Average Error: 0.0 → 0.0
Time: 3.2s
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 r303985 = x;
        double r303986 = 2.0;
        double r303987 = r303985 * r303986;
        double r303988 = r303985 * r303985;
        double r303989 = r303987 + r303988;
        double r303990 = y;
        double r303991 = r303990 * r303990;
        double r303992 = r303989 + r303991;
        return r303992;
}
↓
double f(double x, double y) {
        double r303993 = x;
        double r303994 = 2.0;
        double r303995 = r303994 + r303993;
        double r303996 = y;
        double r303997 = r303996 * r303996;
        double r303998 = fma(r303993, r303995, r303997);
        return r303998;
}

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 2019323 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2 x) (* x x)))

  (+ (+ (* x 2) (* x x)) (* y y)))