Average Error: 0.0 → 0.0
Time: 2.0s
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 r549379 = 2.0;
        double r549380 = x;
        double r549381 = r549380 * r549380;
        double r549382 = y;
        double r549383 = r549380 * r549382;
        double r549384 = r549381 + r549383;
        double r549385 = r549379 * r549384;
        return r549385;
}
↓
double f(double x, double y) {
        double r549386 = x;
        double r549387 = y;
        double r549388 = r549386 * r549387;
        double r549389 = fma(r549386, r549386, r549388);
        double r549390 = 2.0;
        double r549391 = r549389 * r549390;
        return r549391;
}

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 2020027 +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))))