Average Error: 0.0 → 0.0
Time: 3.9s
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 r488652 = 2.0;
        double r488653 = x;
        double r488654 = r488653 * r488653;
        double r488655 = y;
        double r488656 = r488653 * r488655;
        double r488657 = r488654 + r488656;
        double r488658 = r488652 * r488657;
        return r488658;
}
↓
double f(double x, double y) {
        double r488659 = x;
        double r488660 = y;
        double r488661 = r488659 * r488660;
        double r488662 = fma(r488659, r488659, r488661);
        double r488663 = 2.0;
        double r488664 = r488662 * r488663;
        return r488664;
}

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 2019354 +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))))