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n_exp2f.c revision 1.2.4.2 
      1  1.2.4.2  yamt /*-
      2  1.2.4.2  yamt  * Copyright (c) 2005 David Schultz <das (at) FreeBSD.ORG>
      3  1.2.4.2  yamt  * All rights reserved.
      4  1.2.4.2  yamt  *
      5  1.2.4.2  yamt  * Redistribution and use in source and binary forms, with or without
      6  1.2.4.2  yamt  * modification, are permitted provided that the following conditions
      7  1.2.4.2  yamt  * are met:
      8  1.2.4.2  yamt  * 1. Redistributions of source code must retain the above copyright
      9  1.2.4.2  yamt  *    notice, this list of conditions and the following disclaimer.
     10  1.2.4.2  yamt  * 2. Redistributions in binary form must reproduce the above copyright
     11  1.2.4.2  yamt  *    notice, this list of conditions and the following disclaimer in the
     12  1.2.4.2  yamt  *    documentation and/or other materials provided with the distribution.
     13  1.2.4.2  yamt  *
     14  1.2.4.2  yamt  * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
     15  1.2.4.2  yamt  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     16  1.2.4.2  yamt  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
     17  1.2.4.2  yamt  * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
     18  1.2.4.2  yamt  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
     19  1.2.4.2  yamt  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
     20  1.2.4.2  yamt  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     21  1.2.4.2  yamt  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
     22  1.2.4.2  yamt  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
     23  1.2.4.2  yamt  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
     24  1.2.4.2  yamt  * SUCH DAMAGE.
     25  1.2.4.2  yamt  */
     26  1.2.4.2  yamt 
     27  1.2.4.2  yamt #include <sys/cdefs.h>
     28  1.2.4.2  yamt __RCSID("$NetBSD: n_exp2f.c,v 1.2.4.2 2014/05/22 11:36:57 yamt Exp $");
     29  1.2.4.2  yamt #ifdef __FBSDID
     30  1.2.4.2  yamt __FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $");
     31  1.2.4.2  yamt #endif
     32  1.2.4.2  yamt 
     33  1.2.4.2  yamt #include <stdint.h>
     34  1.2.4.2  yamt #include <float.h>
     35  1.2.4.2  yamt #include <string.h>
     36  1.2.4.2  yamt 
     37  1.2.4.2  yamt #include "math.h"
     38  1.2.4.2  yamt 
     39  1.2.4.2  yamt #define	TBLBITS	4
     40  1.2.4.2  yamt #define	TBLSIZE	(1 << TBLBITS)
     41  1.2.4.2  yamt 
     42  1.2.4.2  yamt static const float
     43  1.2.4.2  yamt     huge    = 0x1p100f,
     44  1.2.4.2  yamt     redux   = 0x1.8p23f / TBLSIZE,
     45  1.2.4.2  yamt     P1	    = 0x1.62e430p-1f,
     46  1.2.4.2  yamt     P2	    = 0x1.ebfbe0p-3f,
     47  1.2.4.2  yamt     P3	    = 0x1.c6b348p-5f,
     48  1.2.4.2  yamt     P4	    = 0x1.3b2c9cp-7f;
     49  1.2.4.2  yamt 
     50  1.2.4.2  yamt static volatile float twom100 = 0x1p-100f;
     51  1.2.4.2  yamt 
     52  1.2.4.2  yamt static const double exp2ft[TBLSIZE] = {
     53  1.2.4.2  yamt 	0x1.6a09e667f3bcdp-1,
     54  1.2.4.2  yamt 	0x1.7a11473eb0187p-1,
     55  1.2.4.2  yamt 	0x1.8ace5422aa0dbp-1,
     56  1.2.4.2  yamt 	0x1.9c49182a3f090p-1,
     57  1.2.4.2  yamt 	0x1.ae89f995ad3adp-1,
     58  1.2.4.2  yamt 	0x1.c199bdd85529cp-1,
     59  1.2.4.2  yamt 	0x1.d5818dcfba487p-1,
     60  1.2.4.2  yamt 	0x1.ea4afa2a490dap-1,
     61  1.2.4.2  yamt 	0x1.0000000000000p+0,
     62  1.2.4.2  yamt 	0x1.0b5586cf9890fp+0,
     63  1.2.4.2  yamt 	0x1.172b83c7d517bp+0,
     64  1.2.4.2  yamt 	0x1.2387a6e756238p+0,
     65  1.2.4.2  yamt 	0x1.306fe0a31b715p+0,
     66  1.2.4.2  yamt 	0x1.3dea64c123422p+0,
     67  1.2.4.2  yamt 	0x1.4bfdad5362a27p+0,
     68  1.2.4.2  yamt 	0x1.5ab07dd485429p+0,
     69  1.2.4.2  yamt };
     70  1.2.4.2  yamt 
     71  1.2.4.2  yamt /*
     72  1.2.4.2  yamt  * exp2f(x): compute the base 2 exponential of x
     73  1.2.4.2  yamt  *
     74  1.2.4.2  yamt  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
     75  1.2.4.2  yamt  *
     76  1.2.4.2  yamt  * Method: (equally-spaced tables)
     77  1.2.4.2  yamt  *
     78  1.2.4.2  yamt  *   Reduce x:
     79  1.2.4.2  yamt  *     x = 2**k + y, for integer k and |y| <= 1/2.
     80  1.2.4.2  yamt  *     Thus we have exp2f(x) = 2**k * exp2(y).
     81  1.2.4.2  yamt  *
     82  1.2.4.2  yamt  *   Reduce y:
     83  1.2.4.2  yamt  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
     84  1.2.4.2  yamt  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
     85  1.2.4.2  yamt  *     with |z| <= 2**-(TBLSIZE+1).
     86  1.2.4.2  yamt  *
     87  1.2.4.2  yamt  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
     88  1.2.4.2  yamt  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
     89  1.2.4.2  yamt  *   Using double precision for everything except the reduction makes
     90  1.2.4.2  yamt  *   roundoff error insignificant and simplifies the scaling step.
     91  1.2.4.2  yamt  *
     92  1.2.4.2  yamt  *   This method is due to Tang, but I do not use his suggested parameters:
     93  1.2.4.2  yamt  *
     94  1.2.4.2  yamt  *	Tang, P.  Table-driven Implementation of the Exponential Function
     95  1.2.4.2  yamt  *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
     96  1.2.4.2  yamt  */
     97  1.2.4.2  yamt float
     98  1.2.4.2  yamt exp2f(float x)
     99  1.2.4.2  yamt {
    100  1.2.4.2  yamt 	double tv, twopk, u, z;
    101  1.2.4.2  yamt 	float t;
    102  1.2.4.2  yamt 	uint32_t hx, ix, i0;
    103  1.2.4.2  yamt 	int32_t k, temp;
    104  1.2.4.2  yamt 
    105  1.2.4.2  yamt 	/* Filter out exceptional cases. */
    106  1.2.4.2  yamt 	memcpy(&hx, &x, sizeof(hx));
    107  1.2.4.2  yamt 	ix = hx & 0x7fffffff;		/* high word of |x| */
    108  1.2.4.2  yamt 	if(ix >= 0x43000000) {			/* |x| >= 128 */
    109  1.2.4.2  yamt 		if(x >= 0x1.0p7f)
    110  1.2.4.2  yamt 			return (huge * huge);	/* overflow */
    111  1.2.4.2  yamt 		if(x <= -0x1.2cp7f)
    112  1.2.4.2  yamt 			return (twom100 * twom100); /* underflow */
    113  1.2.4.2  yamt 	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
    114  1.2.4.2  yamt 		return (1.0f + x);
    115  1.2.4.2  yamt 	}
    116  1.2.4.2  yamt 
    117  1.2.4.2  yamt 	/* Reduce x, computing z, i0, and k. */
    118  1.2.4.2  yamt 	i0 = x + redux;
    119  1.2.4.2  yamt 	memcpy(&t, &i0, sizeof(t));
    120  1.2.4.2  yamt 	i0 += TBLSIZE / 2;
    121  1.2.4.2  yamt 	k = (i0 >> TBLBITS) << 20;
    122  1.2.4.2  yamt 	i0 &= TBLSIZE - 1;
    123  1.2.4.2  yamt 	t -= redux;
    124  1.2.4.2  yamt 	z = x - t;
    125  1.2.4.2  yamt 	temp = 0x3ff00000+k;
    126  1.2.4.2  yamt 	twopk = 0.0;
    127  1.2.4.2  yamt 	memcpy(&twopk, &temp, sizeof(temp));
    128  1.2.4.2  yamt 
    129  1.2.4.2  yamt 	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
    130  1.2.4.2  yamt 	tv = exp2ft[i0];
    131  1.2.4.2  yamt 	u = tv * z;
    132  1.2.4.2  yamt 	tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
    133  1.2.4.2  yamt 
    134  1.2.4.2  yamt 	/* Scale by 2**(k>>20). */
    135  1.2.4.2  yamt 	return (tv * twopk);
    136  1.2.4.2  yamt }
    137