/* Compute x * y + z as ternary operation. Copyright (C) 2010-2012 Free Software Foundation, Inc. This file is part of the GNU C Library. Contributed by Jakub Jelinek , 2010. The GNU C Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. The GNU C Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU C Library; if not, see . */ #include "quadmath-imp.h" #include #include #ifdef HAVE_FENV_H # include # if defined HAVE_FEHOLDEXCEPT && defined HAVE_FESETROUND \ && defined HAVE_FEUPDATEENV && defined HAVE_FETESTEXCEPT \ && defined FE_TOWARDZERO && defined FE_INEXACT # define USE_FENV_H # endif #endif /* This implementation uses rounding to odd to avoid problems with double rounding. See a paper by Boldo and Melquiond: http://www.lri.fr/~melquion/doc/08-tc.pdf */ __float128 fmaq (__float128 x, __float128 y, __float128 z) { ieee854_float128 u, v, w; int adjust = 0; u.value = x; v.value = y; w.value = z; if (__builtin_expect (u.ieee.exponent + v.ieee.exponent >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) || __builtin_expect (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) || __builtin_expect (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent + v.ieee.exponent <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG, 0)) { /* If z is Inf, but x and y are finite, the result should be z rather than NaN. */ if (w.ieee.exponent == 0x7fff && u.ieee.exponent != 0x7fff && v.ieee.exponent != 0x7fff) return (z + x) + y; /* If z is zero and x are y are nonzero, compute the result as x * y to avoid the wrong sign of a zero result if x * y underflows to 0. */ if (z == 0 && x != 0 && y != 0) return x * y; /* If x or y or z is Inf/NaN, or if x * y is zero, compute as x * y + z. */ if (u.ieee.exponent == 0x7fff || v.ieee.exponent == 0x7fff || w.ieee.exponent == 0x7fff || x == 0 || y == 0) return x * y + z; /* If fma will certainly overflow, compute as x * y. */ if (u.ieee.exponent + v.ieee.exponent > 0x7fff + IEEE854_FLOAT128_BIAS) return x * y; /* If x * y is less than 1/4 of FLT128_DENORM_MIN, neither the result nor whether there is underflow depends on its exact value, only on its sign. */ if (u.ieee.exponent + v.ieee.exponent < IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG - 2) { int neg = u.ieee.negative ^ v.ieee.negative; __float128 tiny = neg ? -0x1p-16494Q : 0x1p-16494Q; if (w.ieee.exponent >= 3) return tiny + z; /* Scaling up, adding TINY and scaling down produces the correct result, because in round-to-nearest mode adding TINY has no effect and in other modes double rounding is harmless. But it may not produce required underflow exceptions. */ v.value = z * 0x1p114Q + tiny; if (TININESS_AFTER_ROUNDING ? v.ieee.exponent < 115 : (w.ieee.exponent == 0 || (w.ieee.exponent == 1 && w.ieee.negative != neg && w.ieee.mant_low == 0 && w.ieee.mant_high == 0))) { volatile __float128 force_underflow = x * y; (void) force_underflow; } return v.value * 0x1p-114Q; } if (u.ieee.exponent + v.ieee.exponent >= 0x7fff + IEEE854_FLOAT128_BIAS - FLT128_MANT_DIG) { /* Compute 1p-113 times smaller result and multiply at the end. */ if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent -= FLT128_MANT_DIG; else v.ieee.exponent -= FLT128_MANT_DIG; /* If x + y exponent is very large and z exponent is very small, it doesn't matter if we don't adjust it. */ if (w.ieee.exponent > FLT128_MANT_DIG) w.ieee.exponent -= FLT128_MANT_DIG; adjust = 1; } else if (w.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) { /* Similarly. If z exponent is very large and x and y exponents are very small, adjust them up to avoid spurious underflows, rather than down. */ if (u.ieee.exponent + v.ieee.exponent <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) { if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent += 2 * FLT128_MANT_DIG + 2; else v.ieee.exponent += 2 * FLT128_MANT_DIG + 2; } else if (u.ieee.exponent > v.ieee.exponent) { if (u.ieee.exponent > FLT128_MANT_DIG) u.ieee.exponent -= FLT128_MANT_DIG; } else if (v.ieee.exponent > FLT128_MANT_DIG) v.ieee.exponent -= FLT128_MANT_DIG; w.ieee.exponent -= FLT128_MANT_DIG; adjust = 1; } else if (u.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) { u.ieee.exponent -= FLT128_MANT_DIG; if (v.ieee.exponent) v.ieee.exponent += FLT128_MANT_DIG; else v.value *= 0x1p113Q; } else if (v.ieee.exponent >= 0x7fff - FLT128_MANT_DIG) { v.ieee.exponent -= FLT128_MANT_DIG; if (u.ieee.exponent) u.ieee.exponent += FLT128_MANT_DIG; else u.value *= 0x1p113Q; } else /* if (u.ieee.exponent + v.ieee.exponent <= IEEE854_FLOAT128_BIAS + FLT128_MANT_DIG) */ { if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent += 2 * FLT128_MANT_DIG; else v.ieee.exponent += 2 * FLT128_MANT_DIG; if (w.ieee.exponent <= 4 * FLT128_MANT_DIG + 4) { if (w.ieee.exponent) w.ieee.exponent += 2 * FLT128_MANT_DIG; else w.value *= 0x1p226Q; adjust = -1; } /* Otherwise x * y should just affect inexact and nothing else. */ } x = u.value; y = v.value; z = w.value; } /* Ensure correct sign of exact 0 + 0. */ if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0)) return x * y + z; #ifdef USE_FENV_H fenv_t env; feholdexcept (&env); fesetround (FE_TONEAREST); #endif /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ #define C ((1LL << (FLT128_MANT_DIG + 1) / 2) + 1) __float128 x1 = x * C; __float128 y1 = y * C; __float128 m1 = x * y; x1 = (x - x1) + x1; y1 = (y - y1) + y1; __float128 x2 = x - x1; __float128 y2 = y - y1; __float128 m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ __float128 a1 = z + m1; __float128 t1 = a1 - z; __float128 t2 = a1 - t1; t1 = m1 - t1; t2 = z - t2; __float128 a2 = t1 + t2; #ifdef USE_FENV_H feclearexcept (FE_INEXACT); #endif /* If the result is an exact zero, ensure it has the correct sign. */ if (a1 == 0 && m2 == 0) { #ifdef USE_FENV_H feupdateenv (&env); #endif /* Ensure that round-to-nearest value of z + m1 is not reused. */ asm volatile ("" : "=m" (z) : "m" (z)); return z + m1; } #ifdef USE_FENV_H fesetround (FE_TOWARDZERO); #endif /* Perform m2 + a2 addition with round to odd. */ u.value = a2 + m2; if (__builtin_expect (adjust == 0, 1)) { #ifdef USE_FENV_H if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff) u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); #endif /* Result is a1 + u.value. */ return a1 + u.value; } else if (__builtin_expect (adjust > 0, 1)) { #ifdef USE_FENV_H if ((u.ieee.mant_low & 1) == 0 && u.ieee.exponent != 0x7fff) u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); #endif /* Result is a1 + u.value, scaled up. */ return (a1 + u.value) * 0x1p113Q; } else { #ifdef USE_FENV_H if ((u.ieee.mant_low & 1) == 0) u.ieee.mant_low |= fetestexcept (FE_INEXACT) != 0; #endif v.value = a1 + u.value; /* Ensure the addition is not scheduled after fetestexcept call. */ asm volatile ("" : : "m" (v.value)); #ifdef USE_FENV_H int j = fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); #else int j = 0; #endif /* Ensure the following computations are performed in default rounding mode instead of just reusing the round to zero computation. */ asm volatile ("" : "=m" (u) : "m" (u)); /* If a1 + u.value is exact, the only rounding happens during scaling down. */ if (j == 0) return v.value * 0x1p-226Q; /* If result rounded to zero is not subnormal, no double rounding will occur. */ if (v.ieee.exponent > 226) return (a1 + u.value) * 0x1p-226Q; /* If v.value * 0x1p-226Q with round to zero is a subnormal above or equal to FLT128_MIN / 2, then v.value * 0x1p-226Q shifts mantissa down just by 1 bit, which means v.ieee.mant_low |= j would change the round bit, not sticky or guard bit. v.value * 0x1p-226Q never normalizes by shifting up, so round bit plus sticky bit should be already enough for proper rounding. */ if (v.ieee.exponent == 226) { /* If the exponent would be in the normal range when rounding to normal precision with unbounded exponent range, the exact result is known and spurious underflows must be avoided on systems detecting tininess after rounding. */ if (TININESS_AFTER_ROUNDING) { w.value = a1 + u.value; if (w.ieee.exponent == 227) return w.value * 0x1p-226Q; } /* v.ieee.mant_low & 2 is LSB bit of the result before rounding, v.ieee.mant_low & 1 is the round bit and j is our sticky bit. */ w.value = 0.0Q; w.ieee.mant_low = ((v.ieee.mant_low & 3) << 1) | j; w.ieee.negative = v.ieee.negative; v.ieee.mant_low &= ~3U; v.value *= 0x1p-226Q; w.value *= 0x1p-2Q; return v.value + w.value; } v.ieee.mant_low |= j; return v.value * 0x1p-226Q; } }