Average Error: 0.0 → 0.0
Time: 1.3s
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 r477373 = 2.0;
        double r477374 = x;
        double r477375 = r477374 * r477374;
        double r477376 = y;
        double r477377 = r477374 * r477376;
        double r477378 = r477375 + r477377;
        double r477379 = r477373 * r477378;
        return r477379;
}
↓
double f(double x, double y) {
        double r477380 = x;
        double r477381 = y;
        double r477382 = r477380 * r477381;
        double r477383 = fma(r477380, r477380, r477382);
        double r477384 = 2.0;
        double r477385 = r477383 * r477384;
        return r477385;
}

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 2020036 +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))))