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