Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)
double f(double x, double y) {
        double r17138372 = x;
        double r17138373 = 2.0;
        double r17138374 = r17138372 * r17138373;
        double r17138375 = r17138372 * r17138372;
        double r17138376 = r17138374 + r17138375;
        double r17138377 = y;
        double r17138378 = r17138377 * r17138377;
        double r17138379 = r17138376 + r17138378;
        return r17138379;
}
↓
double f(double x, double y) {
        double r17138380 = y;
        double r17138381 = x;
        double r17138382 = 2.0;
        double r17138383 = r17138382 + r17138381;
        double r17138384 = r17138381 * r17138383;
        double r17138385 = fma(r17138380, r17138380, r17138384);
        return r17138385;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[y \cdot y + \left(2 \cdot x + x \cdot x\right)\]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \left(2 + x\right) \cdot x\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, y, x \cdot \left(2 + x\right)\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))