Average Error: 0.0 → 0.0
Time: 537.0ms
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 r533358 = 2.0;
        double r533359 = x;
        double r533360 = r533359 * r533359;
        double r533361 = y;
        double r533362 = r533359 * r533361;
        double r533363 = r533360 + r533362;
        double r533364 = r533358 * r533363;
        return r533364;
}
↓
double f(double x, double y) {
        double r533365 = x;
        double r533366 = y;
        double r533367 = r533365 * r533366;
        double r533368 = fma(r533365, r533365, r533367);
        double r533369 = 2.0;
        double r533370 = r533368 * r533369;
        return r533370;
}

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 2020062 +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))))