2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
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.
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
27 #include <sys/cdefs.h>
28 __FBSDID("$FreeBSD$");
33 #include "math_private.h"
36 #define TBLSIZE (1 << TBLBITS)
39 redux = 0x1.8p23f / TBLSIZE,
49 static const double exp2ft[TBLSIZE] = {
69 * exp2f(x): compute the base 2 exponential of x
71 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
73 * Method: (equally-spaced tables)
76 * x = 2**k + y, for integer k and |y| <= 1/2.
77 * Thus we have exp2f(x) = 2**k * exp2(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).
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.
89 * This method is due to Tang, but I do not use his suggested parameters:
91 * Tang, P. Table-driven Implementation of the Exponential Function
92 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
97 double tv, twopk, u, z;
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 */
110 return (0.0); /* x is -Inf */
113 return (huge * huge); /* overflow */
115 return (twom100 * twom100); /* underflow */
116 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
120 /* Reduce x, computing z, i0, and k. */
121 STRICT_ASSIGN(float, t, x + redux);
122 GET_FLOAT_WORD(i0, t);
124 k = (i0 >> TBLBITS) << 20;
128 INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
130 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
133 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
135 /* Scale by 2**(k>>20). */