/* 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 #include #include #include #include /* 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 */ long double __fmal (long double x, long double y, long double z) { union ieee854_long_double u, v, w; int adjust = 0; u.d = x; v.d = y; w.d = z; if (__builtin_expect (u.ieee.exponent + v.ieee.exponent >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0) || __builtin_expect (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0) || __builtin_expect (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent + v.ieee.exponent <= IEEE854_LONG_DOUBLE_BIAS + LDBL_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 fma will certainly overflow, or if x * y is less than half of LDBL_DENORM_MIN, compute as x * y + z. */ if (u.ieee.exponent == 0x7fff || v.ieee.exponent == 0x7fff || w.ieee.exponent == 0x7fff || u.ieee.exponent + v.ieee.exponent > 0x7fff + IEEE854_LONG_DOUBLE_BIAS || u.ieee.exponent + v.ieee.exponent < IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG - 2) return x * y + z; if (u.ieee.exponent + v.ieee.exponent >= 0x7fff + IEEE854_LONG_DOUBLE_BIAS - LDBL_MANT_DIG) { /* Compute 1p-113 times smaller result and multiply at the end. */ if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent -= LDBL_MANT_DIG; else v.ieee.exponent -= LDBL_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 > LDBL_MANT_DIG) w.ieee.exponent -= LDBL_MANT_DIG; adjust = 1; } else if (w.ieee.exponent >= 0x7fff - LDBL_MANT_DIG) { /* Similarly. If z exponent is very large and x and y exponents are very small, it doesn't matter if we don't adjust it. */ if (u.ieee.exponent > v.ieee.exponent) { if (u.ieee.exponent > LDBL_MANT_DIG) u.ieee.exponent -= LDBL_MANT_DIG; } else if (v.ieee.exponent > LDBL_MANT_DIG) v.ieee.exponent -= LDBL_MANT_DIG; w.ieee.exponent -= LDBL_MANT_DIG; adjust = 1; } else if (u.ieee.exponent >= 0x7fff - LDBL_MANT_DIG) { u.ieee.exponent -= LDBL_MANT_DIG; if (v.ieee.exponent) v.ieee.exponent += LDBL_MANT_DIG; else v.d *= 0x1p113L; } else if (v.ieee.exponent >= 0x7fff - LDBL_MANT_DIG) { v.ieee.exponent -= LDBL_MANT_DIG; if (u.ieee.exponent) u.ieee.exponent += LDBL_MANT_DIG; else u.d *= 0x1p113L; } else /* if (u.ieee.exponent + v.ieee.exponent <= IEEE854_LONG_DOUBLE_BIAS + LDBL_MANT_DIG) */ { if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent += 2 * LDBL_MANT_DIG; else v.ieee.exponent += 2 * LDBL_MANT_DIG; if (w.ieee.exponent <= 4 * LDBL_MANT_DIG + 4) { if (w.ieee.exponent) w.ieee.exponent += 2 * LDBL_MANT_DIG; else w.d *= 0x1p226L; adjust = -1; } /* Otherwise x * y should just affect inexact and nothing else. */ } x = u.d; y = v.d; z = w.d; } /* Ensure correct sign of exact 0 + 0. */ if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0)) return x * y + z; /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ #define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1) long double x1 = x * C; long double y1 = y * C; long double m1 = x * y; x1 = (x - x1) + x1; y1 = (y - y1) + y1; long double x2 = x - x1; long double y2 = y - y1; long double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ long double a1 = z + m1; long double t1 = a1 - z; long double t2 = a1 - t1; t1 = m1 - t1; t2 = z - t2; long double a2 = t1 + t2; fenv_t env; feholdexcept (&env); fesetround (FE_TOWARDZERO); /* Perform m2 + a2 addition with round to odd. */ u.d = a2 + m2; if (__builtin_expect (adjust == 0, 1)) { if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); /* Result is a1 + u.d. */ return a1 + u.d; } else if (__builtin_expect (adjust > 0, 1)) { if ((u.ieee.mantissa3 & 1) == 0 && u.ieee.exponent != 0x7fff) u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); /* Result is a1 + u.d, scaled up. */ return (a1 + u.d) * 0x1p113L; } else { if ((u.ieee.mantissa3 & 1) == 0) u.ieee.mantissa3 |= fetestexcept (FE_INEXACT) != 0; v.d = a1 + u.d; /* Ensure the addition is not scheduled after fetestexcept call. */ math_force_eval (v.d); int j = fetestexcept (FE_INEXACT) != 0; feupdateenv (&env); /* 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.d is exact, the only rounding happens during scaling down. */ if (j == 0) return v.d * 0x1p-226L; /* If result rounded to zero is not subnormal, no double rounding will occur. */ if (v.ieee.exponent > 226) return (a1 + u.d) * 0x1p-226L; /* If v.d * 0x1p-226L with round to zero is a subnormal above or equal to LDBL_MIN / 2, then v.d * 0x1p-226L shifts mantissa down just by 1 bit, which means v.ieee.mantissa3 |= j would change the round bit, not sticky or guard bit. v.d * 0x1p-226L never normalizes by shifting up, so round bit plus sticky bit should be already enough for proper rounding. */ if (v.ieee.exponent == 226) { /* v.ieee.mantissa3 & 2 is LSB bit of the result before rounding, v.ieee.mantissa3 & 1 is the round bit and j is our sticky bit. In round-to-nearest 001 rounds down like 00, 011 rounds up, even though 01 rounds down (thus we need to adjust), 101 rounds down like 10 and 111 rounds up like 11. */ if ((v.ieee.mantissa3 & 3) == 1) { v.d *= 0x1p-226L; if (v.ieee.negative) return v.d - 0x1p-16494L /* __LDBL_DENORM_MIN__ */; else return v.d + 0x1p-16494L /* __LDBL_DENORM_MIN__ */; } else return v.d * 0x1p-226L; } v.ieee.mantissa3 |= j; return v.d * 0x1p-226L; } } weak_alias (__fmal, fmal)