diff options
Diffstat (limited to 'lib/msun/ld128/s_expl.c')
-rw-r--r-- | lib/msun/ld128/s_expl.c | 195 |
1 files changed, 195 insertions, 0 deletions
diff --git a/lib/msun/ld128/s_expl.c b/lib/msun/ld128/s_expl.c index e7fc5e5..4d5e5bc 100644 --- a/lib/msun/ld128/s_expl.c +++ b/lib/msun/ld128/s_expl.c @@ -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); +} |