/*- * 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. */ /** * Compute the exponential of x for Intel 80-bit format. This is based on: * * PTP Tang, "Table-driven implementation of the exponential function * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15, * 144-157 (1989). * * where the 32 table entries have been expanded to INTERVALS (see below). */ #include <float.h> #ifdef __i386__ #include <ieeefp.h> #endif #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 union IEEEl2bits /* log(2**16384 - 0.5) rounded towards zero: */ /* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */ o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L), #define o_threshold (o_thresholdu.e) /* log(2**(-16381-64-1)) rounded towards zero: */ u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L); #define u_threshold (u_thresholdu.e) 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, -NaN or unsupported */ RETURNF(-1 / x); RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ } if (x > o_threshold) RETURNF(huge * huge); if (x < u_threshold) RETURNF(tiny * tiny); } else if (ix < BIAS - 75) { /* |x| < 0x1p-75 (includes pseudos) */ 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. */ 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); } } /** * 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 ~[-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), 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 */ RETURNF(-1 / x - 1); RETURNF(x + x); /* x is +Inf, +NaN or unsupported */ } 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 <= -64 */ RETURNF(tiny - 1); /* good for x < -65ln2 - eps */ } ENTERI(); if (T1 < x && x < T2) { if (ix < BIAS - 74) { /* |x| < 0x1p-74 (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_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 = rnintl(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). */ 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 = 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); }
