1/* Single-precision floating point square root. 2 Copyright (C) 1997, 2002 Free Software Foundation, Inc. 3 This file is part of the GNU C Library. 4 5 The GNU C Library is free software; you can redistribute it and/or 6 modify it under the terms of the GNU Lesser General Public 7 License as published by the Free Software Foundation; either 8 version 2.1 of the License, or (at your option) any later version. 9 10 The GNU C Library is distributed in the hope that it will be useful, 11 but WITHOUT ANY WARRANTY; without even the implied warranty of 12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 13 Lesser General Public License for more details. 14 15 You should have received a copy of the GNU Lesser General Public 16 License along with the GNU C Library; if not, write to the Free 17 Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 18 02111-1307 USA. */ 19 20#include <math.h> 21#include <math_private.h> 22#include <fenv_libc.h> 23#include <inttypes.h> 24 25static const double almost_half = 0.5000000000000001; /* 0.5 + 2^-53 */ 26static const uint32_t a_nan = 0x7fc00000; 27static const uint32_t a_inf = 0x7f800000; 28static const float two108 = 3.245185536584267269e+32; 29static const float twom54 = 5.551115123125782702e-17; 30extern const float __t_sqrt[1024]; 31 32/* The method is based on a description in 33 Computation of elementary functions on the IBM RISC System/6000 processor, 34 P. W. Markstein, IBM J. Res. Develop, 34(1) 1990. 35 Basically, it consists of two interleaved Newton-Rhapson approximations, 36 one to find the actual square root, and one to find its reciprocal 37 without the expense of a division operation. The tricky bit here 38 is the use of the POWER/PowerPC multiply-add operation to get the 39 required accuracy with high speed. 40 41 The argument reduction works by a combination of table lookup to 42 obtain the initial guesses, and some careful modification of the 43 generated guesses (which mostly runs on the integer unit, while the 44 Newton-Rhapson is running on the FPU). */ 45double 46__sqrt(double x) 47{ 48 const float inf = *(const float *)&a_inf; 49 /* x = f_wash(x); *//* This ensures only one exception for SNaN. */ 50 if (x > 0) 51 { 52 if (x != inf) 53 { 54 /* Variables named starting with 's' exist in the 55 argument-reduced space, so that 2 > sx >= 0.5, 56 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... . 57 Variables named ending with 'i' are integer versions of 58 floating-point values. */ 59 double sx; /* The value of which we're trying to find the 60 square root. */ 61 double sg,g; /* Guess of the square root of x. */ 62 double sd,d; /* Difference between the square of the guess and x. */ 63 double sy; /* Estimate of 1/2g (overestimated by 1ulp). */ 64 double sy2; /* 2*sy */ 65 double e; /* Difference between y*g and 1/2 (se = e * fsy). */ 66 double shx; /* == sx * fsg */ 67 double fsg; /* sg*fsg == g. */ 68 fenv_t fe; /* Saved floating-point environment (stores rounding 69 mode and whether the inexact exception is 70 enabled). */ 71 uint32_t xi0, xi1, sxi, fsgi; 72 const float *t_sqrt; 73 74 fe = fegetenv_register(); 75 EXTRACT_WORDS (xi0,xi1,x); 76 relax_fenv_state(); 77 sxi = (xi0 & 0x3fffffff) | 0x3fe00000; 78 INSERT_WORDS (sx, sxi, xi1); 79 t_sqrt = __t_sqrt + (xi0 >> (52-32-8-1) & 0x3fe); 80 sg = t_sqrt[0]; 81 sy = t_sqrt[1]; 82 83 /* Here we have three Newton-Rhapson iterations each of a 84 division and a square root and the remainder of the 85 argument reduction, all interleaved. */ 86 sd = -(sg*sg - sx); 87 fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000; 88 sy2 = sy + sy; 89 sg = sy*sd + sg; /* 16-bit approximation to sqrt(sx). */ 90 INSERT_WORDS (fsg, fsgi, 0); 91 e = -(sy*sg - almost_half); 92 sd = -(sg*sg - sx); 93 if ((xi0 & 0x7ff00000) == 0) 94 goto denorm; 95 sy = sy + e*sy2; 96 sg = sg + sy*sd; /* 32-bit approximation to sqrt(sx). */ 97 sy2 = sy + sy; 98 e = -(sy*sg - almost_half); 99 sd = -(sg*sg - sx); 100 sy = sy + e*sy2; 101 shx = sx * fsg; 102 sg = sg + sy*sd; /* 64-bit approximation to sqrt(sx), 103 but perhaps rounded incorrectly. */ 104 sy2 = sy + sy; 105 g = sg * fsg; 106 e = -(sy*sg - almost_half); 107 d = -(g*sg - shx); 108 sy = sy + e*sy2; 109 fesetenv_register (fe); 110 return g + sy*d; 111 denorm: 112 /* For denormalised numbers, we normalise, calculate the 113 square root, and return an adjusted result. */ 114 fesetenv_register (fe); 115 return __sqrt(x * two108) * twom54; 116 } 117 } 118 else if (x < 0) 119 { 120#ifdef FE_INVALID_SQRT 121 feraiseexcept (FE_INVALID_SQRT); 122 /* For some reason, some PowerPC processors don't implement 123 FE_INVALID_SQRT. I guess no-one ever thought they'd be 124 used for square roots... :-) */ 125 if (!fetestexcept (FE_INVALID)) 126#endif 127 feraiseexcept (FE_INVALID); 128#ifndef _IEEE_LIBM 129 if (_LIB_VERSION != _IEEE_) 130 x = __kernel_standard(x,x,26); 131 else 132#endif 133 x = *(const float*)&a_nan; 134 } 135 return f_wash(x); 136} 137 138weak_alias (__sqrt, sqrt) 139/* Strictly, this is wrong, but the only places where _ieee754_sqrt is 140 used will not pass in a negative result. */ 141strong_alias(__sqrt,__ieee754_sqrt) 142 143#ifdef NO_LONG_DOUBLE 144weak_alias (__sqrt, __sqrtl) 145weak_alias (__sqrt, sqrtl) 146#endif 147