Average Error: 0.0 → 0.0
Time: 947.0ms
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 r171835 = x;
        double r171836 = y;
        double r171837 = r171835 * r171836;
        double r171838 = r171837 + r171835;
        double r171839 = r171838 + r171836;
        return r171839;
}
↓
double f(double x, double y) {
        double r171840 = y;
        double r171841 = x;
        double r171842 = fma(r171841, r171840, r171841);
        double r171843 = r171840 + r171842;
        return r171843;
}

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 2020034 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))