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 <sys/cdefs.h> 28 __FBSDID("$FreeBSD$"); 29 30 #include <fenv.h> 31 #include <float.h> 32 33 #include "fpmath.h" 34 #include "math.h" 35 36 /* Return (x + ulp) for normal positive x. Assumes no overflow. */ 37 static inline long double 38 inc(long double x) 39 { 40 union IEEEl2bits u; 41 42 u.e = x; 43 if (++u.bits.manl == 0) { 44 if (++u.bits.manh == 0) { 45 u.bits.exp++; 46 u.bits.manh |= LDBL_NBIT; 47 } 48 } 49 return (u.e); 50 } 51 52 /* Return (x - ulp) for normal positive x. Assumes no underflow. */ 53 static inline long double 54 dec(long double x) 55 { 56 union IEEEl2bits u; 57 58 u.e = x; 59 if (u.bits.manl-- == 0) { 60 if (u.bits.manh-- == LDBL_NBIT) { 61 u.bits.exp--; 62 u.bits.manh |= LDBL_NBIT; 63 } 64 } 65 return (u.e); 66 } 67 68 #pragma STDC FENV_ACCESS ON 69 70 /* 71 * This is slow, but simple and portable. You should use hardware sqrt 72 * if possible. 73 */ 74 75 long double 76 sqrtl(long double x) 77 { 78 union IEEEl2bits u; 79 int k, r; 80 long double lo, xn; 81 fenv_t env; 82 83 u.e = x; 84 85 /* If x = NaN, then sqrt(x) = NaN. */ 86 /* If x = Inf, then sqrt(x) = Inf. */ 87 /* If x = -Inf, then sqrt(x) = NaN. */ 88 if (u.bits.exp == LDBL_MAX_EXP * 2 - 1) 89 return (x * x + x); 90 91 /* If x = +-0, then sqrt(x) = +-0. */ 92 if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0) 93 return (x); 94 95 /* If x < 0, then raise invalid and return NaN */ 96 if (u.bits.sign) 97 return ((x - x) / (x - x)); 98 99 feholdexcept(&env); 100 101 if (u.bits.exp == 0) { 102 /* Adjust subnormal numbers. */ 103 u.e *= 0x1.0p514; 104 k = -514; 105 } else { 106 k = 0; 107 } 108 /* 109 * u.e is a normal number, so break it into u.e = e*2^n where 110 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n. 111 */ 112 if ((u.bits.exp - 0x3ffe) & 1) { /* n is odd. */ 113 k += u.bits.exp - 0x3fff; /* 2k = n - 1. */ 114 u.bits.exp = 0x3fff; /* u.e in [1,2). */ 115 } else { 116 k += u.bits.exp - 0x4000; /* 2k = n - 2. */ 117 u.bits.exp = 0x4000; /* u.e in [2,4). */ 118 } 119 120 /* 121 * Newton's iteration. 122 * Split u.e into a high and low part to achieve additional precision. 123 */ 124 xn = sqrt(u.e); /* 53-bit estimate of sqrtl(x). */ 125 #if LDBL_MANT_DIG > 100 126 xn = (xn + (u.e / xn)) * 0.5; /* 106-bit estimate. */ 127 #endif 128 lo = u.e; 129 u.bits.manl = 0; /* Zero out lower bits. */ 130 lo = (lo - u.e) / xn; /* Low bits divided by xn. */ 131 xn = xn + (u.e / xn); /* High portion of estimate. */ 132 u.e = xn + lo; /* Combine everything. */ 133 u.bits.exp += (k >> 1) - 1; 134 135 feclearexcept(FE_INEXACT); 136 r = fegetround(); 137 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */ 138 xn = x / u.e; /* Chopped quotient (inexact?). */ 139 140 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */ 141 if (xn == u.e) { 142 fesetenv(&env); 143 return (u.e); 144 } 145 /* Round correctly for inputs like x = y**2 - ulp. */ 146 xn = dec(xn); /* xn = xn - ulp. */ 147 } 148 149 if (r == FE_TONEAREST) { 150 xn = inc(xn); /* xn = xn + ulp. */ 151 } else if (r == FE_UPWARD) { 152 u.e = inc(u.e); /* u.e = u.e + ulp. */ 153 xn = inc(xn); /* xn = xn + ulp. */ 154 } 155 u.e = u.e + xn; /* Chopped sum. */ 156 feupdateenv(&env); /* Restore env and raise inexact */ 157 u.bits.exp--; 158 return (u.e); 159 } 160