diff options
Diffstat (limited to 'lib/msun/ld80/s_expl.c')
| -rw-r--r-- | lib/msun/ld80/s_expl.c | 275 |
1 files changed, 45 insertions, 230 deletions
diff --git a/lib/msun/ld80/s_expl.c b/lib/msun/ld80/s_expl.c index ec748d3..3147d35 100644 --- a/lib/msun/ld80/s_expl.c +++ b/lib/msun/ld80/s_expl.c @@ -48,16 +48,15 @@ __FBSDID("$FreeBSD$"); #include "fpmath.h" #include "math.h" #include "math_private.h" +#include "k_expl.h" -#define INTERVALS 128 -#define LOG2_INTERVALS 7 -#define BIAS (LDBL_MAX_EXP - 1) +/* XXX Prevent compilers from erroneously constant folding these: */ +static const volatile long double +huge = 0x1p10000L, +tiny = 0x1p-10000L; static const long double -huge = 0x1p10000L, twom10000 = 0x1p-10000L; -/* XXX Prevent gcc from erroneously constant folding this: */ -static volatile const long double tiny = 0x1p-10000L; static const union IEEEl2bits /* log(2**16384 - 0.5) rounded towards zero: */ @@ -68,178 +67,16 @@ o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); #define u_threshold (u_thresholdu.e) -static const double -/* - * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must - * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest - * bits zero so that multiplication of it by n is exact. - */ -INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */ -L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */ -L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */ -/* - * Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]: - * |exp(x) - p(x)| < 2**-77.2 - * (0.002708 is ln2/(2*INTERVALS) rounded up a little). - */ -A2 = 0.5, -A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */ -A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */ -A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */ -A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */ - -/* - * 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where - * the first 53 bits of the significand are stored in hi and the next 53 - * bits are in lo. Tang's paper states that the trailing 6 bits of hi must - * be zero for his algorithm in both single and double precision, because - * the table is re-used in the implementation of expm1() where a floating - * point addition involving hi must be exact. Here hi is double, so - * converting it to long double gives 11 trailing zero bits. - */ -static const struct { - double hi; - double lo; -} tbl[INTERVALS] = { - 0x1p+0, 0x0p+0, - 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54, - 0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53, - 0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53, - 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55, - 0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53, - 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57, - 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54, - 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54, - 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54, - 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59, - 0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53, - 0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53, - 0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53, - 0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53, - 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55, - 0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53, - 0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53, - 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55, - 0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53, - 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54, - 0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53, - 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55, - 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55, - 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54, - 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55, - 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55, - 0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53, - 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55, - 0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53, - 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54, - 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56, - 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55, - 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55, - 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54, - 0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53, - 0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53, - 0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53, - 0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53, - 0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53, - 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55, - 0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53, - 0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53, - 0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53, - 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59, - 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54, - 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56, - 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54, - 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56, - 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54, - 0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53, - 0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53, - 0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53, - 0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53, - 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54, - 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55, - 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54, - 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60, - 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54, - 0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53, - 0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53, - 0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53, - 0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53, - 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57, - 0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53, - 0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53, - 0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53, - 0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53, - 0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53, - 0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53, - 0x1.75feb564267c8p+0, 0x1.7edd354674916p-53, - 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54, - 0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53, - 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54, - 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56, - 0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53, - 0x1.82589994cce12p+0, 0x1.159f115f56694p-53, - 0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53, - 0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53, - 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54, - 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54, - 0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53, - 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55, - 0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53, - 0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53, - 0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53, - 0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53, - 0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53, - 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56, - 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56, - 0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53, - 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54, - 0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53, - 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54, - 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54, - 0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53, - 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54, - 0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53, - 0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53, - 0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53, - 0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53, - 0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53, - 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55, - 0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53, - 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55, - 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54, - 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54, - 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56, - 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56, - 0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53, - 0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53, - 0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53, - 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55, - 0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53, - 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54, - 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54, - 0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53, - 0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53, - 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55, - 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54, - 0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53, - 0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53, - 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54, - 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54, - 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54, - 0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53, - 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55, - 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57 -}; - long double expl(long double x) { - union IEEEl2bits u, v; - long double fn, q, r, r1, r2, t, twopk, twopkp10000; - long double z; - int k, n, n2; + union IEEEl2bits u; + long double hi, lo, t, twopk; + int k; uint16_t hx, ix; + DOPRINT_START(&x); + /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; @@ -247,59 +84,32 @@ expl(long double x) if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */ - return (-1 / x); - return (x + x); /* x is +Inf, +NaN or unsupported */ + RETURNP(-1 / x); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); if (x < u_threshold) - return (tiny * tiny); - } else if (ix < BIAS - 65) { /* |x| < 0x1p-65 (includes pseudos) */ - return (1 + x); /* 1 with inexact iff x != 0 */ + RETURNP(tiny * tiny); + } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ + RETURN2P(1, x); /* 1 with inexact iff x != 0 */ } ENTERI(); - /* 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; - r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */ -#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; - /* Depend on the sign bit being propagated: */ - k = n >> LOG2_INTERVALS; - r1 = x - fn * L1; - r2 = fn * -L2; - - /* Prepare scale factors. */ - v.e = 1; - if (k >= LDBL_MIN_EXP) { - v.xbits.expsign = BIAS + k; - twopk = v.e; - } else { - v.xbits.expsign = BIAS + k + 10000; - twopkp10000 = v.e; - } - - /* Evaluate 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; - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + twopk = 1; + __k_expl(x, &hi, &lo, &k); + t = SUM2P(hi, lo); /* Scale by 2**k. */ if (k >= LDBL_MIN_EXP) { if (k == LDBL_MAX_EXP) RETURNI(t * 2 * 0x1p16383L); + SET_LDBL_EXPSIGN(twopk, BIAS + k); RETURNI(t * twopk); } else { - RETURNI(t * twopkp10000 * twom10000); + SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); + RETURNI(t * twopk * twom10000); } } @@ -326,8 +136,11 @@ 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 + * Domain [-0.1659, 0.1659], range ~[-2.6155e-22, 2.5507e-23]: + * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.6 + * + * XXX the coeffs aren't very carefully rounded, and I get 2.8 more bits, + * but unlike for ld128 we can't drop any terms. */ static const union IEEEl2bits B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-1L), @@ -353,6 +166,8 @@ expm1l(long double x) int k, n, n2; uint16_t hx, ix; + DOPRINT_START(&x); + /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; @@ -360,11 +175,11 @@ expm1l(long double x) 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 */ + RETURNP(-1 / x - 1); + RETURNP(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) - return (huge * huge); + RETURNP(huge * huge); /* * expm1l() never underflows, but it must avoid * unrepresentable large negative exponents. We used a @@ -373,15 +188,15 @@ expm1l(long double x) * in the same way as large ones here. */ if (hx & 0x8000) /* x <= -64 */ - return (tiny - 1); /* good for x < -65ln2 - eps */ + RETURN2P(tiny, -1); /* good for x < -65ln2 - eps */ } ENTERI(); if (T1 < x && x < T2) { - if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */ + if (ix < BIAS - 74) { /* |x| < 0x1p-74 (includes pseudos) */ /* x (rounded) with inexact if x != 0: */ - RETURNI(x == 0 ? x : + RETURNPI(x == 0 ? x : (0x1p100 * x + fabsl(x)) * 0x1p-100); } @@ -402,9 +217,9 @@ expm1l(long double x) 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)); + RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q); else - RETURNI(hx2_lo + q + hx2_hi + x); + RETURN2PI(x, hx2_lo + q + hx2_hi); } /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ @@ -438,21 +253,21 @@ expm1l(long double x) 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); + t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); RETURNI(t); } if (k == -1) { - t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 + - (tbl[n2].hi - 2); + t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + + tbl[n2].hi * r1); RETURNI(t / 2); } if (k < -7) { - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk - 1); } if (k > 2 * LDBL_MANT_DIG - 1) { - t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); if (k == LDBL_MAX_EXP) RETURNI(t * 2 * 0x1p16383L - 1); RETURNI(t * twopk - 1); @@ -462,8 +277,8 @@ expm1l(long double x) twomk = v.e; if (k > LDBL_MANT_DIG - 1) - t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi; + t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); else - t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk); + t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk); } |
