Average Error: 0.0 → 0.0
Time: 566.0ms
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 r91843 = x;
        double r91844 = y;
        double r91845 = r91843 * r91844;
        double r91846 = r91845 + r91843;
        double r91847 = r91846 + r91844;
        return r91847;
}
↓
double f(double x, double y) {
        double r91848 = y;
        double r91849 = x;
        double r91850 = fma(r91849, r91848, r91849);
        double r91851 = r91848 + r91850;
        return r91851;
}

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 2019354 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))