1 /*- 2 * Copyright (c) 2007 Steven G. Kargl 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice unmodified, this list of conditions, and the following 10 * disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 25 */ 26 27 #include "cdefs-compat.h" 28 //__FBSDID("$FreeBSD: src/lib/msun/src/e_sqrtl.c,v 1.1 2008/03/02 01:47:58 das Exp $"); 29 30 #include <float.h> 31 #include <openlibm_fenv.h> 32 #include <openlibm_math.h> 33 34 #include "fpmath.h" 35 #include "math_private.h" 36 37 /* Return (x + ulp) for normal positive x. Assumes no overflow. */ 38 static inline long double 39 inc(long double x) 40 { 41 union IEEEl2bits u; 42 43 u.e = x; 44 if (++u.bits.manl == 0) { 45 if (++u.bits.manh == 0) { 46 u.bits.exp++; 47 u.bits.manh |= LDBL_NBIT; 48 } 49 } 50 return (u.e); 51 } 52 53 /* Return (x - ulp) for normal positive x. Assumes no underflow. */ 54 static inline long double 55 dec(long double x) 56 { 57 union IEEEl2bits u; 58 59 u.e = x; 60 if (u.bits.manl-- == 0) { 61 if (u.bits.manh-- == LDBL_NBIT) { 62 u.bits.exp--; 63 u.bits.manh |= LDBL_NBIT; 64 } 65 } 66 return (u.e); 67 } 68 69 #ifndef __GNUC__ 70 #pragma STDC FENV_ACCESS ON 71 #endif 72 73 /* 74 * This is slow, but simple and portable. You should use hardware sqrt 75 * if possible. 76 */ 77 78 DLLEXPORT long double 79 sqrtl(long double x) 80 { 81 union IEEEl2bits u; 82 int k, r; 83 long double lo, xn; 84 fenv_t env; 85 86 u.e = x; 87 88 /* If x = NaN, then sqrt(x) = NaN. */ 89 /* If x = Inf, then sqrt(x) = Inf. */ 90 /* If x = -Inf, then sqrt(x) = NaN. */ 91 if (u.bits.exp == LDBL_MAX_EXP * 2 - 1) 92 return (x * x + x); 93 94 /* If x = +-0, then sqrt(x) = +-0. */ 95 if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0) 96 return (x); 97 98 /* If x < 0, then raise invalid and return NaN */ 99 if (u.bits.sign) 100 return ((x - x) / (x - x)); 101 102 feholdexcept(&env); 103 104 if (u.bits.exp == 0) { 105 /* Adjust subnormal numbers. */ 106 u.e *= 0x1.0p514; 107 k = -514; 108 } else { 109 k = 0; 110 } 111 /* 112 * u.e is a normal number, so break it into u.e = e*2^n where 113 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n. 114 */ 115 if ((u.bits.exp - 0x3ffe) & 1) { /* n is odd. */ 116 k += u.bits.exp - 0x3fff; /* 2k = n - 1. */ 117 u.bits.exp = 0x3fff; /* u.e in [1,2). */ 118 } else { 119 k += u.bits.exp - 0x4000; /* 2k = n - 2. */ 120 u.bits.exp = 0x4000; /* u.e in [2,4). */ 121 } 122 123 /* 124 * Newton's iteration. 125 * Split u.e into a high and low part to achieve additional precision. 126 */ 127 xn = sqrt(u.e); /* 53-bit estimate of sqrtl(x). */ 128 #if LDBL_MANT_DIG > 100 129 xn = (xn + (u.e / xn)) * 0.5; /* 106-bit estimate. */ 130 #endif 131 lo = u.e; 132 u.bits.manl = 0; /* Zero out lower bits. */ 133 lo = (lo - u.e) / xn; /* Low bits divided by xn. */ 134 xn = xn + (u.e / xn); /* High portion of estimate. */ 135 u.e = xn + lo; /* Combine everything. */ 136 u.bits.exp += (k >> 1) - 1; 137 138 feclearexcept(FE_INEXACT); 139 r = fegetround(); 140 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */ 141 xn = x / u.e; /* Chopped quotient (inexact?). */ 142 143 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */ 144 if (xn == u.e) { 145 fesetenv(&env); 146 return (u.e); 147 } 148 /* Round correctly for inputs like x = y**2 - ulp. */ 149 xn = dec(xn); /* xn = xn - ulp. */ 150 } 151 152 if (r == FE_TONEAREST) { 153 xn = inc(xn); /* xn = xn + ulp. */ 154 } else if (r == FE_UPWARD) { 155 u.e = inc(u.e); /* u.e = u.e + ulp. */ 156 xn = inc(xn); /* xn = xn + ulp. */ 157 } 158 u.e = u.e + xn; /* Chopped sum. */ 159 feupdateenv(&env); /* Restore env and raise inexact */ 160 u.bits.exp--; 161 return (u.e); 162 } 163