Average Error: 0.0 → 0.0
Time: 775.0ms
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 r127737 = x;
        double r127738 = y;
        double r127739 = r127737 * r127738;
        double r127740 = r127739 + r127737;
        double r127741 = r127740 + r127738;
        return r127741;
}
↓
double f(double x, double y) {
        double r127742 = y;
        double r127743 = x;
        double r127744 = fma(r127743, r127742, r127743);
        double r127745 = r127742 + r127744;
        return r127745;
}

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 2020057 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))