Average Error: 0.0 → 0.0
Time: 669.0ms
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 r481408 = 2.0;
        double r481409 = x;
        double r481410 = r481409 * r481409;
        double r481411 = y;
        double r481412 = r481409 * r481411;
        double r481413 = r481410 + r481412;
        double r481414 = r481408 * r481413;
        return r481414;
}
↓
double f(double x, double y) {
        double r481415 = x;
        double r481416 = y;
        double r481417 = r481415 * r481416;
        double r481418 = fma(r481415, r481415, r481417);
        double r481419 = 2.0;
        double r481420 = r481418 * r481419;
        return r481420;
}

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 2020060 +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))))