Average Error: 0.0 → 0.0
Time: 755.0ms
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 r509652 = 2.0;
        double r509653 = x;
        double r509654 = r509653 * r509653;
        double r509655 = y;
        double r509656 = r509653 * r509655;
        double r509657 = r509654 + r509656;
        double r509658 = r509652 * r509657;
        return r509658;
}
↓
double f(double x, double y) {
        double r509659 = x;
        double r509660 = y;
        double r509661 = r509659 * r509660;
        double r509662 = fma(r509659, r509659, r509661);
        double r509663 = 2.0;
        double r509664 = r509662 * r509663;
        return r509664;
}

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 2020001 +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))))