1/*-
2 * Copyright (c) 2005 David Schultz <das@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>
| 1/*-
2 * Copyright (c) 2005 David Schultz <das@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: head/lib/msun/src/s_exp2f.c 144648 2005-04-05 02:57:15Z das $");
| 28__FBSDID("$FreeBSD: head/lib/msun/src/s_exp2f.c 175468 2008-01-18 22:19:04Z das $");
|
29
30#include "math.h"
31#include "math_private.h"
32
33#define TBLBITS 4
34#define TBLSIZE (1 << TBLBITS)
35
36static const float
37 huge = 0x1p100f,
| 29
30#include "math.h"
31#include "math_private.h"
32
33#define TBLBITS 4
34#define TBLSIZE (1 << TBLBITS)
35
36static const float
37 huge = 0x1p100f,
|
38 twom100 = 0x1p-100f,
| |
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
| 38 redux = 0x1.8p23f / TBLSIZE,
39 P1 = 0x1.62e430p-1f,
40 P2 = 0x1.ebfbe0p-3f,
41 P3 = 0x1.c6b348p-5f,
42 P4 = 0x1.3b2c9cp-7f;
43
|
| 44static volatile float twom100 = 0x1p-100f;
45
|
45static const double exp2ft[TBLSIZE] = {
46 0x1.6a09e667f3bcdp-1,
47 0x1.7a11473eb0187p-1,
48 0x1.8ace5422aa0dbp-1,
49 0x1.9c49182a3f090p-1,
50 0x1.ae89f995ad3adp-1,
51 0x1.c199bdd85529cp-1,
52 0x1.d5818dcfba487p-1,
53 0x1.ea4afa2a490dap-1,
54 0x1.0000000000000p+0,
55 0x1.0b5586cf9890fp+0,
56 0x1.172b83c7d517bp+0,
57 0x1.2387a6e756238p+0,
58 0x1.306fe0a31b715p+0,
59 0x1.3dea64c123422p+0,
60 0x1.4bfdad5362a27p+0,
61 0x1.5ab07dd485429p+0,
62};
63
64/*
65 * exp2f(x): compute the base 2 exponential of x
66 *
67 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
68 *
69 * Method: (equally-spaced tables)
70 *
71 * Reduce x:
72 * x = 2**k + y, for integer k and |y| <= 1/2.
73 * Thus we have exp2f(x) = 2**k * exp2(y).
74 *
75 * Reduce y:
76 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
77 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
78 * with |z| <= 2**-(TBLSIZE+1).
79 *
80 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
81 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
82 * Using double precision in the final calculation avoids roundoff error.
83 *
84 * This method is due to Tang, but I do not use his suggested parameters:
85 *
86 * Tang, P. Table-driven Implementation of the Exponential Function
87 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
88 */
89float
90exp2f(float x)
91{
92 double tv;
93 float r, z;
94 volatile float t; /* prevent gcc from using too much precision */
95 uint32_t hx, hr, ix, i0;
96 int32_t k;
97
98 /* Filter out exceptional cases. */
99 GET_FLOAT_WORD(hx,x);
100 ix = hx & 0x7fffffff; /* high word of |x| */
101 if(ix >= 0x43000000) { /* |x| >= 128 */
102 if(ix >= 0x7f800000) {
103 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
104 return (x); /* x is NaN or +Inf */
105 else
106 return (0.0); /* x is -Inf */
107 }
108 if(x >= 0x1.0p7f)
109 return (huge * huge); /* overflow */
110 if(x <= -0x1.2cp7f)
111 return (twom100 * twom100); /* underflow */
112 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
113 return (1.0f + x);
114 }
115
116 /* Reduce x, computing z, i0, and k. */
117 t = x + redux;
118 GET_FLOAT_WORD(i0, t);
119 i0 += TBLSIZE / 2;
120 k = (i0 >> TBLBITS) << 23;
121 i0 &= TBLSIZE - 1;
122 t -= redux;
123 z = x - t;
124
125 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
126 tv = exp2ft[i0];
127 r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
128
129 /* Scale by 2**(k>>23). */
130 if(k >= -125 << 23) {
131 if (k != 0) {
132 GET_FLOAT_WORD(hr, r);
133 SET_FLOAT_WORD(r, hr + k);
134 }
135 return (r);
136 } else {
137 GET_FLOAT_WORD(hr, r);
138 SET_FLOAT_WORD(r, hr + (k + (100 << 23)));
139 return (r * twom100);
140 }
141}
| 46static const double exp2ft[TBLSIZE] = {
47 0x1.6a09e667f3bcdp-1,
48 0x1.7a11473eb0187p-1,
49 0x1.8ace5422aa0dbp-1,
50 0x1.9c49182a3f090p-1,
51 0x1.ae89f995ad3adp-1,
52 0x1.c199bdd85529cp-1,
53 0x1.d5818dcfba487p-1,
54 0x1.ea4afa2a490dap-1,
55 0x1.0000000000000p+0,
56 0x1.0b5586cf9890fp+0,
57 0x1.172b83c7d517bp+0,
58 0x1.2387a6e756238p+0,
59 0x1.306fe0a31b715p+0,
60 0x1.3dea64c123422p+0,
61 0x1.4bfdad5362a27p+0,
62 0x1.5ab07dd485429p+0,
63};
64
65/*
66 * exp2f(x): compute the base 2 exponential of x
67 *
68 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
69 *
70 * Method: (equally-spaced tables)
71 *
72 * Reduce x:
73 * x = 2**k + y, for integer k and |y| <= 1/2.
74 * Thus we have exp2f(x) = 2**k * exp2(y).
75 *
76 * Reduce y:
77 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
78 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
79 * with |z| <= 2**-(TBLSIZE+1).
80 *
81 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
82 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
83 * Using double precision in the final calculation avoids roundoff error.
84 *
85 * This method is due to Tang, but I do not use his suggested parameters:
86 *
87 * Tang, P. Table-driven Implementation of the Exponential Function
88 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
89 */
90float
91exp2f(float x)
92{
93 double tv;
94 float r, z;
95 volatile float t; /* prevent gcc from using too much precision */
96 uint32_t hx, hr, ix, i0;
97 int32_t k;
98
99 /* Filter out exceptional cases. */
100 GET_FLOAT_WORD(hx,x);
101 ix = hx & 0x7fffffff; /* high word of |x| */
102 if(ix >= 0x43000000) { /* |x| >= 128 */
103 if(ix >= 0x7f800000) {
104 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
105 return (x); /* x is NaN or +Inf */
106 else
107 return (0.0); /* x is -Inf */
108 }
109 if(x >= 0x1.0p7f)
110 return (huge * huge); /* overflow */
111 if(x <= -0x1.2cp7f)
112 return (twom100 * twom100); /* underflow */
113 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
114 return (1.0f + x);
115 }
116
117 /* Reduce x, computing z, i0, and k. */
118 t = x + redux;
119 GET_FLOAT_WORD(i0, t);
120 i0 += TBLSIZE / 2;
121 k = (i0 >> TBLBITS) << 23;
122 i0 &= TBLSIZE - 1;
123 t -= redux;
124 z = x - t;
125
126 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
127 tv = exp2ft[i0];
128 r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
129
130 /* Scale by 2**(k>>23). */
131 if(k >= -125 << 23) {
132 if (k != 0) {
133 GET_FLOAT_WORD(hr, r);
134 SET_FLOAT_WORD(r, hr + k);
135 }
136 return (r);
137 } else {
138 GET_FLOAT_WORD(hr, r);
139 SET_FLOAT_WORD(r, hr + (k + (100 << 23)));
140 return (r * twom100);
141 }
142}
|