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 1.2 2014/03/16 22:30:43 dsl 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     redux   = 0x1.8p23f / TBLSIZE,
  43     P1      = 0x1.62e430p-1f,
  44     P2      = 0x1.ebfbe0p-3f,
  45     P3      = 0x1.c6b348p-5f,
  46     P4      = 0x1.3b2c9cp-7f;
  47
  48 /*
  49  * For out of range values we need to generate the appropriate
  50  * underflow or overflow trap as well as generating infinity or zero.
  51  * This means we have to get the fpu to execute an instruction that
  52  * will generate the trap (and not have the compiler optimise it away).
  53  * This is normally done by calculating 'huge * huge' or 'tiny * tiny'.
  54  *
  55  * i386 is particularly problematic.
  56  * The 'float' result is returned on the x87 stack, so is 'long double'.
  57  * If we just multiply two 'float' values the caller will see 0x1p+/-200
  58  * (not 0 or infinity).
  59  * If we use 'double' the compiler does a store-load which will convert the
  60  * value and generate the required exception.
  61  */
  62 #ifdef __i386__
  63 static volatile double overflow = 0x1p+1000;
  64 static volatile double underflow = 0x1p-1000;
  65 #else
  66 static volatile float huge = 0x1p+100;
  67 static volatile float tiny = 0x1p-100;
  68 #define overflow (huge * huge)
  69 #define underflow (tiny * tiny)
  70 #endif
  71
  72 static const double exp2ft[TBLSIZE] = {
  73         0x1.6a09e667f3bcdp-1,
  74         0x1.7a11473eb0187p-1,
  75         0x1.8ace5422aa0dbp-1,
  76         0x1.9c49182a3f090p-1,
  77         0x1.ae89f995ad3adp-1,
  78         0x1.c199bdd85529cp-1,
  79         0x1.d5818dcfba487p-1,
  80         0x1.ea4afa2a490dap-1,
  81         0x1.0000000000000p+0,
  82         0x1.0b5586cf9890fp+0,
  83         0x1.172b83c7d517bp+0,
  84         0x1.2387a6e756238p+0,
  85         0x1.306fe0a31b715p+0,
  86         0x1.3dea64c123422p+0,
  87         0x1.4bfdad5362a27p+0,
  88         0x1.5ab07dd485429p+0,
  89 };
  90
  91 /*
  92  * exp2f(x): compute the base 2 exponential of x
  93  *
  94  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  95  *
  96  * Method: (equally-spaced tables)
  97  *
  98  *   Reduce x:
  99  *     x = 2**k + y, for integer k and |y| <= 1/2.
 100  *     Thus we have exp2f(x) = 2**k * exp2(y).
 101  *
 102  *   Reduce y:
 103  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
 104  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
 105  *     with |z| <= 2**-(TBLSIZE+1).
 106  *
 107  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
 108  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
 109  *   Using double precision for everything except the reduction makes
 110  *   roundoff error insignificant and simplifies the scaling step.
 111  *
 112  *   This method is due to Tang, but I do not use his suggested parameters:
 113  *
 114  *      Tang, P.  Table-driven Implementation of the Exponential Function
 115  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
 116  */
 117 float
 118 exp2f(float x)
 119 {
 120         double tv, twopk, u, z;
 121         float t;
 122         uint32_t hx, ix, i0;
 123         int32_t k;
 124
 125         /* Filter out exceptional cases. */
 126         GET_FLOAT_WORD(hx, x);
 127         ix = hx & 0x7fffffff;           /* high word of |x| */
 128         if(ix >= 0x43000000) {                  /* |x| >= 128 */
 129                 if(ix >= 0x7f800000) {
 130                         if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
 131                                 return (x + x); /* x is NaN or +Inf */
 132                         else
 133                                 return (0.0);   /* x is -Inf */
 134                 }
 135                 if(x >= 0x1.0p7f)
 136                         return overflow;        /* +infinity with overflow */
 137                 if(x <= -0x1.2cp7f)
 138                         return underflow;       /* zero with underflow */
 139         } else if (ix <= 0x33000000) {          /* |x| <= 0x1p-25 */
 140                 return (1.0f + x);
 141         }
 142
 143         /* Reduce x, computing z, i0, and k. */
 144         STRICT_ASSIGN(float, t, x + redux);
 145         GET_FLOAT_WORD(i0, t);
 146         i0 += TBLSIZE / 2;
 147         k = (i0 >> TBLBITS) << 20;
 148         i0 &= TBLSIZE - 1;
 149         t -= redux;
 150         z = x - t;
 151         INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
 152
 153         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 154         tv = exp2ft[i0];
 155         u = tv * z;
 156         tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
 157
 158         /* Scale by 2**(k>>20). */
 159         return (tv * twopk);
 160 }