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1 /* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */ | 1 /* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */ |
2 /*- | 2 /*- |
3 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> | 3 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> |
4 * All rights reserved. | 4 * All rights reserved. |
5 * | 5 * |
6 * Redistribution and use in source and binary forms, with or without | 6 * Redistribution and use in source and binary forms, with or without |
7 * modification, are permitted provided that the following conditions | 7 * modification, are permitted provided that the following conditions |
8 * are met: | 8 * are met: |
9 * 1. Redistributions of source code must retain the above copyright | 9 * 1. Redistributions of source code must retain the above copyright |
10 * notice, this list of conditions and the following disclaimer. | 10 * notice, this list of conditions and the following disclaimer. |
(...skipping 11 matching lines...) Expand all Loading... |
22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT | 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY | 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
25 * SUCH DAMAGE. | 25 * SUCH DAMAGE. |
26 */ | 26 */ |
27 | 27 |
28 #include "libm.h" | 28 #include "libm.h" |
29 | 29 |
30 #define TBLSIZE 16 | 30 #define TBLSIZE 16 |
31 | 31 |
32 static const float | 32 static const float redux = 0x1.8p23f / TBLSIZE, P1 = 0x1.62e430p-1f, |
33 redux = 0x1.8p23f / TBLSIZE, | 33 P2 = 0x1.ebfbe0p-3f, P3 = 0x1.c6b348p-5f, |
34 P1 = 0x1.62e430p-1f, | 34 P4 = 0x1.3b2c9cp-7f; |
35 P2 = 0x1.ebfbe0p-3f, | |
36 P3 = 0x1.c6b348p-5f, | |
37 P4 = 0x1.3b2c9cp-7f; | |
38 | 35 |
39 static const double exp2ft[TBLSIZE] = { | 36 static const double exp2ft[TBLSIZE] = { |
40 0x1.6a09e667f3bcdp-1, | 37 0x1.6a09e667f3bcdp-1, 0x1.7a11473eb0187p-1, 0x1.8ace5422aa0dbp-1, |
41 0x1.7a11473eb0187p-1, | 38 0x1.9c49182a3f090p-1, 0x1.ae89f995ad3adp-1, 0x1.c199bdd85529cp-1, |
42 0x1.8ace5422aa0dbp-1, | 39 0x1.d5818dcfba487p-1, 0x1.ea4afa2a490dap-1, 0x1.0000000000000p+0, |
43 0x1.9c49182a3f090p-1, | 40 0x1.0b5586cf9890fp+0, 0x1.172b83c7d517bp+0, 0x1.2387a6e756238p+0, |
44 0x1.ae89f995ad3adp-1, | 41 0x1.306fe0a31b715p+0, 0x1.3dea64c123422p+0, 0x1.4bfdad5362a27p+0, |
45 0x1.c199bdd85529cp-1, | 42 0x1.5ab07dd485429p+0, |
46 0x1.d5818dcfba487p-1, | |
47 0x1.ea4afa2a490dap-1, | |
48 0x1.0000000000000p+0, | |
49 0x1.0b5586cf9890fp+0, | |
50 0x1.172b83c7d517bp+0, | |
51 0x1.2387a6e756238p+0, | |
52 0x1.306fe0a31b715p+0, | |
53 0x1.3dea64c123422p+0, | |
54 0x1.4bfdad5362a27p+0, | |
55 0x1.5ab07dd485429p+0, | |
56 }; | 43 }; |
57 | 44 |
58 /* | 45 /* |
59 * exp2f(x): compute the base 2 exponential of x | 46 * exp2f(x): compute the base 2 exponential of x |
60 * | 47 * |
61 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. | 48 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. |
62 * | 49 * |
63 * Method: (equally-spaced tables) | 50 * Method: (equally-spaced tables) |
64 * | 51 * |
65 * Reduce x: | 52 * Reduce x: |
66 * x = k + y, for integer k and |y| <= 1/2. | 53 * x = k + y, for integer k and |y| <= 1/2. |
67 * Thus we have exp2f(x) = 2**k * exp2(y). | 54 * Thus we have exp2f(x) = 2**k * exp2(y). |
68 * | 55 * |
69 * Reduce y: | 56 * Reduce y: |
70 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. | 57 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. |
71 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), | 58 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), |
72 * with |z| <= 2**-(TBLSIZE+1). | 59 * with |z| <= 2**-(TBLSIZE+1). |
73 * | 60 * |
74 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a | 61 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a |
75 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. | 62 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33. |
76 * Using double precision for everything except the reduction makes | 63 * Using double precision for everything except the reduction makes |
77 * roundoff error insignificant and simplifies the scaling step. | 64 * roundoff error insignificant and simplifies the scaling step. |
78 * | 65 * |
79 * This method is due to Tang, but I do not use his suggested parameters: | 66 * This method is due to Tang, but I do not use his suggested parameters: |
80 * | 67 * |
81 * Tang, P. Table-driven Implementation of the Exponential Function | 68 * Tang, P. Table-driven Implementation of the Exponential Function |
82 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). | 69 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). |
83 */ | 70 */ |
84 float exp2f(float x) | 71 float exp2f(float x) { |
85 { | 72 double_t t, r, z; |
86 » double_t t, r, z; | 73 union { |
87 » union {float f; uint32_t i;} u = {x}; | 74 float f; |
88 » union {double f; uint64_t i;} uk; | 75 uint32_t i; |
89 » uint32_t ix, i0, k; | 76 } u = {x}; |
| 77 union { |
| 78 double f; |
| 79 uint64_t i; |
| 80 } uk; |
| 81 uint32_t ix, i0, k; |
90 | 82 |
91 » /* Filter out exceptional cases. */ | 83 /* Filter out exceptional cases. */ |
92 » ix = u.i & 0x7fffffff; | 84 ix = u.i & 0x7fffffff; |
93 » if (ix > 0x42fc0000) { /* |x| > 126 */ | 85 if (ix > 0x42fc0000) { /* |x| > 126 */ |
94 » » if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */ | 86 if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */ |
95 » » » x *= 0x1p127f; | 87 x *= 0x1p127f; |
96 » » » return x; | 88 return x; |
97 » » } | 89 } |
98 » » if (u.i >= 0x80000000) { /* x < -126 */ | 90 if (u.i >= 0x80000000) { /* x < -126 */ |
99 » » » if (u.i >= 0xc3160000 || (u.i & 0x0000ffff)) | 91 if (u.i >= 0xc3160000 || (u.i & 0x0000ffff)) |
100 » » » » FORCE_EVAL(-0x1p-149f/x); | 92 FORCE_EVAL(-0x1p-149f / x); |
101 » » » if (u.i >= 0xc3160000) /* x <= -150 */ | 93 if (u.i >= 0xc3160000) /* x <= -150 */ |
102 » » » » return 0; | 94 return 0; |
103 » » } | 95 } |
104 » } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ | 96 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ |
105 » » return 1.0f + x; | 97 return 1.0f + x; |
106 » } | 98 } |
107 | 99 |
108 » /* Reduce x, computing z, i0, and k. */ | 100 /* Reduce x, computing z, i0, and k. */ |
109 » u.f = x + redux; | 101 u.f = x + redux; |
110 » i0 = u.i; | 102 i0 = u.i; |
111 » i0 += TBLSIZE / 2; | 103 i0 += TBLSIZE / 2; |
112 » k = i0 / TBLSIZE; | 104 k = i0 / TBLSIZE; |
113 » uk.i = (uint64_t)(0x3ff + k)<<52; | 105 uk.i = (uint64_t)(0x3ff + k) << 52; |
114 » i0 &= TBLSIZE - 1; | 106 i0 &= TBLSIZE - 1; |
115 » u.f -= redux; | 107 u.f -= redux; |
116 » z = x - u.f; | 108 z = x - u.f; |
117 » /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ | 109 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ |
118 » r = exp2ft[i0]; | 110 r = exp2ft[i0]; |
119 » t = r * z; | 111 t = r * z; |
120 » r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4); | 112 r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4); |
121 | 113 |
122 » /* Scale by 2**k */ | 114 /* Scale by 2**k */ |
123 » return r * uk.f; | 115 return r * uk.f; |
124 } | 116 } |
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