usr/src/lib/libm/common/Q/expl.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29
  30 /*
  31  * expl(x)
  32  * Table driven method
  33  * Written by K.C. Ng, November 1988.
  34  * Algorithm :
  35  *      1. Argument Reduction: given the input x, find r and integer k
  36  *         and j such that
  37  *                   x = (32k+j)*ln2 + r,  |r| <= (1/64)*ln2 .
  38  *
  39  *      2. expl(x) = 2^k * (2^(j/32) + 2^(j/32)*expm1(r))
  40  *         Note:
  41  *         a. expm1(r) = (2r)/(2-R), R = r - r^2*(t1 + t2*r^2)
  42  *         b. 2^(j/32) is represented as
  43  *                      _TBL_expl_hi[j]+_TBL_expl_lo[j]
  44  *         where
  45  *              _TBL_expl_hi[j] = 2^(j/32) rounded
  46  *              _TBL_expl_lo[j] = 2^(j/32) - _TBL_expl_hi[j].
  47  *
  48  * Special cases:
  49  *      expl(INF) is INF, expl(NaN) is NaN;
  50  *      expl(-INF)=  0;
  51  *      for finite argument, only expl(0)=1 is exact.
  52  *
  53  * Accuracy:
  54  *      according to an error analysis, the error is always less than
  55  *      an ulp (unit in the last place).
  56  *
  57  * Misc. info.
  58  *      For 113 bit long double
  59  *              if x >  1.135652340629414394949193107797076342845e+4
  60  *      then expl(x) overflow;
  61  *              if x < -1.143346274333629787883724384345262150341e+4
  62  *      then expl(x) underflow
  63  *
  64  * Constants:
  65  * Only decimal values are given. We assume that the compiler will convert
  66  * from decimal to binary accurately enough to produce the correct
  67  * hexadecimal values.
  68  */
  69
  70 #pragma weak __expl = expl
  71
  72 #include "libm.h"
  73
  74 extern const long double _TBL_expl_hi[], _TBL_expl_lo[];
  75
  76 static const long double
  77 one             =  1.0L,
  78 two             =  2.0L,
  79 ln2_64          =  1.083042469624914545964425189778400898568e-2L,
  80 ovflthreshold   =  1.135652340629414394949193107797076342845e+4L,
  81 unflthreshold   = -1.143346274333629787883724384345262150341e+4L,
  82 invln2_32       =  4.616624130844682903551758979206054839765e+1L,
  83 ln2_32hi        =  2.166084939249829091928849858592451515688e-2L,
  84 ln2_32lo        =  5.209643502595475652782654157501186731779e-27L;
  85
  86 /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */
  87 static const long double
  88 t1 =   1.666666666666666666666666666660876387437e-1L,
  89 t2 =  -2.777777777777777777777707812093173478756e-3L,
  90 t3 =   6.613756613756613482074280932874221202424e-5L,
  91 t4 =  -1.653439153392139954169609822742235851120e-6L,
  92 t5 =   4.175314851769539751387852116610973796053e-8L;
  93
  94 long double
  95 expl(long double x) {
  96         int *px = (int *) &x, ix, j, k, m;
  97         long double t, r;
  98
  99         ix = px[0];                             /* high word of x */
 100         if (ix >= 0x7fff0000)
 101                 return (x + x);                 /* NaN of +inf */
 102         if (((unsigned) ix) >= 0xffff0000)
 103                 return (-one / x);              /* NaN or -inf */
 104         if ((ix & 0x7fffffff) < 0x3fc30000) {
 105                 if ((int) x < 1)
 106                         return (one + x);       /* |x|<2^-60 */
 107         }
 108         if (ix > 0) {
 109                 if (x > ovflthreshold)
 110                         return (scalbnl(x, 20000));
 111                 k = (int) (invln2_32 * (x + ln2_64));
 112         } else {
 113                 if (x < unflthreshold)
 114                         return (scalbnl(-x, -40000));
 115                 k = (int) (invln2_32 * (x - ln2_64));
 116         }
 117         j  = k&0x1f;
 118         m  = k>>5;
 119         t  = (long double) k;
 120         x  = (x - t * ln2_32hi) - t * ln2_32lo;
 121         t  = x * x;
 122         r  = (x - t * (t1 + t * (t2 + t * (t3 + t * (t4 + t * t5))))) - two;
 123         x  = _TBL_expl_hi[j] - ((_TBL_expl_hi[j] * (x + x)) / r -
 124                 _TBL_expl_lo[j]);
 125         return (scalbnl(x, m));
 126 }