/* Compute x * y + z as ternary operation. Copyright (C) 2010, 2011 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 */ double __fma (double x, double y, double z) { union ieee754_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 >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) || __builtin_expect (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) || __builtin_expect (w.ieee.exponent >= 0x7ff - DBL_MANT_DIG, 0) || __builtin_expect (u.ieee.exponent + v.ieee.exponent <= IEEE754_DOUBLE_BIAS + DBL_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 == 0x7ff && u.ieee.exponent != 0x7ff && v.ieee.exponent != 0x7ff) return (z + 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 DBL_DENORM_MIN, compute as x * y + z. */ if (u.ieee.exponent == 0x7ff || v.ieee.exponent == 0x7ff || w.ieee.exponent == 0x7ff || u.ieee.exponent + v.ieee.exponent > 0x7ff + IEEE754_DOUBLE_BIAS || u.ieee.exponent + v.ieee.exponent < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2) return x * y + z; if (u.ieee.exponent + v.ieee.exponent >= 0x7ff + IEEE754_DOUBLE_BIAS - DBL_MANT_DIG) { /* Compute 1p-53 times smaller result and multiply at the end. */ if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent -= DBL_MANT_DIG; else v.ieee.exponent -= DBL_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 > DBL_MANT_DIG) w.ieee.exponent -= DBL_MANT_DIG; adjust = 1; } else if (w.ieee.exponent >= 0x7ff - DBL_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 > DBL_MANT_DIG) u.ieee.exponent -= DBL_MANT_DIG; } else if (v.ieee.exponent > DBL_MANT_DIG) v.ieee.exponent -= DBL_MANT_DIG; w.ieee.exponent -= DBL_MANT_DIG; adjust = 1; } else if (u.ieee.exponent >= 0x7ff - DBL_MANT_DIG) { u.ieee.exponent -= DBL_MANT_DIG; if (v.ieee.exponent) v.ieee.exponent += DBL_MANT_DIG; else v.d *= 0x1p53; } else if (v.ieee.exponent >= 0x7ff - DBL_MANT_DIG) { v.ieee.exponent -= DBL_MANT_DIG; if (u.ieee.exponent) u.ieee.exponent += DBL_MANT_DIG; else u.d *= 0x1p53; } else /* if (u.ieee.exponent + v.ieee.exponent <= IEEE754_DOUBLE_BIAS + DBL_MANT_DIG) */ { if (u.ieee.exponent > v.ieee.exponent) u.ieee.exponent += 2 * DBL_MANT_DIG; else v.ieee.exponent += 2 * DBL_MANT_DIG; if (w.ieee.exponent <= 4 * DBL_MANT_DIG + 4) { if (w.ieee.exponent) w.ieee.exponent += 2 * DBL_MANT_DIG; else w.d *= 0x1p106; adjust = -1; } /* Otherwise x * y should just affect inexact and nothing else. */ } x = u.d; y = v.d; z = w.d; } /* Multiplication m1 + m2 = x * y using Dekker's algorithm. */ #define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1) double x1 = x * C; double y1 = y * C; double m1 = x * y; x1 = (x - x1) + x1; y1 = (y - y1) + y1; double x2 = x - x1; double y2 = y - y1; double m2 = (((x1 * y1 - m1) + x1 * y2) + x2 * y1) + x2 * y2; /* Addition a1 + a2 = z + m1 using Knuth's algorithm. */ double a1 = z + m1; double t1 = a1 - z; double t2 = a1 - t1; t1 = m1 - t1; t2 = z - t2; double a2 = t1 + t2; fenv_t env; libc_feholdexcept_setround (&env, FE_TOWARDZERO); /* Perform m2 + a2 addition with round to odd. */ u.d = a2 + m2; if (__builtin_expect (adjust < 0, 0)) { if ((u.ieee.mantissa1 & 1) == 0) u.ieee.mantissa1 |= libc_fetestexcept (FE_INEXACT) != 0; v.d = a1 + u.d; } /* Reset rounding mode and test for inexact simultaneously. */ int j = libc_feupdateenv_test (&env, FE_INEXACT) != 0; if (__builtin_expect (adjust == 0, 1)) { if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff) u.ieee.mantissa1 |= j; /* Result is a1 + u.d. */ return a1 + u.d; } else if (__builtin_expect (adjust > 0, 1)) { if ((u.ieee.mantissa1 & 1) == 0 && u.ieee.exponent != 0x7ff) u.ieee.mantissa1 |= j; /* Result is a1 + u.d, scaled up. */ return (a1 + u.d) * 0x1p53; } else { /* If a1 + u.d is exact, the only rounding happens during scaling down. */ if (j == 0) return v.d * 0x1p-106; /* If result rounded to zero is not subnormal, no double rounding will occur. */ if (v.ieee.exponent > 106) return (a1 + u.d) * 0x1p-106; /* If v.d * 0x1p-106 with round to zero is a subnormal above or equal to DBL_MIN / 2, then v.d * 0x1p-106 shifts mantissa down just by 1 bit, which means v.ieee.mantissa1 |= j would change the round bit, not sticky or guard bit. v.d * 0x1p-106 never normalizes by shifting up, so round bit plus sticky bit should be already enough for proper rounding. */ if (v.ieee.exponent == 106) { /* v.ieee.mantissa1 & 2 is LSB bit of the result before rounding, v.ieee.mantissa1 & 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.mantissa1 & 3) == 1) { v.d *= 0x1p-106; if (v.ieee.negative) return v.d - 0x1p-1074 /* __DBL_DENORM_MIN__ */; else return v.d + 0x1p-1074 /* __DBL_DENORM_MIN__ */; } else return v.d * 0x1p-106; } v.ieee.mantissa1 |= j; return v.d * 0x1p-106; } } #ifndef __fma weak_alias (__fma, fma) #endif #ifdef NO_LONG_DOUBLE strong_alias (__fma, __fmal) weak_alias (__fmal, fmal) #endif