compiler/mlib/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 __FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.1 2005/04/05 02:57:15 das Exp $");
  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     twom100 = 0x1p-100f,
  39     redux   = 0x1.8p23f / TBLSIZE,
  40     P1      = 0x1.62e430p-1f,
  41     P2      = 0x1.ebfbe0p-3f,
  42     P3      = 0x1.c6b348p-5f,
  43     P4      = 0x1.3b2c9cp-7f;
  44
  45 static const double exp2ft[TBLSIZE] = {
  46         0x1.6a09e667f3bcdp-1,
  47         0x1.7a11473eb0187p-1,
  48         0x1.8ace5422aa0dbp-1,
  49         0x1.9c49182a3f090p-1,
  50         0x1.ae89f995ad3adp-1,
  51         0x1.c199bdd85529cp-1,
  52         0x1.d5818dcfba487p-1,
  53         0x1.ea4afa2a490dap-1,
  54         0x1.0000000000000p+0,
  55         0x1.0b5586cf9890fp+0,
  56         0x1.172b83c7d517bp+0,
  57         0x1.2387a6e756238p+0,
  58         0x1.306fe0a31b715p+0,
  59         0x1.3dea64c123422p+0,
  60         0x1.4bfdad5362a27p+0,
  61         0x1.5ab07dd485429p+0,
  62 };
  63
  64 /*
  65  * exp2f(x): compute the base 2 exponential of x
  66  *
  67  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  68  *
  69  * Method: (equally-spaced tables)
  70  *
  71  *   Reduce x:
  72  *     x = 2**k + y, for integer k and |y| <= 1/2.
  73  *     Thus we have exp2f(x) = 2**k * exp2(y).
  74  *
  75  *   Reduce y:
  76  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
  77  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
  78  *     with |z| <= 2**-(TBLSIZE+1).
  79  *
  80  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
  81  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
  82  *   Using double precision in the final calculation avoids roundoff error.
  83  *
  84  *   This method is due to Tang, but I do not use his suggested parameters:
  85  *
  86  *      Tang, P.  Table-driven Implementation of the Exponential Function
  87  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
  88  */
  89 float
  90 exp2f(float x)
  91 {
  92         double tv;
  93         float r, z;
  94         volatile float t;       /* prevent gcc from using too much precision */
  95         uint32_t hx, hr, ix, i0;
  96         int32_t k;
  97
  98         /* Filter out exceptional cases. */
  99         GET_FLOAT_WORD(hx,x);
 100         ix = hx & 0x7fffffff;           /* high word of |x| */
 101         if(ix >= 0x43000000) {                  /* |x| >= 128 */
 102                 if(ix >= 0x7f800000) {
 103                         if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
 104                                 return (x);     /* x is NaN or +Inf */
 105                         else
 106                                 return (0.0);   /* x is -Inf */
 107                 }
 108                 if(x >= 0x1.0p7f)
 109                         return (huge * huge);   /* overflow */
 110                 if(x <= -0x1.2cp7f)
 111                         return (twom100 * twom100); /* underflow */
 112         } else if (ix <= 0x33000000) {          /* |x| <= 0x1p-25 */
 113                 return (1.0f + x);
 114         }
 115
 116         /* Reduce x, computing z, i0, and k. */
 117         t = x + redux;
 118         GET_FLOAT_WORD(i0, t);
 119         i0 += TBLSIZE / 2;
 120         k = (i0 >> TBLBITS) << 23;
 121         i0 &= TBLSIZE - 1;
 122         t -= redux;
 123         z = x - t;
 124
 125         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 126         tv = exp2ft[i0];
 127         r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
 128
 129         /* Scale by 2**(k>>23). */
 130         if(k >= -125 << 23) {
 131                 if (k != 0) {
 132                         GET_FLOAT_WORD(hr, r);
 133                         SET_FLOAT_WORD(r, hr + k);
 134                 }
 135                 return (r);
 136         } else {
 137                 GET_FLOAT_WORD(hr, r);
 138                 SET_FLOAT_WORD(r, hr + (k + (100 << 23)));
 139                 return (r * twom100);
 140         }
 141 }