1/* cbrtq.c 2 * 3 * Cube root, long double precision 4 * 5 * 6 * 7 * SYNOPSIS: 8 * 9 * long double 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#include "quadmath-imp.h" 57 58static const __float128 CBRT2 = 1.259921049894873164767210607278228350570251Q; 59static const __float128 CBRT4 = 1.587401051968199474751705639272308260391493Q; 60static const __float128 CBRT2I = 0.7937005259840997373758528196361541301957467Q; 61static const __float128 CBRT4I = 0.6299605249474365823836053036391141752851257Q; 62 63 64__float128 65cbrtq (__float128 x) 66{ 67 int e, rem, sign; 68 __float128 z; 69 70 if (!finiteq (x)) 71 return x + x; 72 73 if (x == 0) 74 return (x); 75 76 if (x > 0) 77 sign = 1; 78 else 79 { 80 sign = -1; 81 x = -x; 82 } 83 84 z = x; 85 /* extract power of 2, leaving mantissa between 0.5 and 1 */ 86 x = frexpq (x, &e); 87 88 /* Approximate cube root of number between .5 and 1, 89 peak relative error = 1.2e-6 */ 90 x = ((((1.3584464340920900529734e-1Q * x 91 - 6.3986917220457538402318e-1Q) * x 92 + 1.2875551670318751538055e0Q) * x 93 - 1.4897083391357284957891e0Q) * x 94 + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q; 95 96 /* exponent divided by 3 */ 97 if (e >= 0) 98 { 99 rem = e; 100 e /= 3; 101 rem -= 3 * e; 102 if (rem == 1) 103 x *= CBRT2; 104 else if (rem == 2) 105 x *= CBRT4; 106 } 107 else 108 { /* argument less than 1 */ 109 e = -e; 110 rem = e; 111 e /= 3; 112 rem -= 3 * e; 113 if (rem == 1) 114 x *= CBRT2I; 115 else if (rem == 2) 116 x *= CBRT4I; 117 e = -e; 118 } 119 120 /* multiply by power of 2 */ 121 x = ldexpq (x, e); 122 123 /* Newton iteration */ 124 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 125 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 126 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q; 127 128 if (sign < 0) 129 x = -x; 130 return (x); 131} 132