lib/libm/src/s_exp2f.c

   1 /*-
   2  * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
   3  * All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  *
  14  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
  15  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  16  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  17  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
  18  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  19  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  20  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  21  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  22  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  23  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  24  * SUCH DAMAGE.
  25  */
  26
  27 #include <sys/cdefs.h>
  28 __RCSID("$NetBSD: s_exp2f.c,v 1.1 2010/01/11 16:28:39 christos Exp $");
  29 #ifdef __FBSDID
  30 __FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $");
  31 #endif
  32
  33 #include <float.h>
  34
  35 #include "math.h"
  36 #include "math_private.h"
  37
  38 #define TBLBITS 4
  39 #define TBLSIZE (1 << TBLBITS)
  40
  41 static const float
  42     huge    = 0x1p100f,
  43     redux   = 0x1.8p23f / TBLSIZE,
  44     P1      = 0x1.62e430p-1f,
  45     P2      = 0x1.ebfbe0p-3f,
  46     P3      = 0x1.c6b348p-5f,
  47     P4      = 0x1.3b2c9cp-7f;
  48
  49 static volatile float twom100 = 0x1p-100f;
  50
  51 static const double exp2ft[TBLSIZE] = {
  52         0x1.6a09e667f3bcdp-1,
  53         0x1.7a11473eb0187p-1,
  54         0x1.8ace5422aa0dbp-1,
  55         0x1.9c49182a3f090p-1,
  56         0x1.ae89f995ad3adp-1,
  57         0x1.c199bdd85529cp-1,
  58         0x1.d5818dcfba487p-1,
  59         0x1.ea4afa2a490dap-1,
  60         0x1.0000000000000p+0,
  61         0x1.0b5586cf9890fp+0,
  62         0x1.172b83c7d517bp+0,
  63         0x1.2387a6e756238p+0,
  64         0x1.306fe0a31b715p+0,
  65         0x1.3dea64c123422p+0,
  66         0x1.4bfdad5362a27p+0,
  67         0x1.5ab07dd485429p+0,
  68 };
  69
  70 /*
  71  * exp2f(x): compute the base 2 exponential of x
  72  *
  73  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  74  *
  75  * Method: (equally-spaced tables)
  76  *
  77  *   Reduce x:
  78  *     x = 2**k + y, for integer k and |y| <= 1/2.
  79  *     Thus we have exp2f(x) = 2**k * exp2(y).
  80  *
  81  *   Reduce y:
  82  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
  83  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
  84  *     with |z| <= 2**-(TBLSIZE+1).
  85  *
  86  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
  87  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
  88  *   Using double precision for everything except the reduction makes
  89  *   roundoff error insignificant and simplifies the scaling step.
  90  *
  91  *   This method is due to Tang, but I do not use his suggested parameters:
  92  *
  93  *      Tang, P.  Table-driven Implementation of the Exponential Function
  94  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
  95  */
  96 float
  97 exp2f(float x)
  98 {
  99         double tv, twopk, u, z;
 100         float t;
 101         uint32_t hx, ix, i0;
 102         int32_t k;
 103
 104         /* Filter out exceptional cases. */
 105         GET_FLOAT_WORD(hx, x);
 106         ix = hx & 0x7fffffff;           /* high word of |x| */
 107         if(ix >= 0x43000000) {                  /* |x| >= 128 */
 108                 if(ix >= 0x7f800000) {
 109                         if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
 110                                 return (x + x); /* x is NaN or +Inf */
 111                         else
 112                                 return (0.0);   /* x is -Inf */
 113                 }
 114                 if(x >= 0x1.0p7f)
 115                         return (huge * huge);   /* overflow */
 116                 if(x <= -0x1.2cp7f)
 117                         return (twom100 * twom100); /* underflow */
 118         } else if (ix <= 0x33000000) {          /* |x| <= 0x1p-25 */
 119                 return (1.0f + x);
 120         }
 121
 122         /* Reduce x, computing z, i0, and k. */
 123         STRICT_ASSIGN(float, t, x + redux);
 124         GET_FLOAT_WORD(i0, t);
 125         i0 += TBLSIZE / 2;
 126         k = (i0 >> TBLBITS) << 20;
 127         i0 &= TBLSIZE - 1;
 128         t -= redux;
 129         z = x - t;
 130         INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
 131
 132         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 133         tv = exp2ft[i0];
 134         u = tv * z;
 135         tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
 136
 137         /* Scale by 2**(k>>20). */
 138         return (tv * twopk);
 139 }