Average Error: 0.0 → 0.0
Time: 8.9s
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 r22782699 = x;
        double r22782700 = 2.0;
        double r22782701 = r22782699 * r22782700;
        double r22782702 = r22782699 * r22782699;
        double r22782703 = r22782701 + r22782702;
        double r22782704 = y;
        double r22782705 = r22782704 * r22782704;
        double r22782706 = r22782703 + r22782705;
        return r22782706;
}
↓
double f(double x, double y) {
        double r22782707 = y;
        double r22782708 = x;
        double r22782709 = 2.0;
        double r22782710 = r22782709 + r22782708;
        double r22782711 = r22782708 * r22782710;
        double r22782712 = fma(r22782707, r22782707, r22782711);
        return r22782712;
}

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 2019172 +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)))