common/Q/expm1l.c

	expm1l.c revision 1ec68d336ba97cd53f46053ac10401d16014d075
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi/*
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * CDDL HEADER START
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * The contents of this file are subject to the terms of the
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Common Development and Distribution License (the "License").
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * You may not use this file except in compliance with the License.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * or http://www.opensolaris.org/os/licensing.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * See the License for the specific language governing permissions
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * and limitations under the License.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * When distributing Covered Code, include this CDDL HEADER in each
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * If applicable, add the following below this CDDL HEADER, with the
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * fields enclosed by brackets "[]" replaced with your own identifying
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * information: Portions Copyright [yyyy] [name of copyright owner]
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * CDDL HEADER END
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi/*
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi/*
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Use is subject to license terms.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi#pragma weak expm1l = __expm1l
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi#if !defined(__sparc)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi#error Unsupported architecture
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi#endif
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi/*
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * expm1l(x)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Table driven method
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Written by K.C. Ng, June 1995.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Algorithm :
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  1. expm1(x) = x if x<2**-114
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  2. if |x| <= 0.0625 = 1/16, use approximation
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      expm1(x) = x + x*P/(2-P)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * where
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  P = x - z*(P1+z*(P2+z*(P3+z*(P4+z*(P5+z*P6+z*P7))))), z = x*x;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * (this formula is derived from
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  2-P+x = R = x*(exp(x)+1)/(exp(x)-1) ~ 2 + x*x/6 - x^4/360 + ...)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P1 =   1.66666666666666666666666666666638500528074603030e-0001
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P2 =  -2.77777777777777777777777759668391122822266551158e-0003
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P3 =   6.61375661375661375657437408890138814721051293054e-0005
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P4 =  -1.65343915343915303310185228411892601606669528828e-0006
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P5 =   4.17535139755122945763580609663414647067443411178e-0008
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P6 =  -1.05683795988668526689182102605260986731620026832e-0009
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * P7 =   2.67544168821852702827123344217198187229611470514e-0011
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Accuracy: |R-x*(exp(x)+1)/(exp(x)-1)|<=2**-119.13
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  3. For 1/16 < |x| < 1.125, choose x(+-i) ~ +-(i+4.5)/64, i=0,..,67
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *     since
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      exp(x) = exp(xi+(x-xi))= exp(xi)*exp((x-xi))
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *     we have
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      expm1(x) = expm1(xi)+(exp(xi))*(expm1(x-xi))
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *     where
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      |s=x-xi| <= 1/128
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *     and
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  expm1(s)=2s/(2-R), R= s-s^2*(T1+s^2*(T2+s^2*(T3+s^2*(T4+s^2*T5))))
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * T1 =   1.666666666666666666666666666660876387437e-1L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * T2 =  -2.777777777777777777777707812093173478756e-3L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * T3 =   6.613756613756613482074280932874221202424e-5L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * T4 =  -1.653439153392139954169609822742235851120e-6L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * T5 =   4.175314851769539751387852116610973796053e-8L;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  4. For |x| >= 1.125, return exp(x)-1.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      (see algorithm for exp)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Special cases:
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  expm1l(INF) is INF, expm1l(NaN) is NaN;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  expm1l(-INF)= -1;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  for finite argument, only expm1l(0)=0 is exact.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Accuracy:
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  according to an error analysis, the error is always less than
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  2 ulp (unit in the last place).
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Misc. info.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *  For 113 bit long double
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      if x >  1.135652340629414394949193107797076342845e+4
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *      then expm1l(x) overflow;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Constants:
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * Only decimal values are given. We assume that the compiler will convert
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * from decimal to binary accurately enough to produce the correct
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi * hexadecimal values.
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi#include "libm.h"
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchiextern const long double _TBL_expl_hi[], _TBL_expl_lo[];
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchiextern const long double _TBL_expm1lx[], _TBL_expm1l[];
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchistatic const long double
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    zero        = +0.0L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    one     = +1.0L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    two     = +2.0L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    ln2_64      = +1.083042469624914545964425189778400898568e-2L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    ovflthreshold   = +1.135652340629414394949193107797076342845e+4L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    invln2_32   = +4.616624130844682903551758979206054839765e+1L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    ln2_32hi    = +2.166084939249829091928849858592451515688e-2L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    ln2_32lo    = +5.209643502595475652782654157501186731779e-27L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    huge        = +1.0e4000L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    tiny        = +1.0e-4000L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P1 = +1.66666666666666666666666666666638500528074603030e-0001L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P2 = -2.77777777777777777777777759668391122822266551158e-0003L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P3 = +6.61375661375661375657437408890138814721051293054e-0005L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P4 = -1.65343915343915303310185228411892601606669528828e-0006L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P5 = +4.17535139755122945763580609663414647067443411178e-0008L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P6 = -1.05683795988668526689182102605260986731620026832e-0009L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    P7 = +2.67544168821852702827123344217198187229611470514e-0011L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi/* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    T1 = +1.666666666666666666666666666660876387437e-1L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    T2 = -2.777777777777777777777707812093173478756e-3L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    T3 = +6.613756613756613482074280932874221202424e-5L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    T4 = -1.653439153392139954169609822742235851120e-6L,
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    T5 = +4.175314851769539751387852116610973796053e-8L;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchilong double
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchiexpm1l(long double x) {
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    int hx, ix, j, k, m;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    long double t, r, s, w;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    hx = ((int *) &x)[HIXWORD];
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    ix = hx & ~0x80000000;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    if (ix >= 0x7fff0000) {
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (x != x)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            return (x + x); /* NaN */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (x < zero)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            return (-one);  /* -inf */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        return (x); /* +inf */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    }
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    if (ix < 0x3fff4000) {  /* |x| < 1.25 */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (ix < 0x3ffb0000) {  /* |x| < 0.0625 */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            if (ix < 0x3f8d0000) {
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi                if ((int) x == 0)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi                    return (x); /* |x|<2^-114 */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            }
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            t = x * x;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            r = (x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t *
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi                (P5 + t * (P6 + t * P7)))))));
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            return (x + (x * r) / (two - r));
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        }
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        /* compute i = [64*x] */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        m = 0x4009 - (ix >> 16);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        j = ((ix & 0x0000ffff) | 0x10000) >> m; /* j=4,...,67 */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (hx < 0)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            j += 82;            /* negative */
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        s = x - _TBL_expm1lx[j];
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        t = s * s;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        r = s - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))));
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        r = (s + s) / (two - r);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        w = _TBL_expm1l[j];
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        return (w + (w + one) * r);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    }
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    if (hx > 0) {
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (x > ovflthreshold)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            return (huge * huge);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        k = (int) (invln2_32 * (x + ln2_64));
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    } else {
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        if (x < -80.0)
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi            return (tiny - x / x);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        k = (int) (invln2_32 * (x - ln2_64));
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    }
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    j = k & 0x1f;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    m = k >> 5;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    t = (long double) k;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    x = (x - t * ln2_32hi) - t * ln2_32lo;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    t = x * x;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    r = (x - t * (T1 + t * (T2 + t * (T3 + t * (T4 + t * T5))))) - two;
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    x = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r -
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi        _TBL_expl_lo[j]);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi    return (scalbnl(x, m) - one);
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi}
d14abf155341d55053c76eeec58b787a456b753bRobert Mustacchi