Average Error: 0.0 → 0.0
Time: 681.0ms
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 r474467 = 2.0;
        double r474468 = x;
        double r474469 = r474468 * r474468;
        double r474470 = y;
        double r474471 = r474468 * r474470;
        double r474472 = r474469 + r474471;
        double r474473 = r474467 * r474472;
        return r474473;
}
↓
double f(double x, double y) {
        double r474474 = x;
        double r474475 = y;
        double r474476 = r474474 * r474475;
        double r474477 = fma(r474474, r474474, r474476);
        double r474478 = 2.0;
        double r474479 = r474477 * r474478;
        return r474479;
}

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 2020056 +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))))