Average Error: 0.0 → 0.0
Time: 1.7s
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 r462613 = 2.0;
        double r462614 = x;
        double r462615 = r462614 * r462614;
        double r462616 = y;
        double r462617 = r462614 * r462616;
        double r462618 = r462615 + r462617;
        double r462619 = r462613 * r462618;
        return r462619;
}
↓
double f(double x, double y) {
        double r462620 = x;
        double r462621 = y;
        double r462622 = r462620 * r462621;
        double r462623 = fma(r462620, r462620, r462622);
        double r462624 = 2.0;
        double r462625 = r462623 * r462624;
        return r462625;
}

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 2020036 +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))))