src/math/exp2f.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */
   2 /*-
   3  * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
   4  * All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  *
  15  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
  16  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  17  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  18  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
  19  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  20  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  21  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  22  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  23  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  24  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  25  * SUCH DAMAGE.
  26  */
  27
  28 #include "libm.h"
  29
  30 #define TBLSIZE 16
  31
  32 static const float
  33 redux = 0x1.8p23f / TBLSIZE,
  34 P1    = 0x1.62e430p-1f,
  35 P2    = 0x1.ebfbe0p-3f,
  36 P3    = 0x1.c6b348p-5f,
  37 P4    = 0x1.3b2c9cp-7f;
  38
  39 static const double exp2ft[TBLSIZE] = {
  40   0x1.6a09e667f3bcdp-1,
  41   0x1.7a11473eb0187p-1,
  42   0x1.8ace5422aa0dbp-1,
  43   0x1.9c49182a3f090p-1,
  44   0x1.ae89f995ad3adp-1,
  45   0x1.c199bdd85529cp-1,
  46   0x1.d5818dcfba487p-1,
  47   0x1.ea4afa2a490dap-1,
  48   0x1.0000000000000p+0,
  49   0x1.0b5586cf9890fp+0,
  50   0x1.172b83c7d517bp+0,
  51   0x1.2387a6e756238p+0,
  52   0x1.306fe0a31b715p+0,
  53   0x1.3dea64c123422p+0,
  54   0x1.4bfdad5362a27p+0,
  55   0x1.5ab07dd485429p+0,
  56 };
  57
  58 /*
  59  * exp2f(x): compute the base 2 exponential of x
  60  *
  61  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  62  *
  63  * Method: (equally-spaced tables)
  64  *
  65  *   Reduce x:
  66  *     x = k + y, for integer k and |y| <= 1/2.
  67  *     Thus we have exp2f(x) = 2**k * exp2(y).
  68  *
  69  *   Reduce y:
  70  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
  71  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
  72  *     with |z| <= 2**-(TBLSIZE+1).
  73  *
  74  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
  75  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
  76  *   Using double precision for everything except the reduction makes
  77  *   roundoff error insignificant and simplifies the scaling step.
  78  *
  79  *   This method is due to Tang, but I do not use his suggested parameters:
  80  *
  81  *      Tang, P.  Table-driven Implementation of the Exponential Function
  82  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
  83  */
  84 float exp2f(float x)
  85 {
  86         double_t t, r, z;
  87         union {float f; uint32_t i;} u = {x};
  88         union {double f; uint64_t i;} uk;
  89         uint32_t ix, i0, k;
  90
  91         /* Filter out exceptional cases. */
  92         ix = u.i & 0x7fffffff;
  93         if (ix > 0x42fc0000) {  /* |x| > 126 */
  94                 if (ix > 0x7f800000) /* NaN */
  95                         return x;
  96                 if (u.i >= 0x43000000 && u.i < 0x80000000) {  /* x >= 128 */
  97                         x *= 0x1p127f;
  98                         return x;
  99                 }
 100                 if (u.i >= 0x80000000) {  /* x < -126 */
 101                         if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
 102                                 FORCE_EVAL(-0x1p-149f/x);
 103                         if (u.i >= 0xc3160000)  /* x <= -150 */
 104                                 return 0;
 105                 }
 106         } else if (ix <= 0x33000000) {  /* |x| <= 0x1p-25 */
 107                 return 1.0f + x;
 108         }
 109
 110         /* Reduce x, computing z, i0, and k. */
 111         u.f = x + redux;
 112         i0 = u.i;
 113         i0 += TBLSIZE / 2;
 114         k = i0 / TBLSIZE;
 115         uk.i = (uint64_t)(0x3ff + k)<<52;
 116         i0 &= TBLSIZE - 1;
 117         u.f -= redux;
 118         z = x - u.f;
 119         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 120         r = exp2ft[i0];
 121         t = r * z;
 122         r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
 123
 124         /* Scale by 2**k */
 125         return r * uk.f;
 126 }