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Diffstat (limited to 'lib/msun/ld80/s_expl.c')
| -rw-r--r-- | lib/msun/ld80/s_expl.c | 165 |
1 files changed, 165 insertions, 0 deletions
diff --git a/lib/msun/ld80/s_expl.c b/lib/msun/ld80/s_expl.c index 5bec63b..ec748d3 100644 --- a/lib/msun/ld80/s_expl.c +++ b/lib/msun/ld80/s_expl.c @@ -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); +} |
