Average Error: 0.0 → 0.0
Time: 4.2s
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 r502457 = 2.0;
        double r502458 = x;
        double r502459 = r502458 * r502458;
        double r502460 = y;
        double r502461 = r502458 * r502460;
        double r502462 = r502459 + r502461;
        double r502463 = r502457 * r502462;
        return r502463;
}
↓
double f(double x, double y) {
        double r502464 = x;
        double r502465 = y;
        double r502466 = r502464 * r502465;
        double r502467 = fma(r502464, r502464, r502466);
        double r502468 = 2.0;
        double r502469 = r502467 * r502468;
        return r502469;
}

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 2020020 +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))))