1/* Compute x * y + z as ternary operation. 2 Copyright (C) 2010-2018 Free Software Foundation, Inc. 3 This file is part of the GNU C Library. 4 Contributed by Jakub Jelinek <jakub@redhat.com>, 2010. 5 6 The GNU C Library is free software; you can redistribute it and/or 7 modify it under the terms of the GNU Lesser General Public 8 License as published by the Free Software Foundation; either 9 version 2.1 of the License, or (at your option) any later version. 10 11 The GNU C Library is distributed in the hope that it will be useful, 12 but WITHOUT ANY WARRANTY; without even the implied warranty of 13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 14 Lesser General Public License for more details. 15 16 You should have received a copy of the GNU Lesser General Public 17 License along with the GNU C Library; if not, see 18 <http://www.gnu.org/licenses/>. */ 19 20#include "quadmath-imp.h" 21 22/* This implementation uses rounding to odd to avoid problems with 23 double rounding. See a paper by Boldo and Melquiond: 24 http://www.lri.fr/~melquion/doc/08-tc.pdf */ 25 26__float128 27fmaq (__float128 x, __float128 y, __float128 z) 28{ 29 ieee854_float128 u, v, w; 30 int adjust = 0; 31 u.value = x; 32 v.value = y; 33 w.value = z; 34 if (__builtin_expect (u.ieee.exponent + v.ieee.exponent 35 >= 0x7fff + IEEE854_FLOAT128_BIAS 36 - FLT128_MANT_DIG, 0) 37 || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) 38 || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) 39 || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) 40 || __builtin_expect (u.ieee.exponent + v.ieee.exponent 41 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG, 0)) 42 { 43 /* If z is Inf, but x and y are finite, the result should be 44 z rather than NaN. */ 45 if (w.ieee.exponent == 0x7fff 46 && u.ieee.exponent != 0x7fff 47 && v.ieee.exponent != 0x7fff) 48 return (z + x) + y; 49 /* If z is zero and x are y are nonzero, compute the result 50 as x * y to avoid the wrong sign of a zero result if x * y 51 underflows to 0. */ 52 if (z == 0 && x != 0 && y != 0) 53 return x * y; 54 /* If x or y or z is Inf/NaN, or if x * y is zero, compute as 55 x * y + z. */ 56 if (u.ieee.exponent == 0x7fff 57 || v.ieee.exponent == 0x7fff 58 || w.ieee.exponent == 0x7fff 59 || x == 0 60 || y == 0) 61 return x * y + z; 62 /* If fma will certainly overflow, compute as x * y. */ 63 if (u.ieee.exponent + v.ieee.exponent 64 > 0x7fff + IEEE854_FLOAT128_BIAS) 65 return x * y; 66 /* If x * y is less than 1/4 of FLT128_TRUE_MIN, neither the 67 result nor whether there is underflow depends on its exact 68 value, only on its sign. */ 69 if (u.ieee.exponent + v.ieee.exponent 70 < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2) 71 { 72 int neg = u.ieee.negative ^ v.ieee.negative; 73 __float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q; 74 if (w.ieee.exponent >= 3) 75 return tiny + z; 76 /* Scaling up, adding TINY and scaling down produces the 77 correct result, because in round-to-nearest mode adding 78 TINY has no effect and in other modes double rounding is 79 harmless. But it may not produce required underflow 80 exceptions. */ 81 v.value = z * 0x1p114Q + tiny; 82 if (TININESS_AFTER_ROUNDING 83 ? v.ieee.exponent < 115 84 : (w.ieee.exponent == 0 85 || (w.ieee.exponent == 1 86 && w.ieee.negative != neg 87 && w.ieee.mantissa3 == 0 88 && w.ieee.mantissa2 == 0 89 && w.ieee.mantissa1 == 0 90 && w.ieee.mantissa0 == 0))) 91 { 92 __float128 force_underflow = x * y; 93 math_force_eval (force_underflow); 94 } 95 return v.value * 0x1p-114Q; 96 } 97 if (u.ieee.exponent + v.ieee.exponent 98 >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG) 99 { 100 /* Compute 1p-113 times smaller result and multiply 101 at the end. */ 102 if (u.ieee.exponent > v.ieee.exponent) 103 u.ieee.exponent -= FLT128_MANT_DIG; 104 else 105 v.ieee.exponent -= FLT128_MANT_DIG; 106 /* If x + y exponent is very large and z exponent is very small, 107 it doesn't matter if we don't adjust it. */ 108 if (w.ieee.exponent > FLT128_MANT_DIG) 109 w.ieee.exponent -= FLT128_MANT_DIG; 110 adjust = 1; 111 } 112 else if (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) 113 { 114 /* Similarly. 115 If z exponent is very large and x and y exponents are 116 very small, adjust them up to avoid spurious underflows, 117 rather than down. */ 118 if (u.ieee.exponent + v.ieee.exponent 119 <= IEEE854_FLOAT128_BIAS + 2 * FLT128_MANT_DIG) 120 { 121 if (u.ieee.exponent > v.ieee.exponent) 122 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2; 123 else 124 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2; 125 } 126 else if (u.ieee.exponent > v.ieee.exponent) 127 { 128 if (u.ieee.exponent > FLT128_MANT_DIG) 129 u.ieee.exponent -= FLT128_MANT_DIG; 130 } 131 else if (v.ieee.exponent > FLT128_MANT_DIG) 132 v.ieee.exponent -= FLT128_MANT_DIG; 133 w.ieee.exponent -= FLT128_MANT_DIG; 134 adjust = 1; 135 } 136 else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) 137 { 138 u.ieee.exponent -= FLT128_MANT_DIG; 139 if (v.ieee.exponent) 140 v.ieee.exponent += FLT128_MANT_DIG; 141 else 142 v.value *= 0x1p113Q; 143 } 144 else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) 145 { 146 v.ieee.exponent -= FLT128_MANT_DIG; 147 if (u.ieee.exponent) 148 u.ieee.exponent += FLT128_MANT_DIG; 149 else 150 u.value *= 0x1p113Q; 151 } 152 else /* if (u.ieee.exponent + v.ieee.exponent 153 <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */ 154 { 155 if (u.ieee.exponent > v.ieee.exponent) 156 u.ieee.exponent += 2 * FLT128_MANT_DIG + 2; 157 else 158 v.ieee.exponent += 2 * FLT128_MANT_DIG + 2; 159 if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 6) 160 { 161 if (w.ieee.exponent) 162 w.ieee.exponent += 2 * FLT128_MANT_DIG + 2; 163 else 164 w.value *= 0x1p228Q; 165 adjust = -1; 166 } 167 /* Otherwise x * y should just affect inexact 168 and nothing else. */ 169 } 170 x = u.value; 171 y = v.value; 172 z = w.value; 173 } 174 175 /* Ensure correct sign of exact 0 + 0. */ 176 if (__glibc_unlikely ((x == 0 || y == 0) && z == 0)) 177 { 178 x = math_opt_barrier (x); 179 return x * y + z; 180 } 181 182 fenv_t env; 183 feholdexcept (&env); 184 fesetround (FE_TONEAREST); 185 186 /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ 187#define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1) 188 __float128 x1 = x * C; 189 __float128 y1 = y * C; 190 __float128 m1 = x * y; 191 x1 = (x - x1) + x1; 192 y1 = (y - y1) + y1; 193 __float128 x2 = x - x1; 194 __float128 y2 = y - y1; 195 __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; 196 197 /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ 198 __float128 a1 = z + m1; 199 __float128 t1 = a1 - z; 200 __float128 t2 = a1 - t1; 201 t1 = m1 - t1; 202 t2 = z - t2; 203 __float128 a2 = t1 + t2; 204 /* Ensure the arithmetic is not scheduled after feclearexcept call. */ 205 math_force_eval (m2); 206 math_force_eval (a2); 207 feclearexcept (FE_INEXACT); 208 209 /* If the result is an exact zero, ensure it has the correct sign. */ 210 if (a1 == 0 && m2 == 0) 211 { 212 feupdateenv (&env); 213 /* Ensure that round-to-nearest value of z + m1 is not reused. */ 214 z = math_opt_barrier (z); 215 return z + m1; 216 } 217 218 fesetround (FE_TOWARDZERO); 219 /* Perform m2 + a2 addition with round to odd. */ 220 u.value = a2 + m2; 221 222 if (__glibc_likely (adjust == 0)) 223 { 224 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) 225 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; 226 feupdateenv (&env); 227 /* Result is a1 + u.value. */ 228 return a1 + u.value; 229 } 230 else if (__glibc_likely (adjust > 0)) 231 { 232 if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) 233 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; 234 feupdateenv (&env); 235 /* Result is a1 + u.value, scaled up. */ 236 return (a1 + u.value) * 0x1p113Q; 237 } 238 else 239 { 240 if ((u.ieee.mantissa3 & 1) == 0) 241 u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; 242 v.value = a1 + u.value; 243 /* Ensure the addition is not scheduled after fetestexcept call. */ 244 math_force_eval (v.value); 245 int j = fetestexcept (FE_INEXACT) != 0; 246 feupdateenv (&env); 247 /* Ensure the following computations are performed in default rounding 248 mode instead of just reusing the round to zero computation. */ 249 asm volatile ("" : "=m" (u) : "m" (u)); 250 /* If a1 + u.value is exact, the only rounding happens during 251 scaling down. */ 252 if (j == 0) 253 return v.value * 0x1p-228Q; 254 /* If result rounded to zero is not subnormal, no double 255 rounding will occur. */ 256 if (v.ieee.exponent > 228) 257 return (a1 + u.value) * 0x1p-228Q; 258 /* If v.value * 0x1p-228L with round to zero is a subnormal above 259 or equal to FLT128_MIN / 2, then v.value * 0x1p-228L shifts mantissa 260 down just by 1 bit, which means v.ieee.mantissa3 |= j would 261 change the round bit, not sticky or guard bit. 262 v.value * 0x1p-228L never normalizes by shifting up, 263 so round bit plus sticky bit should be already enough 264 for proper rounding. */ 265 if (v.ieee.exponent == 228) 266 { 267 /* If the exponent would be in the normal range when 268 rounding to normal precision with unbounded exponent 269 range, the exact result is known and spurious underflows 270 must be avoided on systems detecting tininess after 271 rounding. */ 272 if (TININESS_AFTER_ROUNDING) 273 { 274 w.value = a1 + u.value; 275 if (w.ieee.exponent == 229) 276 return w.value * 0x1p-228Q; 277 } 278 /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding, 279 v.ieee.mantissa3 & 1 is the round bit and j is our sticky 280 bit. */ 281 w.value = 0; 282 w.ieee.mantissa3 = ((v.ieee.mantissa3 & 3) << 1) | j; 283 w.ieee.negative = v.ieee.negative; 284 v.ieee.mantissa3 &= ~3U; 285 v.value *= 0x1p-228Q; 286 w.value *= 0x1p-2Q; 287 return v.value + w.value; 288 } 289 v.ieee.mantissa3 |= j; 290 return v.value * 0x1p-228Q; 291 } 292} 293