Average Error: 0.0 → 0.0
Time: 4.4s
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 r19412860 = x;
        double r19412861 = 2.0;
        double r19412862 = r19412860 * r19412861;
        double r19412863 = r19412860 * r19412860;
        double r19412864 = r19412862 + r19412863;
        double r19412865 = y;
        double r19412866 = r19412865 * r19412865;
        double r19412867 = r19412864 + r19412866;
        return r19412867;
}
↓
double f(double x, double y) {
        double r19412868 = y;
        double r19412869 = x;
        double r19412870 = 2.0;
        double r19412871 = r19412870 + r19412869;
        double r19412872 = r19412869 * r19412871;
        double r19412873 = fma(r19412868, r19412868, r19412872);
        return r19412873;
}

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 2019169 +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)))