[388] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */
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| 2 | /*-
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| 3 | * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
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| 4 | * All rights reserved.
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| 5 | *
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| 6 | * Redistribution and use in source and binary forms, with or without
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| 7 | * modification, are permitted provided that the following conditions
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| 8 | * are met:
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| 9 | * 1. Redistributions of source code must retain the above copyright
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| 10 | * notice, this list of conditions and the following disclaimer.
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| 11 | * 2. Redistributions in binary form must reproduce the above copyright
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| 12 | * notice, this list of conditions and the following disclaimer in the
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| 13 | * documentation and/or other materials provided with the distribution.
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| 14 | *
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| 15 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
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| 16 | * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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| 17 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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| 18 | * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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| 19 | * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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| 20 | * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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| 21 | * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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| 22 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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| 23 | * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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| 24 | * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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| 25 | * SUCH DAMAGE.
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| 26 | */
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| 27 |
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| 28 | #include "libm.h"
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| 29 |
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| 30 | #define TBLSIZE 16
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| 31 |
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| 32 | static const float
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| 33 | redux = 0x1.8p23f / TBLSIZE,
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| 34 | P1 = 0x1.62e430p-1f,
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| 35 | P2 = 0x1.ebfbe0p-3f,
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| 36 | P3 = 0x1.c6b348p-5f,
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| 37 | P4 = 0x1.3b2c9cp-7f;
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| 38 |
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| 39 | static const double exp2ft[TBLSIZE] = {
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| 40 | 0x1.6a09e667f3bcdp-1,
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| 41 | 0x1.7a11473eb0187p-1,
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| 42 | 0x1.8ace5422aa0dbp-1,
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| 43 | 0x1.9c49182a3f090p-1,
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| 44 | 0x1.ae89f995ad3adp-1,
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| 45 | 0x1.c199bdd85529cp-1,
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| 46 | 0x1.d5818dcfba487p-1,
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| 47 | 0x1.ea4afa2a490dap-1,
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| 48 | 0x1.0000000000000p+0,
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| 49 | 0x1.0b5586cf9890fp+0,
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| 50 | 0x1.172b83c7d517bp+0,
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| 51 | 0x1.2387a6e756238p+0,
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| 52 | 0x1.306fe0a31b715p+0,
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| 53 | 0x1.3dea64c123422p+0,
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| 54 | 0x1.4bfdad5362a27p+0,
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| 55 | 0x1.5ab07dd485429p+0,
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| 56 | };
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| 57 |
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| 58 | /*
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| 59 | * exp2f(x): compute the base 2 exponential of x
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| 60 | *
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| 61 | * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
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| 62 | *
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| 63 | * Method: (equally-spaced tables)
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| 64 | *
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| 65 | * Reduce x:
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| 66 | * x = k + y, for integer k and |y| <= 1/2.
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| 67 | * Thus we have exp2f(x) = 2**k * exp2(y).
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| 68 | *
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| 69 | * Reduce y:
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| 70 | * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
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| 71 | * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
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| 72 | * with |z| <= 2**-(TBLSIZE+1).
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| 73 | *
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| 74 | * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
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| 75 | * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
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| 76 | * Using double precision for everything except the reduction makes
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| 77 | * roundoff error insignificant and simplifies the scaling step.
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| 78 | *
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| 79 | * This method is due to Tang, but I do not use his suggested parameters:
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| 80 | *
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| 81 | * Tang, P. Table-driven Implementation of the Exponential Function
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| 82 | * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
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| 83 | */
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| 84 | float exp2f(float x)
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| 85 | {
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| 86 | double_t t, r, z;
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| 87 | union {float f; uint32_t i;} u = {x};
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| 88 | union {double f; uint64_t i;} uk;
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| 89 | uint32_t ix, i0, k;
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| 90 |
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| 91 | /* Filter out exceptional cases. */
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| 92 | ix = u.i & 0x7fffffff;
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| 93 | if (ix > 0x42fc0000) { /* |x| > 126 */
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| 94 | if (ix > 0x7f800000) /* NaN */
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| 95 | return x;
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| 96 | if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */
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| 97 | x *= 0x1p127f;
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| 98 | return x;
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| 99 | }
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| 100 | if (u.i >= 0x80000000) { /* x < -126 */
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| 101 | if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
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| 102 | FORCE_EVAL(-0x1p-149f/x);
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| 103 | if (u.i >= 0xc3160000) /* x <= -150 */
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| 104 | return 0;
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| 105 | }
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| 106 | } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
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| 107 | return 1.0f + x;
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| 108 | }
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| 109 |
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| 110 | /* Reduce x, computing z, i0, and k. */
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| 111 | u.f = x + redux;
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| 112 | i0 = u.i;
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| 113 | i0 += TBLSIZE / 2;
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| 114 | k = i0 / TBLSIZE;
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| 115 | uk.i = (uint64_t)(0x3ff + k)<<52;
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| 116 | i0 &= TBLSIZE - 1;
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| 117 | u.f -= redux;
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| 118 | z = x - u.f;
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| 119 | /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
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| 120 | r = exp2ft[i0];
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| 121 | t = r * z;
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| 122 | r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
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| 123 |
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| 124 | /* Scale by 2**k */
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| 125 | return r * uk.f;
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| 126 | }
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