lib/libm/noieee_src/n_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: n_exp2f.c,v 1.2 2014/03/23 15:26:47 martin 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 <stdint.h>
  34 #include <float.h>
  35 #include <string.h>
  36
  37 #include "math.h"
  38
  39 #define TBLBITS 4
  40 #define TBLSIZE (1 << TBLBITS)
  41
  42 static const float
  43     huge    = 0x1p100f,
  44     redux   = 0x1.8p23f / TBLSIZE,
  45     P1      = 0x1.62e430p-1f,
  46     P2      = 0x1.ebfbe0p-3f,
  47     P3      = 0x1.c6b348p-5f,
  48     P4      = 0x1.3b2c9cp-7f;
  49
  50 static volatile float twom100 = 0x1p-100f;
  51
  52 static const double exp2ft[TBLSIZE] = {
  53         0x1.6a09e667f3bcdp-1,
  54         0x1.7a11473eb0187p-1,
  55         0x1.8ace5422aa0dbp-1,
  56         0x1.9c49182a3f090p-1,
  57         0x1.ae89f995ad3adp-1,
  58         0x1.c199bdd85529cp-1,
  59         0x1.d5818dcfba487p-1,
  60         0x1.ea4afa2a490dap-1,
  61         0x1.0000000000000p+0,
  62         0x1.0b5586cf9890fp+0,
  63         0x1.172b83c7d517bp+0,
  64         0x1.2387a6e756238p+0,
  65         0x1.306fe0a31b715p+0,
  66         0x1.3dea64c123422p+0,
  67         0x1.4bfdad5362a27p+0,
  68         0x1.5ab07dd485429p+0,
  69 };
  70
  71 /*
  72  * exp2f(x): compute the base 2 exponential of x
  73  *
  74  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
  75  *
  76  * Method: (equally-spaced tables)
  77  *
  78  *   Reduce x:
  79  *     x = 2**k + y, for integer k and |y| <= 1/2.
  80  *     Thus we have exp2f(x) = 2**k * exp2(y).
  81  *
  82  *   Reduce y:
  83  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
  84  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
  85  *     with |z| <= 2**-(TBLSIZE+1).
  86  *
  87  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
  88  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
  89  *   Using double precision for everything except the reduction makes
  90  *   roundoff error insignificant and simplifies the scaling step.
  91  *
  92  *   This method is due to Tang, but I do not use his suggested parameters:
  93  *
  94  *      Tang, P.  Table-driven Implementation of the Exponential Function
  95  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
  96  */
  97 float
  98 exp2f(float x)
  99 {
 100         double tv, twopk, u, z;
 101         float t;
 102         uint32_t hx, ix, i0;
 103         int32_t k, temp;
 104
 105         /* Filter out exceptional cases. */
 106         memcpy(&hx, &x, sizeof(hx));
 107         ix = hx & 0x7fffffff;           /* high word of |x| */
 108         if(ix >= 0x43000000) {                  /* |x| >= 128 */
 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         i0 = x + redux;
 119         memcpy(&t, &i0, sizeof(t));
 120         i0 += TBLSIZE / 2;
 121         k = (i0 >> TBLBITS) << 20;
 122         i0 &= TBLSIZE - 1;
 123         t -= redux;
 124         z = x - t;
 125         temp = 0x3ff00000+k;
 126         twopk = 0.0;
 127         memcpy(&twopk, &temp, sizeof(temp));
 128
 129         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
 130         tv = exp2ft[i0];
 131         u = tv * z;
 132         tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
 133
 134         /* Scale by 2**(k>>20). */
 135         return (tv * twopk);
 136 }