Average Error: 0.0 → 0.0
Time: 988.0ms
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 r135629 = x;
        double r135630 = y;
        double r135631 = r135629 * r135630;
        double r135632 = r135631 + r135629;
        double r135633 = r135632 + r135630;
        return r135633;
}
↓
double f(double x, double y) {
        double r135634 = y;
        double r135635 = x;
        double r135636 = fma(r135635, r135634, r135635);
        double r135637 = r135634 + r135636;
        return r135637;
}

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 2020035 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))