src/math/generic/fmaf.cpp

//===-- 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