Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[2 \cdot \left(x \cdot x + x \cdot y\right)\]
↓
\[\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2\]
2 \cdot \left(x \cdot x + x \cdot y\right)
↓
\mathsf{fma}\left(x, x, x \cdot y\right) \cdot 2
double f(double x, double y) {
        double r613791 = 2.0;
        double r613792 = x;
        double r613793 = r613792 * r613792;
        double r613794 = y;
        double r613795 = r613792 * r613794;
        double r613796 = r613793 + r613795;
        double r613797 = r613791 * r613796;
        return r613797;
}
↓
double f(double x, double y) {
        double r613798 = x;
        double r613799 = y;
        double r613800 = r613798 * r613799;
        double r613801 = fma(r613798, r613798, r613800);
        double r613802 = 2.0;
        double r613803 = r613801 * r613802;
        return r613803;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[\left(x \cdot 2\right) \cdot \left(x + y\right)\]

Derivation

  1. Initial program 0.0

    \[2 \cdot \left(x \cdot x + x \cdot y\right)\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

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