; + else + t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); + RETURNI(t * twopk); +} Modified: head/lib/msun/ld80/s_expl.c ============================================================================== --- head/lib/msun/ld80/s_expl.c Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/ld80/s_expl.c Mon Jun 3 19:51:32 2013 (r251343) @@ -302,3 +302,168 @@ expl(long double x) RETURNI(t * twopkp10000 * twom10000); } } + +/** + * Compute expm1l(x) for Intel 80-bit format. This is based on: + * + * PTP Tang, "Table-driven implementation of the Expm1 function + * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18, + * 211-222 (1992). + */ + +/* + * Our T1 and T2 are chosen to be approximately the points where method + * A and method B have the same accuracy. Tang's T1 and T2 are the + * points where method A's accuracy changes by a full bit. For Tang, + * this drop in accuracy makes method A immediately less accurate than + * method B, but our larger INTERVALS makes method A 2 bits more + * accurate so it remains the most accurate method significantly + * closer to the origin despite losing the full bit in our extended + * range for it. + */ +static const double +T1 = -0.1659, /* ~-30.625/128 * log(2) */ +T2 = 0.1659; /* ~30.625/128 * log(2) */ + +/* + * Domain [-0.1659, 0.1659], range ~[-1.2027e-22, 3.4417e-22]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2 + */ +static const union IEEEl2bits +B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), +B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-2L); + +static const double +B5 = 8.3333333333333245e-3, /* 0x1.111111111110cp-7 */ +B6 = 1.3888888888888861e-3, /* 0x1.6c16c16c16c0ap-10 */ +B7 = 1.9841269841532042e-4, /* 0x1.a01a01a0319f9p-13 */ +B8 = 2.4801587302069236e-5, /* 0x1.a01a01a03cbbcp-16 */ +B9 = 2.7557316558468562e-6, /* 0x1.71de37fd33d67p-19 */ +B10 = 2.7557315829785151e-7, /* 0x1.27e4f91418144p-22 */ +B11 = 2.5063168199779829e-8, /* 0x1.ae94fabdc6b27p-26 */ +B12 = 2.0887164654459567e-9; /* 0x1.1f122d6413fe1p-29 */ + +long double +expm1l(long double x) +{ + union IEEEl2bits u, v; + long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi; + long double x_lo, x2, z; + long double x4; + int k, n, n2; + uint16_t hx, ix; + + /* Filter out exceptional cases. */ + u.e = x; + hx = u.xbits.expsign; + ix = hx & 0x7fff; + if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */ + if (ix == BIAS + LDBL_MAX_EXP) { + if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ + return (-1 / x - 1); + return (x + x); /* x is +Inf, +NaN or unsupported */ + } + if (x > o_threshold) + return (huge * huge); + /* + * expm1l() never underflows, but it must avoid + * unrepresentable large negative exponents. We used a + * much smaller threshold for large |x| above than in + * expl() so as to handle not so large negative exponents + * in the same way as large ones here. + */ + if (hx & 0x8000) /* x <= -64 */ + return (tiny - 1); /* good for x < -65ln2 - eps */ + } + + ENTERI(); + + if (T1 < x && x < T2) { + if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */ + /* x (rounded) with inexact if x != 0: */ + RETURNI(x == 0 ? x : + (0x1p100 * x + fabsl(x)) * 0x1p-100); + } + + x2 = x * x; + x4 = x2 * x2; + q = x4 * (x2 * (x4 * + /* + * XXX the number of terms is no longer good for + * pairwise grouping of all except B3, and the + * grouping is no longer from highest down. + */ + (x2 * B12 + (x * B11 + B10)) + + (x2 * (x * B9 + B8) + (x * B7 + B6))) + + (x * B5 + B4.e)) + x2 * x * B3.e; + + x_hi = (float)x; + x_lo = x - x_hi; + hx2_hi = x_hi * x_hi / 2; + hx2_lo = x_lo * (x + x_hi) / 2; + if (ix >= BIAS - 7) + RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi)); + else + RETURNI(hx2_lo + q + hx2_hi + x); + } + + /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ + /* Use a specialized rint() to get fn. Assume round-to-nearest. */ + fn = x * INV_L + 0x1.8p63 - 0x1.8p63; +#if defined(HAVE_EFFICIENT_IRINTL) + n = irintl(fn); +#elif defined(HAVE_EFFICIENT_IRINT) + n = irint(fn); +#else + n = (int)fn; +#endif + n2 = (unsigned)n % INTERVALS; + k = n >> LOG2_INTERVALS; + r1 = x - fn * L1; + r2 = fn * -L2; + r = r1 + r2; + + /* Prepare scale factor. */ + v.e = 1; + v.xbits.expsign = BIAS + k; + twopk = v.e; + + /* + * Evaluate lower terms of + * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). + */ + z = r * r; + q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6; + + t = (long double)tbl[n2].lo + tbl[n2].hi; + + if (k == 0) { + t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + + (tbl[n2].hi - 1); + RETURNI(t); + } + if (k == -1) { + t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + + (tbl[n2].hi - 2); + RETURNI(t / 2); + } + if (k < -7) { + t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + RETURNI(t * twopk - 1); + } + if (k > 2 * LDBL_MANT_DIG - 1) { + t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + if (k == LDBL_MAX_EXP) + RETURNI(t * 2 * 0x1p16383L - 1); + RETURNI(t * twopk - 1); + } + + v.xbits.expsign = BIAS - k; + twomk = v.e; + + if (k > LDBL_MANT_DIG - 1) + t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; + else + t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); + RETURNI(t * twopk); +} Modified: head/lib/msun/man/exp.3 ============================================================================== --- head/lib/msun/man/exp.3 Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/man/exp.3 Mon Jun 3 19:51:32 2013 (r251343) @@ -28,7 +28,7 @@ .\" from: @(#)exp.3 6.12 (Berkeley) 7/31/91 .\" $FreeBSD$ .\" -.Dd July 10, 2012 +.Dd June 3, 2013 .Dt EXP 3 .Os .Sh NAME @@ -41,6 +41,7 @@ .Nm exp2l , .Nm expm1 , .Nm expm1f , +.Nm expm1l , .Nm pow , .Nm powf .Nd exponential and power functions @@ -64,6 +65,8 @@ .Fn expm1 "double x" .Ft float .Fn expm1f "float x" +.Ft long double +.Fn expm1l "long double x" .Ft double .Fn pow "double x" "double y" .Ft float @@ -88,9 +91,10 @@ functions compute the base 2 exponential .Fa x . .Pp The -.Fn expm1 +.Fn expm1 , +.Fn expm1f , and the -.Fn expm1f +.Fn expm1l functions compute the value exp(x)\-1 accurately even for tiny argument .Fa x . .Pp Modified: head/lib/msun/src/math.h ============================================================================== --- head/lib/msun/src/math.h Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/src/math.h Mon Jun 3 19:51:32 2013 (r251343) @@ -405,6 +405,7 @@ long double copysignl(long double, long long double cosl(long double); long double exp2l(long double); long double expl(long double); +long double expm1l(long double); long double fabsl(long double) __pure2; long double fdiml(long double, long double); long double floorl(long double); @@ -466,7 +467,6 @@ long double atanhl(long double); long double coshl(long double); long double erfcl(long double); long double erfl(long double); -long double expm1l(long double); long double lgammal(long double); long double powl(long double, long double); long double sinhl(long double); Modified: head/lib/msun/src/s_expm1.c ============================================================================== --- head/lib/msun/src/s_expm1.c Mon Jun 3 19:39:37 2013 (r251342) +++ head/lib/msun/src/s_expm1.c Mon Jun 3 19:51:32 2013 (r251343) @@ -216,3 +216,7 @@ expm1(double x) } return y; } + +#if (LDBL_MANT_DIG == 53) +__weak_reference(expm1, expm1l); +#endif
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