Average Error: 0.0 → 0.0
Time: 3.1s
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 r310900 = x;
        double r310901 = 2.0;
        double r310902 = r310900 * r310901;
        double r310903 = r310900 * r310900;
        double r310904 = r310902 + r310903;
        double r310905 = y;
        double r310906 = r310905 * r310905;
        double r310907 = r310904 + r310906;
        return r310907;
}
↓
double f(double x, double y) {
        double r310908 = x;
        double r310909 = 2.0;
        double r310910 = r310909 + r310908;
        double r310911 = y;
        double r310912 = r310911 * r310911;
        double r310913 = fma(r310908, r310910, r310912);
        return r310913;
}

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)))