Average Error: 0.0 → 0.0
Time: 3.9s
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 r2841 = 2.0;
        double r2842 = x;
        double r2843 = r2842 * r2842;
        double r2844 = y;
        double r2845 = r2842 * r2844;
        double r2846 = r2843 + r2845;
        double r2847 = r2841 * r2846;
        return r2847;
}
↓
double f(double x, double y) {
        double r2848 = x;
        double r2849 = y;
        double r2850 = r2848 * r2849;
        double r2851 = fma(r2848, r2848, r2850);
        double r2852 = 2.0;
        double r2853 = r2851 * r2852;
        return r2853;
}

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 2020025 +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))))