Average Error: 0.1 → 0.1
Time: 3.6s
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 r408593 = x;
        double r408594 = r408593 * r408593;
        double r408595 = y;
        double r408596 = r408595 * r408595;
        double r408597 = r408594 + r408596;
        double r408598 = r408597 + r408596;
        double r408599 = r408598 + r408596;
        return r408599;
}
↓
double f(double x, double y) {
        double r408600 = 3.0;
        double r408601 = y;
        double r408602 = r408601 * r408601;
        double r408603 = x;
        double r408604 = r408603 * r408603;
        double r408605 = fma(r408600, r408602, r408604);
        return r408605;
}

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 2020001 +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)))