1/* cbrtq.c 2 * 3 * Cube root, __float128 precision 4 * 5 * 6 * 7 * SYNOPSIS: 8 * 9 * __float128 x, y, cbrtq(); 10 * 11 * y = cbrtq( x ); 12 * 13 * 14 * 15 * DESCRIPTION: 16 * 17 * Returns the cube root of the argument, which may be negative. 18 * 19 * Range reduction involves determining the power of 2 of 20 * the argument. A polynomial of degree 2 applied to the 21 * mantissa, and multiplication by the cube root of 1, 2, or 4 22 * approximates the root to within about 0.1%. Then Newton's 23 * iteration is used three times to converge to an accurate 24 * result. 25 * 26 * 27 * 28 * ACCURACY: 29 * 30 * Relative error: 31 * arithmetic domain # trials peak rms 32 * IEEE -8,8 100000 1.3e-34 3.9e-35 33 * IEEE exp(+-707) 100000 1.3e-34 4.3e-35 34 * 35 */ 36 37/* 38Cephes Math Library Release 2.2: January, 1991 39Copyright 1984, 1991 by Stephen L. Moshier 40Adapted for glibc October, 2001. 41 42 This library is free software; you can redistribute it and/or 43 modify it under the terms of the GNU Lesser General Public 44 License as published by the Free Software Foundation; either 45 version 2.1 of the License, or (at your option) any later version. 46 47 This library is distributed in the hope that it will be useful, 48 but WITHOUT ANY WARRANTY; without even the implied warranty of 49 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 50 Lesser General Public License for more details. 51 52 You should have received a copy of the GNU Lesser General Public 53 License along with this library; if not, see 54 <http://www.gnu.org/licenses/>. */ 55 56 57#include "quadmath-imp.h" 58 59static const long double CBRT2 = 1.259921049894873164767210607278228350570251Q; 60static const long double CBRT4 = 1.587401051968199474751705639272308260391493Q; 61static const long double CBRT2I = 0.7937005259840997373758528196361541301957467Q; 62static const long double CBRT4I = 0.6299605249474365823836053036391141752851257Q; 63 64 65__float128 66cbrtq ( __float128 x) 67{ 68 int e, rem, sign; 69 __float128 z; 70 71 if (!finiteq (x)) 72 return x + x; 73 74 if (x == 0) 75 return (x); 76 77 if (x > 0) 78 sign = 1; 79 else 80 { 81 sign = -1; 82 x = -x; 83 } 84 85 z = x; 86 /* extract power of 2, leaving mantissa between 0.5 and 1 */ 87 x = frexpq (x, &e); 88 89 /* Approximate cube root of number between .5 and 1, 90 peak relative error = 1.2e-6 */ 91 x = ((((1.3584464340920900529734e-1Q * x 92 - 6.3986917220457538402318e-1Q) * x 93 + 1.2875551670318751538055e0Q) * x 94 - 1.4897083391357284957891e0Q) * x 95 + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q; 96 97 /* exponent divided by 3 */ 98 if (e >= 0) 99 { 100 rem = e; 101 e /= 3; 102 rem -= 3 * e; 103 if (rem == 1) 104 x *= CBRT2; 105 else if (rem == 2) 106 x *= CBRT4; 107 } 108 else 109 { /* argument less than 1 */ 110 e = -e; 111 rem = e; 112 e /= 3; 113 rem -= 3 * e; 114 if (rem == 1) 115 x *= CBRT2I; 116 else if (rem == 2) 117 x *= CBRT4I; 118 e = -e; 119 } 120 121 /* multiply by power of 2 */ 122 x = ldexpq (x, e); 123 124 /* Newton iteration */ 125 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 126 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 127 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 128 129 if (sign < 0) 130 x = -x; 131 return (x); 132} 133