/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2006 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ #pragma weak __expm1l = expm1l #if !defined(__sparc) #error Unsupported architecture #endif /* * expm1l(x) * * Table driven method * Written by K.C. Ng, June 1995. * Algorithm : * 1. expm1(x) = x if x<2**-114 * 2. if |x| <= 0.0625 = 1/16, use approximation * expm1(x) = x + x*P/(2-P) * where * P = x - z*(P1+z*(P2+z*(P3+z*(P4+z*(P5+z*P6+z*P7))))), z = x*x; * (this formula is derived from * 2-P+x = R = x*(exp(x)+1)/(exp(x)-1) ~ 2 + x*x/6 - x^4/360 + ...) * * P1 = 1.66666666666666666666666666666638500528074603030e-0001 * P2 = -2.77777777777777777777777759668391122822266551158e-0003 * P3 = 6.61375661375661375657437408890138814721051293054e-0005 * P4 = -1.65343915343915303310185228411892601606669528828e-0006 * P5 = 4.17535139755122945763580609663414647067443411178e-0008 * P6 = -1.05683795988668526689182102605260986731620026832e-0009 * P7 = 2.67544168821852702827123344217198187229611470514e-0011 * * Accuracy: |R-x*(exp(x)+1)/(exp(x)-1)|<=2**-119.13 * * 3. For 1/16 < |x| < 1.125, choose x(+-i) ~ +-(i+4.5)/64, i=0,..,67 * since * exp(x) = exp(xi+(x-xi))= exp(xi)*exp((x-xi)) * we have * expm1(x) = expm1(xi)+(exp(xi))*(expm1(x-xi)) * where * |s=x-xi| <= 1/128 * and * expm1(s)=2s/(2-R), R= s-s^2*(T1+s^2*(T2+s^2*(T3+s^2*(T4+s^2*T5)))) * * T1 = 1.666666666666666666666666666660876387437e-1L, * T2 = -2.777777777777777777777707812093173478756e-3L, * T3 = 6.613756613756613482074280932874221202424e-5L, * T4 = -1.653439153392139954169609822742235851120e-6L, * T5 = 4.175314851769539751387852116610973796053e-8L; * * 4. For |x| >= 1.125, return exp(x)-1. * (see algorithm for exp) * * Special cases: * expm1l(INF) is INF, expm1l(NaN) is NaN; * expm1l(-INF)= -1; * for finite argument, only expm1l(0)=0 is exact. * * Accuracy: * according to an error analysis, the error is always less than * 2 ulp (unit in the last place). * * Misc. info. * For 113 bit long double * if x > 1.135652340629414394949193107797076342845e+4 * then expm1l(x) overflow; * * Constants: * Only decimal values are given. We assume that the compiler will convert * from decimal to binary accurately enough to produce the correct * hexadecimal values. */ #include "libm.h" extern const long double _TBL_expl_hi[], _TBL_expl_lo[]; extern const long double _TBL_expm1lx[], _TBL_expm1l[]; static const long double zero = +0.0L, one = +1.0L, two = +2.0L, ln2_64 = +1.083042469624914545964425189778400898568e-2L, ovflthreshold = +1.135652340629414394949193107797076342845e+4L, invln2_32 = +4.616624130844682903551758979206054839765e+1L, ln2_32hi = +2.166084939249829091928849858592451515688e-2L, ln2_32lo = +5.209643502595475652782654157501186731779e-27L, huge = +1.0e4000L, tiny = +1.0e-4000L, P1 = +1.66666666666666666666666666666638500528074603030e-0001L, P2 = -2.77777777777777777777777759668391122822266551158e-0003L, P3 = +6.61375661375661375657437408890138814721051293054e-0005L, P4 = -1.65343915343915303310185228411892601606669528828e-0006L, P5 = +4.17535139755122945763580609663414647067443411178e-0008L, P6 = -1.05683795988668526689182102605260986731620026832e-0009L, P7 = +2.67544168821852702827123344217198187229611470514e-0011L, /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */ T1 = +1.666666666666666666666666666660876387437e-1L, T2 = -2.777777777777777777777707812093173478756e-3L, T3 = +6.613756613756613482074280932874221202424e-5L, T4 = -1.653439153392139954169609822742235851120e-6L, T5 = +4.175314851769539751387852116610973796053e-8L; long double expm1l(long double x) { int hx, ix, j, k, m; long double t, r, s, w; hx = ((int *) &x)[HIXWORD]; ix = hx & ~0x80000000; if (ix >= 0x7fff0000) { if (x != x) return (x + x); /* NaN */ if (x < zero) return (-one); /* -inf */ return (x); /* +inf */ } if (ix < 0x3fff4000) { /* |x| < 1.25 */ if (ix < 0x3ffb0000) { /* |x| < 0.0625 */ if (ix < 0x3f8d0000) { if ((int) x == 0) return (x); /* |x|<2^-114 */ } t = x * x; r = (x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * (P5 + t * (P6 + t * P7))))))); return (x + (x * r) / (two - r)); } /* compute i = [64*x] */ m = 0x4009 - (ix >> 16); j = ((ix & 0x0000ffff) | 0x10000) >> m; /* j=4,...,67 */ if (hx < 0) j += 82; /* negative */ s = x - _TBL_expm1lx[j]; t = s * s; r = s - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5)))); r = (s + s) / (two - r); w = _TBL_expm1l[j]; return (w + (w + one) * r); } if (hx > 0) { if (x > ovflthreshold) return (huge * huge); k = (int) (invln2_32 * (x + ln2_64)); } else { if (x < -80.0) return (tiny - x / x); k = (int) (invln2_32 * (x - ln2_64)); } j = k & 0x1f; m = k >> 5; t = (long double) k; x = (x - t * ln2_32hi) - t * ln2_32lo; t = x * x; r = (x - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))))) - two; x = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r - _TBL_expl_lo[j]); return (scalbnl(x, m) - one); }
