[352] | 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log2l.c */
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| 2 | /*
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| 3 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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| 4 | *
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| 5 | * Permission to use, copy, modify, and distribute this software for any
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| 6 | * purpose with or without fee is hereby granted, provided that the above
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| 7 | * copyright notice and this permission notice appear in all copies.
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| 8 | *
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| 9 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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| 10 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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| 11 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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| 12 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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| 13 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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| 14 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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| 15 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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| 16 | */
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| 17 | /*
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| 18 | * Base 2 logarithm, long double precision
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| 19 | *
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| 20 | *
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| 21 | * SYNOPSIS:
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| 22 | *
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| 23 | * long double x, y, log2l();
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| 24 | *
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| 25 | * y = log2l( x );
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| 26 | *
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| 27 | *
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| 28 | * DESCRIPTION:
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| 29 | *
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| 30 | * Returns the base 2 logarithm of x.
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| 31 | *
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| 32 | * The argument is separated into its exponent and fractional
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| 33 | * parts. If the exponent is between -1 and +1, the (natural)
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| 34 | * logarithm of the fraction is approximated by
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| 35 | *
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| 36 | * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
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| 37 | *
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| 38 | * Otherwise, setting z = 2(x-1)/x+1),
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| 39 | *
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| 40 | * log(x) = z + z**3 P(z)/Q(z).
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| 41 | *
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| 42 | *
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| 43 | * ACCURACY:
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| 44 | *
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| 45 | * Relative error:
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| 46 | * arithmetic domain # trials peak rms
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| 47 | * IEEE 0.5, 2.0 30000 9.8e-20 2.7e-20
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| 48 | * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20
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| 49 | *
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| 50 | * In the tests over the interval exp(+-10000), the logarithms
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| 51 | * of the random arguments were uniformly distributed over
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| 52 | * [-10000, +10000].
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| 53 | */
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| 54 |
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| 55 | #include "libm.h"
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| 56 |
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| 57 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 58 | long double log2l(long double x)
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| 59 | {
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| 60 | return log2(x);
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| 61 | }
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| 62 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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| 63 | /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
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| 64 | * 1/sqrt(2) <= x < sqrt(2)
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| 65 | * Theoretical peak relative error = 6.2e-22
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| 66 | */
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| 67 | static const long double P[] = {
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| 68 | 4.9962495940332550844739E-1L,
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| 69 | 1.0767376367209449010438E1L,
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| 70 | 7.7671073698359539859595E1L,
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| 71 | 2.5620629828144409632571E2L,
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| 72 | 4.2401812743503691187826E2L,
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| 73 | 3.4258224542413922935104E2L,
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| 74 | 1.0747524399916215149070E2L,
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| 75 | };
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| 76 | static const long double Q[] = {
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| 77 | /* 1.0000000000000000000000E0,*/
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| 78 | 2.3479774160285863271658E1L,
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| 79 | 1.9444210022760132894510E2L,
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| 80 | 7.7952888181207260646090E2L,
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| 81 | 1.6911722418503949084863E3L,
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| 82 | 2.0307734695595183428202E3L,
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| 83 | 1.2695660352705325274404E3L,
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| 84 | 3.2242573199748645407652E2L,
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| 85 | };
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| 86 |
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| 87 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
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| 88 | * where z = 2(x-1)/(x+1)
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| 89 | * 1/sqrt(2) <= x < sqrt(2)
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| 90 | * Theoretical peak relative error = 6.16e-22
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| 91 | */
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| 92 | static const long double R[4] = {
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| 93 | 1.9757429581415468984296E-3L,
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| 94 | -7.1990767473014147232598E-1L,
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| 95 | 1.0777257190312272158094E1L,
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| 96 | -3.5717684488096787370998E1L,
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| 97 | };
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| 98 | static const long double S[4] = {
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| 99 | /* 1.00000000000000000000E0L,*/
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| 100 | -2.6201045551331104417768E1L,
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| 101 | 1.9361891836232102174846E2L,
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| 102 | -4.2861221385716144629696E2L,
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| 103 | };
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| 104 | /* log2(e) - 1 */
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| 105 | #define LOG2EA 4.4269504088896340735992e-1L
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| 106 |
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| 107 | #define SQRTH 0.70710678118654752440L
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| 108 |
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| 109 | long double log2l(long double x)
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| 110 | {
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| 111 | long double y, z;
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| 112 | int e;
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| 113 |
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| 114 | if (isnan(x))
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| 115 | return x;
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| 116 | if (x == INFINITY)
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| 117 | return x;
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| 118 | if (x <= 0.0) {
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| 119 | if (x == 0.0)
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| 120 | return -1/(x*x); /* -inf with divbyzero */
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| 121 | return 0/0.0f; /* nan with invalid */
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| 122 | }
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| 123 |
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| 124 | /* separate mantissa from exponent */
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| 125 | /* Note, frexp is used so that denormal numbers
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| 126 | * will be handled properly.
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| 127 | */
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| 128 | x = frexpl(x, &e);
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| 129 |
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| 130 | /* logarithm using log(x) = z + z**3 P(z)/Q(z),
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| 131 | * where z = 2(x-1)/x+1)
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| 132 | */
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| 133 | if (e > 2 || e < -2) {
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| 134 | if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
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| 135 | e -= 1;
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| 136 | z = x - 0.5;
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| 137 | y = 0.5 * z + 0.5;
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| 138 | } else { /* 2 (x-1)/(x+1) */
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| 139 | z = x - 0.5;
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| 140 | z -= 0.5;
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| 141 | y = 0.5 * x + 0.5;
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| 142 | }
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| 143 | x = z / y;
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| 144 | z = x*x;
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| 145 | y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
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| 146 | goto done;
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| 147 | }
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| 148 |
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| 149 | /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
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| 150 | if (x < SQRTH) {
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| 151 | e -= 1;
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| 152 | x = 2.0*x - 1.0;
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| 153 | } else {
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| 154 | x = x - 1.0;
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| 155 | }
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| 156 | z = x*x;
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| 157 | y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
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| 158 | y = y - 0.5*z;
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| 159 |
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| 160 | done:
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| 161 | /* Multiply log of fraction by log2(e)
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| 162 | * and base 2 exponent by 1
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| 163 | *
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| 164 | * ***CAUTION***
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| 165 | *
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| 166 | * This sequence of operations is critical and it may
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| 167 | * be horribly defeated by some compiler optimizers.
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| 168 | */
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| 169 | z = y * LOG2EA;
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| 170 | z += x * LOG2EA;
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| 171 | z += y;
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| 172 | z += x;
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| 173 | z += e;
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| 174 | return z;
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| 175 | }
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| 176 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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| 177 | // TODO: broken implementation to make things compile
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| 178 | long double log2l(long double x)
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| 179 | {
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| 180 | return log2(x);
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| 181 | }
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| 182 | #endif
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