Average Error: 0.0 → 0.0
Time: 772.0ms
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(x, y, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(x, y, x\right)
double f(double x, double y) {
        double r99871 = x;
        double r99872 = y;
        double r99873 = r99871 * r99872;
        double r99874 = r99873 + r99871;
        double r99875 = r99874 + r99872;
        return r99875;
}
↓
double f(double x, double y) {
        double r99876 = y;
        double r99877 = x;
        double r99878 = fma(r99877, r99876, r99877);
        double r99879 = r99876 + r99878;
        return r99879;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))