Average Error: 0.0 → 0.0
Time: 2.0s
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 r454362 = 2.0;
        double r454363 = x;
        double r454364 = r454363 * r454363;
        double r454365 = y;
        double r454366 = r454363 * r454365;
        double r454367 = r454364 + r454366;
        double r454368 = r454362 * r454367;
        return r454368;
}
↓
double f(double x, double y) {
        double r454369 = x;
        double r454370 = y;
        double r454371 = r454369 * r454370;
        double r454372 = fma(r454369, r454369, r454371);
        double r454373 = 2.0;
        double r454374 = r454372 * r454373;
        return r454374;
}

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 2020021 +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))))