Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, 2 + x, y \cdot y\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(x, 2 + x, y \cdot y\right)
double f(double x, double y) {
        double r353585 = x;
        double r353586 = 2.0;
        double r353587 = r353585 * r353586;
        double r353588 = r353585 * r353585;
        double r353589 = r353587 + r353588;
        double r353590 = y;
        double r353591 = r353590 * r353590;
        double r353592 = r353589 + r353591;
        return r353592;
}
↓
double f(double x, double y) {
        double r353593 = x;
        double r353594 = 2.0;
        double r353595 = r353594 + r353593;
        double r353596 = y;
        double r353597 = r353596 * r353596;
        double r353598 = fma(r353593, r353595, r353597);
        return r353598;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019198 +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)))