Average Error: 0.0 → 0.0
Time: 2.3s
Precision: 64
\[2 \cdot \left(x \cdot x + x \cdot y\right)\]
↓
\[\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]
2 \cdot \left(x \cdot x + x \cdot y\right)
↓
\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2
double f(double x, double y) {
        double r603842 = 2.0;
        double r603843 = x;
        double r603844 = r603843 * r603843;
        double r603845 = y;
        double r603846 = r603843 * r603845;
        double r603847 = r603844 + r603846;
        double r603848 = r603842 * r603847;
        return r603848;
}
↓
double f(double x, double y) {
        double r603849 = x;
        double r603850 = y;
        double r603851 = r603849 * r603850;
        double r603852 = fma(r603849, r603849, r603851);
        double r603853 = 2.0;
        double r603854 = r603852 * r603853;
        return r603854;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\left(x \cdot 2\right) \cdot \left(x + y\right)\]

Derivation

  1. Initial program 0.0

    \[2 \cdot \left(x \cdot x + x \cdot y\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (* x 2) (+ x y))

  (* 2 (+ (* x x) (* x y))))