Average Error: 0.0 → 0.0
Time: 22.8s
Precision: 64
\[\left(x \cdot y + x\right) + y\]
↓
\[y + \mathsf{fma}\left(y, x, x\right)\]
\left(x \cdot y + x\right) + y
↓
y + \mathsf{fma}\left(y, x, x\right)
double f(double x, double y) {
        double r3603678 = x;
        double r3603679 = y;
        double r3603680 = r3603678 * r3603679;
        double r3603681 = r3603680 + r3603678;
        double r3603682 = r3603681 + r3603679;
        return r3603682;
}
↓
double f(double x, double y) {
        double r3603683 = y;
        double r3603684 = x;
        double r3603685 = fma(r3603683, r3603684, r3603684);
        double r3603686 = r3603683 + r3603685;
        return r3603686;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  (+ (+ (* x y) x) y))