| //===-- Implementation of fmaf function -----------------------------------===// |
| // |
| // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. |
| // See https://llvm.org/LICENSE.txt for license information. |
| // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception |
| // |
| //===----------------------------------------------------------------------===// |
| |
| #include "src/math/fmaf.h" |
| #include "src/__support/common.h" |
| |
| #include "utils/FPUtil/FEnv.h" |
| #include "utils/FPUtil/FPBits.h" |
| |
| namespace __llvm_libc { |
| |
| LLVM_LIBC_FUNCTION(float, fmaf, (float x, float y, float z)){ |
| // Product is exact. |
| double prod = static_cast<double>(x) * static_cast<double>(y); |
| double z_d = static_cast<double>(z); |
| double sum = prod + z_d; |
| fputil::FPBits<double> bit_prod(prod), bitz(z_d), bit_sum(sum); |
| |
| if (!(bit_sum.isInfOrNaN() || bit_sum.isZero())) { |
| // Since the sum is computed in double precision, rounding might happen |
| // (for instance, when bitz.exponent > bit_prod.exponent + 5, or |
| // bit_prod.exponent > bitz.exponent + 40). In that case, when we round |
| // the sum back to float, double rounding error might occur. |
| // A concrete example of this phenomenon is as follows: |
| // x = y = 1 + 2^(-12), z = 2^(-53) |
| // The exact value of x*y + z is 1 + 2^(-11) + 2^(-24) + 2^(-53) |
| // So when rounding to float, fmaf(x, y, z) = 1 + 2^(-11) + 2^(-23) |
| // On the other hand, with the default rounding mode, |
| // double(x*y + z) = 1 + 2^(-11) + 2^(-24) |
| // and casting again to float gives us: |
| // float(double(x*y + z)) = 1 + 2^(-11). |
| // |
| // In order to correct this possible double rounding error, first we use |
| // Dekker's 2Sum algorithm to find t such that sum - t = prod + z exactly, |
| // assuming the (default) rounding mode is round-to-the-nearest, |
| // tie-to-even. Moreover, t satisfies the condition that t < eps(sum), |
| // i.e., t.exponent < sum.exponent - 52. So if t is not 0, meaning rounding |
| // occurs when computing the sum, we just need to use t to adjust (any) last |
| // bit of sum, so that the sticky bits used when rounding sum to float are |
| // correct (when it matters). |
| fputil::FPBits<double> t( |
| (bit_prod.exponent >= bitz.exponent) |
| ? ((static_cast<double>(bit_sum) - bit_prod) - bitz) |
| : ((static_cast<double>(bit_sum) - bitz) - bit_prod)); |
| |
| // Update sticky bits if t != 0.0 and the least (52 - 23 - 1 = 28) bits are |
| // zero. |
| if (!t.isZero() && ((bit_sum.mantissa & 0xfff'ffffULL) == 0)) { |
| if (bit_sum.sign != t.sign) { |
| ++bit_sum.mantissa; |
| } else if (bit_sum.mantissa) { |
| --bit_sum.mantissa; |
| } |
| } |
| } |
| |
| return static_cast<float>(static_cast<double>(bit_sum)); |
| } |
| |
| } // namespace __llvm_libc |