Average Error: 0.0 → 0.0
Time: 3.6s
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 r537574 = 2.0;
        double r537575 = x;
        double r537576 = r537575 * r537575;
        double r537577 = y;
        double r537578 = r537575 * r537577;
        double r537579 = r537576 + r537578;
        double r537580 = r537574 * r537579;
        return r537580;
}
↓
double f(double x, double y) {
        double r537581 = x;
        double r537582 = y;
        double r537583 = r537581 * r537582;
        double r537584 = fma(r537581, r537581, r537583);
        double r537585 = 2.0;
        double r537586 = r537584 * r537585;
        return r537586;
}

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 2020033 +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))))