[352] | 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/s_log1pl.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 | * Relative error logarithm
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| 19 | * Natural logarithm of 1+x, long double precision
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| 20 | *
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| 21 | *
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| 22 | * SYNOPSIS:
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| 23 | *
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| 24 | * long double x, y, log1pl();
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| 25 | *
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| 26 | * y = log1pl( x );
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| 27 | *
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| 28 | *
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| 29 | * DESCRIPTION:
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| 30 | *
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| 31 | * Returns the base e (2.718...) logarithm of 1+x.
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| 32 | *
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| 33 | * The argument 1+x is separated into its exponent and fractional
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| 34 | * parts. If the exponent is between -1 and +1, the logarithm
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| 35 | * of the fraction is approximated by
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| 36 | *
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| 37 | * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
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| 38 | *
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| 39 | * Otherwise, setting z = 2(x-1)/x+1),
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| 40 | *
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| 41 | * log(x) = z + z^3 P(z)/Q(z).
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| 42 | *
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| 43 | *
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| 44 | * ACCURACY:
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| 45 | *
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| 46 | * Relative error:
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| 47 | * arithmetic domain # trials peak rms
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| 48 | * IEEE -1.0, 9.0 100000 8.2e-20 2.5e-20
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| 49 | */
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| 50 |
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| 51 | #include "libm.h"
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| 52 |
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| 53 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 54 | long double log1pl(long double x)
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| 55 | {
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| 56 | return log1p(x);
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| 57 | }
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| 58 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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| 59 | /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
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| 60 | * 1/sqrt(2) <= x < sqrt(2)
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| 61 | * Theoretical peak relative error = 2.32e-20
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| 62 | */
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| 63 | static const long double P[] = {
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| 64 | 4.5270000862445199635215E-5L,
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| 65 | 4.9854102823193375972212E-1L,
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| 66 | 6.5787325942061044846969E0L,
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| 67 | 2.9911919328553073277375E1L,
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| 68 | 6.0949667980987787057556E1L,
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| 69 | 5.7112963590585538103336E1L,
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| 70 | 2.0039553499201281259648E1L,
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| 71 | };
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| 72 | static const long double Q[] = {
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| 73 | /* 1.0000000000000000000000E0,*/
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| 74 | 1.5062909083469192043167E1L,
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| 75 | 8.3047565967967209469434E1L,
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| 76 | 2.2176239823732856465394E2L,
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| 77 | 3.0909872225312059774938E2L,
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| 78 | 2.1642788614495947685003E2L,
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| 79 | 6.0118660497603843919306E1L,
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| 80 | };
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| 81 |
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| 82 | /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
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| 83 | * where z = 2(x-1)/(x+1)
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| 84 | * 1/sqrt(2) <= x < sqrt(2)
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| 85 | * Theoretical peak relative error = 6.16e-22
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| 86 | */
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| 87 | static const long double R[4] = {
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| 88 | 1.9757429581415468984296E-3L,
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| 89 | -7.1990767473014147232598E-1L,
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| 90 | 1.0777257190312272158094E1L,
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| 91 | -3.5717684488096787370998E1L,
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| 92 | };
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| 93 | static const long double S[4] = {
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| 94 | /* 1.00000000000000000000E0L,*/
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| 95 | -2.6201045551331104417768E1L,
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| 96 | 1.9361891836232102174846E2L,
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| 97 | -4.2861221385716144629696E2L,
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| 98 | };
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| 99 | static const long double C1 = 6.9314575195312500000000E-1L;
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| 100 | static const long double C2 = 1.4286068203094172321215E-6L;
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| 101 |
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| 102 | #define SQRTH 0.70710678118654752440L
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| 103 |
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| 104 | long double log1pl(long double xm1)
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| 105 | {
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| 106 | long double x, y, z;
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| 107 | int e;
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| 108 |
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| 109 | if (isnan(xm1))
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| 110 | return xm1;
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| 111 | if (xm1 == INFINITY)
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| 112 | return xm1;
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| 113 | if (xm1 == 0.0)
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| 114 | return xm1;
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| 115 |
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| 116 | x = xm1 + 1.0;
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| 117 |
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| 118 | /* Test for domain errors. */
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| 119 | if (x <= 0.0) {
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| 120 | if (x == 0.0)
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| 121 | return -1/(x*x); /* -inf with divbyzero */
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| 122 | return 0/0.0f; /* nan with invalid */
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| 123 | }
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| 124 |
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| 125 | /* Separate mantissa from exponent.
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| 126 | Use frexp so that denormal numbers will be handled properly. */
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| 127 | x = frexpl(x, &e);
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| 128 |
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| 129 | /* logarithm using log(x) = z + z^3 P(z)/Q(z),
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| 130 | where z = 2(x-1)/x+1) */
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| 131 | if (e > 2 || e < -2) {
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| 132 | if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
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| 133 | e -= 1;
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| 134 | z = x - 0.5;
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| 135 | y = 0.5 * z + 0.5;
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| 136 | } else { /* 2 (x-1)/(x+1) */
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| 137 | z = x - 0.5;
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| 138 | z -= 0.5;
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| 139 | y = 0.5 * x + 0.5;
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| 140 | }
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| 141 | x = z / y;
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| 142 | z = x*x;
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| 143 | z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
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| 144 | z = z + e * C2;
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| 145 | z = z + x;
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| 146 | z = z + e * C1;
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| 147 | return z;
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| 148 | }
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| 149 |
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| 150 | /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
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| 151 | if (x < SQRTH) {
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| 152 | e -= 1;
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| 153 | if (e != 0)
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| 154 | x = 2.0 * x - 1.0;
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| 155 | else
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| 156 | x = xm1;
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| 157 | } else {
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| 158 | if (e != 0)
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| 159 | x = x - 1.0;
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| 160 | else
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| 161 | x = xm1;
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| 162 | }
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| 163 | z = x*x;
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| 164 | y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
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| 165 | y = y + e * C2;
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| 166 | z = y - 0.5 * z;
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| 167 | z = z + x;
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| 168 | z = z + e * C1;
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| 169 | return z;
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| 170 | }
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| 171 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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| 172 | // TODO: broken implementation to make things compile
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| 173 | long double log1pl(long double x)
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| 174 | {
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| 175 | return log1p(x);
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| 176 | }
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| 177 | #endif
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