Average Error: 0.1 → 0.1
Time: 2.1s
Precision: 64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)
double f(double x, double y) {
        double r487826 = x;
        double r487827 = r487826 * r487826;
        double r487828 = y;
        double r487829 = r487828 * r487828;
        double r487830 = r487827 + r487829;
        double r487831 = r487830 + r487829;
        double r487832 = r487831 + r487829;
        return r487832;
}
↓
double f(double x, double y) {
        double r487833 = 3.0;
        double r487834 = y;
        double r487835 = r487834 * r487834;
        double r487836 = x;
        double r487837 = r487836 * r487836;
        double r487838 = fma(r487833, r487835, r487837);
        return r487838;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right)\]

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot y, x \cdot x\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(3, y \cdot y, x \cdot x\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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