/*- * SPDX-License-Identifier: BSD-2-Clause * * Copyright (c) 2009-2013 Steven G. Kargl * All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the above copyright * notice unmodified, this list of conditions, and the following * disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. * * Optimized by Bruce D. Evans. */ /* * ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments. */ #include <float.h> #include "fpmath.h" #include "math.h" #include "math_private.h" #include "k_expl.h" /* XXX Prevent compilers from erroneously constant folding these: */ static const volatile long double huge = 0x1p10000L, tiny = 0x1p-10000L; static const long double twom10000 = 0x1p-10000L; static const long double /* log(2**16384 - 0.5) rounded towards zero: */ /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ o_threshold = 11356.523406294143949491931077970763428L, /* log(2**(-16381-64-1)) rounded towards zero: */ u_threshold = -11433.462743336297878837243843452621503L; long double expl(long double x) { union IEEEl2bits u; long double hi, lo, t, twopk; int k; uint16_t hx, ix; /* Filter out exceptional cases. */ u.e = x; hx = u.xbits.expsign; ix = hx & 0x7fff; if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */ if (ix == BIAS + LDBL_MAX_EXP) { if (hx & 0x8000) /* x is -Inf or -NaN */ RETURNF(-1 / x); RETURNF(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) RETURNF(huge * huge); if (x < u_threshold) RETURNF(tiny * tiny); } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */ RETURNF(1 + x); /* 1 with inexact iff x != 0 */ } ENTERI(); twopk = 1; __k_expl(x, &hi, &lo, &k); t = SUM2P(hi, lo); /* Scale by 2**k. */ /* * XXX sparc64 multiplication was so slow that scalbnl() is faster, * but performance on aarch64 and riscv hasn't yet been quantified. */ if (k >= LDBL_MIN_EXP) { if (k == LDBL_MAX_EXP) RETURNI(t * 2 * 0x1p16383L); SET_LDBL_EXPSIGN(twopk, BIAS + k); RETURNI(t * twopk); } else { SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000); RETURNI(t * twopk * 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. * * XXX we now do this more to (partially) balance the number of terms * in the C and D polys than to avoid checking the condition in both * intervals. * * XXX these micro-optimizations are excessive. */ 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 * * XXX none of the long double C or D coeffs except C10 is correctly printed. * If you re-print their values in %.35Le format, the result is always * different. For example, the last 2 digits in C3 should be 59, not 67. * 67 is apparently from rounding an extra-precision value to 36 decimal * places. */ 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; /* * XXX this has 1 more coeff than needed. * XXX can start the double coeffs but not the double mults at C10. * With my coeffs (C10-C17 double; s = best_s): * Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]: * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 */ 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; /* * XXX this has 1 more coeff than needed. * XXX can start the double coeffs but not the double mults at D11. * With my coeffs (D11-D16 double): * Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]: * |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65 */ 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 */ RETURNF(-1 / x - 1); RETURNF(x + x); /* x is +Inf or +NaN */ } if (x > o_threshold) RETURNF(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 */ RETURNF(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_hi + x_hi) + (hx2_lo + x_lo + q)); else RETURNI(x + (hx2_lo + q + hx2_hi)); } /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */ fn = rnint((double)x * INV_L); n = irint(fn); 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 = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t); } if (k == -1) { t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1); RETURNI(t / 2); } if (k < -7) { t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk - 1); } if (k > 2 * LDBL_MANT_DIG - 1) { 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); } v.xbits.expsign = BIAS - k; twomk = v.e; if (k > LDBL_MANT_DIG - 1) t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1)); else t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1)); RETURNI(t * twopk); }
