/* 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 #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 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 zero, 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 || x == 0 || y == 0) return x * y + z; /* If x * y is less than 1/4 of DBL_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 < IEEE754_DOUBLE_BIAS - DBL_MANT_DIG - 2) { int neg = u.ieee.negative ^ v.ieee.negative; double tiny = neg ? -0x1p-1074 : 0x1p-1074; 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.d = z * 0x1p54 + tiny; if (TININESS_AFTER_ROUNDING ? v.ieee.exponent < 55 : (w.ieee.exponent == 0 || (w.ieee.exponent == 1 && w.ieee.negative != neg && w.ieee.mantissa1 == 0 && w.ieee.mantissa0 == 0))) { volatile double force_underflow = x * y; (void) force_underflow; } return v.d * 0x1p-54; } 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; } /* Ensure correct sign of exact 0 + 0. */ if (__builtin_expect ((x == 0 || y == 0) && z == 0, 0)) return x * y + z; fenv_t env; libc_feholdexcept_setround (&env, FE_TONEAREST); /* 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; feclearexcept (FE_INEXACT); /* If the result is an exact zero, ensure it has the correct sign. */ if (a1 == 0 && m2 == 0) { libc_feupdateenv (&env); /* Ensure that round-to-nearest value of z + m1 is not reused. */ asm volatile ("" : "=m" (z) : "m" (z)); return z + m1; } libc_fesetround (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; /* Ensure the addition is not scheduled after fetestexcept call. */ math_force_eval (v.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) { /* 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.d = a1 + u.d; if (w.ieee.exponent == 107) return w.d * 0x1p-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. */ w.d = 0.0; w.ieee.mantissa1 = ((v.ieee.mantissa1 & 3) << 1) | j; w.ieee.negative = v.ieee.negative; v.ieee.mantissa1 &= ~3U; v.d *= 0x1p-106; w.d *= 0x1p-2; return v.d + w.d; } 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