Average Error: 0.0 → 0.0
Time: 4.5s
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 r829687 = 2.0;
        double r829688 = x;
        double r829689 = r829688 * r829688;
        double r829690 = y;
        double r829691 = r829688 * r829690;
        double r829692 = r829689 + r829691;
        double r829693 = r829687 * r829692;
        return r829693;
}
↓
double f(double x, double y) {
        double r829694 = x;
        double r829695 = y;
        double r829696 = r829694 * r829695;
        double r829697 = fma(r829694, r829694, r829696);
        double r829698 = 2.0;
        double r829699 = r829697 * r829698;
        return r829699;
}

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 2019362 +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))))