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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