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s_exp2f.cpp (4228B)

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