Average Error: 0.0 → 0.0
Time: 966.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[\mathsf{fma}\left(x + 1, y, x\right)\]
\left(x \cdot y + x\right) + y
↓
\mathsf{fma}\left(x + 1, y, x\right)
double f(double x, double y) {
        double r111864 = x;
        double r111865 = y;
        double r111866 = r111864 * r111865;
        double r111867 = r111866 + r111864;
        double r111868 = r111867 + r111865;
        return r111868;
}
↓
double f(double x, double y) {
        double r111869 = x;
        double r111870 = 1.0;
        double r111871 = r111869 + r111870;
        double r111872 = y;
        double r111873 = fma(r111871, r111872, r111869);
        return r111873;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot y + x\right) + y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{y + \mathsf{fma}\left(x, y, x\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{x + \left(y + x \cdot y\right)}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x + 1, y, x\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x + 1, y, x\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))