contrib/openbsd_libm/src/s_exp2f.c

   1 /*      $OpenBSD: s_exp2f.c,v 1.2 2012/12/05 23:20:04 deraadt Exp $     */
   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 <float.h>
  29
  30 #include "math.h"
  31 #include "math_private.h"
  32
  33 #define TBLBITS 4
  34 #define TBLSIZE (1 << TBLBITS)
  35
  36 static const float
  37     huge    = 0x1p100f,
  38     redux   = 0x1.8p23f / TBLSIZE,
  39     P1      = 0x1.62e430p-1f,
  40     P2      = 0x1.ebfbe0p-3f,
  41     P3      = 0x1.c6b348p-5f,
  42     P4      = 0x1.3b2c9cp-7f;
  43
  44 static volatile float twom100 = 0x1p-100f;
  45
  46 static const double exp2ft[TBLSIZE] = {
  47         0x1.6a09e667f3bcdp-1,
  48         0x1.7a11473eb0187p-1,
  49         0x1.8ace5422aa0dbp-1,
  50         0x1.9c49182a3f090p-1,
  51         0x1.ae89f995ad3adp-1,
  52         0x1.c199bdd85529cp-1,
  53         0x1.d5818dcfba487p-1,
  54         0x1.ea4afa2a490dap-1,
  55         0x1.0000000000000p+0,
  56         0x1.0b5586cf9890fp+0,
  57         0x1.172b83c7d517bp+0,
  58         0x1.2387a6e756238p+0,
  59         0x1.306fe0a31b715p+0,
  60         0x1.3dea64c123422p+0,
  61         0x1.4bfdad5362a27p+0,
  62         0x1.5ab07dd485429p+0,
  63 };
  64
  65 /*
  66  * exp2f(x): compute the base 2 exponential of x
  67  *
  68  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  69  *
  70  * Method: (equally-spaced tables)
  71  *
  72  *   Reduce x:
  73  *     x = 2**k + y, for integer k and |y| <= 1/2.
  74  *     Thus we have exp2f(x) = 2**k * exp2(y).
  75  *
  76  *   Reduce y:
  77  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
  78  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
  79  *     with |z| <= 2**-(TBLSIZE+1).
  80  *
  81  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
  82  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
  83  *   Using double precision for everything except the reduction makes
  84  *   roundoff error insignificant and simplifies the scaling step.
  85  *
  86  *   This method is due to Tang, but I do not use his suggested parameters:
  87  *
  88  *      Tang, P.  Table-driven Implementation of the Exponential Function
  89  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
  90  */
  91 float
  92 exp2f(float x)
  93 {
  94         double tv, twopk, u, z;
  95         float t;
  96         uint32_t hx, ix, i0;
  97         int32_t k;
  98
  99         /* Filter out exceptional cases. */
 100         GET_FLOAT_WORD(hx, x);
 101         ix = hx & 0x7fffffff;           /* high word of |x| */
 102         if(ix >= 0x43000000) {                  /* |x| >= 128 */
 103                 if(ix >= 0x7f800000) {
 104                         if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
 105                                 return (x + x); /* x is NaN or +Inf */
 106                         else
 107                                 return (0.0);   /* x is -Inf */
 108                 }
 109                 if(x >= 0x1.0p7f)
 110                         return (huge * huge);   /* overflow */
 111                 if(x <= -0x1.2cp7f)
 112                         return (twom100 * twom100); /* underflow */
 113         } else if (ix <= 0x33000000) {          /* |x| <= 0x1p-25 */
 114                 return (1.0f + x);
 115         }
 116
 117         /* Reduce x, computing z, i0, and k. */
 118         STRICT_ASSIGN(float, t, x + redux);
 119         GET_FLOAT_WORD(i0, t);
 120         i0 += TBLSIZE / 2;
 121         k = (i0 >> TBLBITS) << 20;
 122         i0 &= TBLSIZE - 1;
 123         t -= redux;
 124         z = x - t;
 125         INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
 126
 127         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 128         tv = exp2ft[i0];
 129         u = tv * z;
 130         tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
 131
 132         /* Scale by 2**(k>>20). */
 133         return (tv * twopk);
 134 }