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1 /* cbrtq.c
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2 *
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145
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3 * Cube root, long double precision
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4 *
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5 *
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6 *
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7 * SYNOPSIS:
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8 *
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9 * long double x, y, cbrtq();
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10 *
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11 * y = cbrtq( x );
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12 *
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13 *
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14 *
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15 * DESCRIPTION:
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16 *
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17 * Returns the cube root of the argument, which may be negative.
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18 *
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19 * Range reduction involves determining the power of 2 of
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20 * the argument. A polynomial of degree 2 applied to the
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21 * mantissa, and multiplication by the cube root of 1, 2, or 4
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22 * approximates the root to within about 0.1%. Then Newton's
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23 * iteration is used three times to converge to an accurate
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24 * result.
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25 *
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26 *
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27 *
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28 * ACCURACY:
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29 *
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30 * Relative error:
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31 * arithmetic domain # trials peak rms
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32 * IEEE -8,8 100000 1.3e-34 3.9e-35
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33 * IEEE exp(+-707) 100000 1.3e-34 4.3e-35
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34 *
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35 */
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36
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37 /*
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38 Cephes Math Library Release 2.2: January, 1991
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39 Copyright 1984, 1991 by Stephen L. Moshier
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40 Adapted for glibc October, 2001.
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41
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42 This library is free software; you can redistribute it and/or
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43 modify it under the terms of the GNU Lesser General Public
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44 License as published by the Free Software Foundation; either
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45 version 2.1 of the License, or (at your option) any later version.
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46
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47 This library is distributed in the hope that it will be useful,
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48 but WITHOUT ANY WARRANTY; without even the implied warranty of
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49 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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50 Lesser General Public License for more details.
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51
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52 You should have received a copy of the GNU Lesser General Public
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53 License along with this library; if not, see
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54 <http://www.gnu.org/licenses/>. */
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55
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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56 #include "quadmath-imp.h"
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57
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58 static const __float128 CBRT2 = 1.259921049894873164767210607278228350570251Q;
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59 static const __float128 CBRT4 = 1.587401051968199474751705639272308260391493Q;
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60 static const __float128 CBRT2I = 0.7937005259840997373758528196361541301957467Q;
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61 static const __float128 CBRT4I = 0.6299605249474365823836053036391141752851257Q;
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62
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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63
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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64 __float128
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65 cbrtq (__float128 x)
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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66 {
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67 int e, rem, sign;
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68 __float128 z;
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69
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70 if (!finiteq (x))
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71 return x + x;
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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72
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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73 if (x == 0)
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74 return (x);
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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75
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76 if (x > 0)
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77 sign = 1;
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78 else
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79 {
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80 sign = -1;
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81 x = -x;
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82 }
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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83
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84 z = x;
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85 /* extract power of 2, leaving mantissa between 0.5 and 1 */
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86 x = frexpq (x, &e);
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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87
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88 /* Approximate cube root of number between .5 and 1,
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89 peak relative error = 1.2e-6 */
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90 x = ((((1.3584464340920900529734e-1Q * x
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91 - 6.3986917220457538402318e-1Q) * x
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92 + 1.2875551670318751538055e0Q) * x
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93 - 1.4897083391357284957891e0Q) * x
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94 + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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95
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96 /* exponent divided by 3 */
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97 if (e >= 0)
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98 {
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99 rem = e;
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100 e /= 3;
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101 rem -= 3 * e;
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102 if (rem == 1)
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103 x *= CBRT2;
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104 else if (rem == 2)
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105 x *= CBRT4;
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106 }
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107 else
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108 { /* argument less than 1 */
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109 e = -e;
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110 rem = e;
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111 e /= 3;
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112 rem -= 3 * e;
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113 if (rem == 1)
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114 x *= CBRT2I;
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115 else if (rem == 2)
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116 x *= CBRT4I;
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117 e = -e;
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118 }
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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119
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120 /* multiply by power of 2 */
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121 x = ldexpq (x, e);
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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122
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123 /* Newton iteration */
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124 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
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125 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
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126 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
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127
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128 if (sign < 0)
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129 x = -x;
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130 return (x);
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68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
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131 }
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