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 r391799 = x;
        double r391800 = 2.0;
        double r391801 = r391799 * r391800;
        double r391802 = r391799 * r391799;
        double r391803 = r391801 + r391802;
        double r391804 = y;
        double r391805 = r391804 * r391804;
        double r391806 = r391803 + r391805;
        return r391806;
}
↓
double f(double x, double y) {
        double r391807 = y;
        double r391808 = x;
        double r391809 = 2.0;
        double r391810 = r391809 + r391808;
        double r391811 = r391808 * r391810;
        double r391812 = fma(r391807, r391807, r391811);
        return r391812;
}

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 2019196 +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)))