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      1 /*-
      2  * Copyright (c) 2005 David Schultz <das (at) 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("$FreeBSD$");
     29 
     30 #include <float.h>
     31 
     32 #include "math.h"
     33 #include "math_private.h"
     34 
     35 #define	TBLBITS	4
     36 #define	TBLSIZE	(1 << TBLBITS)
     37 
     38 static const float
     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
     46     huge    = 0x1p100f,
     47     twom100 = 0x1p-100f;
     48 
     49 static const double exp2ft[TBLSIZE] = {
     50 	0x1.6a09e667f3bcdp-1,
     51 	0x1.7a11473eb0187p-1,
     52 	0x1.8ace5422aa0dbp-1,
     53 	0x1.9c49182a3f090p-1,
     54 	0x1.ae89f995ad3adp-1,
     55 	0x1.c199bdd85529cp-1,
     56 	0x1.d5818dcfba487p-1,
     57 	0x1.ea4afa2a490dap-1,
     58 	0x1.0000000000000p+0,
     59 	0x1.0b5586cf9890fp+0,
     60 	0x1.172b83c7d517bp+0,
     61 	0x1.2387a6e756238p+0,
     62 	0x1.306fe0a31b715p+0,
     63 	0x1.3dea64c123422p+0,
     64 	0x1.4bfdad5362a27p+0,
     65 	0x1.5ab07dd485429p+0,
     66 };
     67 
     68 /*
     69  * exp2f(x): compute the base 2 exponential of x
     70  *
     71  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
     72  *
     73  * Method: (equally-spaced tables)
     74  *
     75  *   Reduce x:
     76  *     x = 2**k + y, for integer k and |y| <= 1/2.
     77  *     Thus we have exp2f(x) = 2**k * exp2(y).
     78  *
     79  *   Reduce y:
     80  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
     81  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
     82  *     with |z| <= 2**-(TBLSIZE+1).
     83  *
     84  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
     85  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
     86  *   Using double precision for everything except the reduction makes
     87  *   roundoff error insignificant and simplifies the scaling step.
     88  *
     89  *   This method is due to Tang, but I do not use his suggested parameters:
     90  *
     91  *	Tang, P.  Table-driven Implementation of the Exponential Function
     92  *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
     93  */
     94 float
     95 exp2f(float x)
     96 {
     97 	double tv, twopk, u, z;
     98 	float t;
     99 	uint32_t hx, ix, i0;
    100 	int32_t k;
    101 
    102 	/* Filter out exceptional cases. */
    103 	GET_FLOAT_WORD(hx, x);
    104 	ix = hx & 0x7fffffff;		/* high word of |x| */
    105 	if(ix >= 0x43000000) {			/* |x| >= 128 */
    106 		if(ix >= 0x7f800000) {
    107 			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
    108 				return (x + x);	/* x is NaN or +Inf */
    109 			else
    110 				return (0.0);	/* x is -Inf */
    111 		}
    112 		if(x >= 0x1.0p7f)
    113 			return (huge * huge);	/* overflow */
    114 		if(x <= -0x1.2cp7f)
    115 			return (twom100 * twom100); /* underflow */
    116 	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
    117 		return (1.0f + x);
    118 	}
    119 
    120 	/* Reduce x, computing z, i0, and k. */
    121 	STRICT_ASSIGN(float, t, x + redux);
    122 	GET_FLOAT_WORD(i0, t);
    123 	i0 += TBLSIZE / 2;
    124 	k = (i0 >> TBLBITS) << 20;
    125 	i0 &= TBLSIZE - 1;
    126 	t -= redux;
    127 	z = x - t;
    128 	INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
    129 
    130 	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
    131 	tv = exp2ft[i0];
    132 	u = tv * z;
    133 	tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
    134 
    135 	/* Scale by 2**(k>>20). */
    136 	return (tv * twopk);
    137 }
    138