gonzui

    1: /*      $OpenBSD: s_exp2f.c,v 1.1 2008/07/24 09:40:16 martynas 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 <sys/cdefs.h>
   29: #include <float.h>
   30: 
   31: #include "math.h"
   32: #include "math_private.h"
   33: 
   34: #define TBLBITS 4
   35: #define TBLSIZE (1 << TBLBITS)
   36: 
   37: static const float
   38:     huge    = 0x1p100f,
   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 volatile float twom100 = 0x1p-100f;
   46: 
   47: static const double exp2ft[TBLSIZE] = {
   48:         0x1.6a09e667f3bcdp-1,
   49:         0x1.7a11473eb0187p-1,
   50:         0x1.8ace5422aa0dbp-1,
   51:         0x1.9c49182a3f090p-1,
   52:         0x1.ae89f995ad3adp-1,
   53:         0x1.c199bdd85529cp-1,
   54:         0x1.d5818dcfba487p-1,
   55:         0x1.ea4afa2a490dap-1,
   56:         0x1.0000000000000p+0,
   57:         0x1.0b5586cf9890fp+0,
   58:         0x1.172b83c7d517bp+0,
   59:         0x1.2387a6e756238p+0,
   60:         0x1.306fe0a31b715p+0,
   61:         0x1.3dea64c123422p+0,
   62:         0x1.4bfdad5362a27p+0,
   63:         0x1.5ab07dd485429p+0,
   64: };
   65:         
   66: /*
   67:  * exp2f(x): compute the base 2 exponential of x
   68:  *
   69:  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
   70:  *
   71:  * Method: (equally-spaced tables)
   72:  *
   73:  *   Reduce x:
   74:  *     x = 2**k + y, for integer k and |y| <= 1/2.
   75:  *     Thus we have exp2f(x) = 2**k * exp2(y).
   76:  *
   77:  *   Reduce y:
   78:  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
   79:  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
   80:  *     with |z| <= 2**-(TBLSIZE+1).
   81:  *
   82:  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
   83:  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
   84:  *   Using double precision for everything except the reduction makes
   85:  *   roundoff error insignificant and simplifies the scaling step.
   86:  *
   87:  *   This method is due to Tang, but I do not use his suggested parameters:
   88:  *
   89:  *      Tang, P.  Table-driven Implementation of the Exponential Function
   90:  *      in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
   91:  */
   92: float
   93: exp2f(float x)
   94: {
   95:         double tv, twopk, u, z;
   96:         float t;
   97:         uint32_t hx, ix, i0;
   98:         int32_t k;
   99: 
  100:         /* Filter out exceptional cases. */
  101:         GET_FLOAT_WORD(hx, x);
  102:         ix = hx & 0x7fffffff;          /* high word of |x| */
  103:         if(ix >= 0x43000000) {                 /* |x| >= 128 */
  104:                 if(ix >= 0x7f800000) {
  105:                         if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
  106:                                 return (x + x);     /* x is NaN or +Inf */
  107:                         else 
  108:                                 return (0.0);       /* x is -Inf */
  109:                 }
  110:                 if(x >= 0x1.0p7f)
  111:                         return (huge * huge);        /* overflow */
  112:                 if(x <= -0x1.2cp7f)
  113:                         return (twom100 * twom100); /* underflow */
  114:         } else if (ix <= 0x33000000) {         /* |x| <= 0x1p-25 */
  115:                 return (1.0f + x);
  116:         }
  117: 
  118:         /* Reduce x, computing z, i0, and k. */
  119:         STRICT_ASSIGN(float, t, x + redux);
  120:         GET_FLOAT_WORD(i0, t);
  121:         i0 += TBLSIZE / 2;
  122:         k = (i0 >> TBLBITS) << 20;
  123:         i0 &= TBLSIZE - 1;
  124:         t -= redux;
  125:         z = x - t;
  126:         INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
  127: 
  128:         /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
  129:         tv = exp2ft[i0];
  130:         u = tv * z;
  131:         tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
  132: 
  133:         /* Scale by 2**(k>>20). */
  134:         return (tv * twopk);
  135: }