Average Error: 0.0 → 0.0
Time: 727.0ms
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 r521936 = 2.0;
        double r521937 = x;
        double r521938 = r521937 * r521937;
        double r521939 = y;
        double r521940 = r521937 * r521939;
        double r521941 = r521938 + r521940;
        double r521942 = r521936 * r521941;
        return r521942;
}
↓
double f(double x, double y) {
        double r521943 = x;
        double r521944 = y;
        double r521945 = r521943 * r521944;
        double r521946 = fma(r521943, r521943, r521945);
        double r521947 = 2.0;
        double r521948 = r521946 * r521947;
        return r521948;
}

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 2020001 +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))))