Average Error: 0.0 → 0.0
Time: 6.3s
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 r451810 = x;
        double r451811 = 2.0;
        double r451812 = r451810 * r451811;
        double r451813 = r451810 * r451810;
        double r451814 = r451812 + r451813;
        double r451815 = y;
        double r451816 = r451815 * r451815;
        double r451817 = r451814 + r451816;
        return r451817;
}
↓
double f(double x, double y) {
        double r451818 = y;
        double r451819 = x;
        double r451820 = 2.0;
        double r451821 = r451820 + r451819;
        double r451822 = r451819 * r451821;
        double r451823 = fma(r451818, r451818, r451822);
        return r451823;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + \left(2 \cdot x + {y}^{2}\right)}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

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

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