Average Error: 0.0 → 0.0
Time: 9.6s
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 r326413 = x;
        double r326414 = 2.0;
        double r326415 = r326413 * r326414;
        double r326416 = r326413 * r326413;
        double r326417 = r326415 + r326416;
        double r326418 = y;
        double r326419 = r326418 * r326418;
        double r326420 = r326417 + r326419;
        return r326420;
}
↓
double f(double x, double y) {
        double r326421 = y;
        double r326422 = x;
        double r326423 = 2.0;
        double r326424 = r326423 + r326422;
        double r326425 = r326422 * r326424;
        double r326426 = fma(r326421, r326421, r326425);
        return r326426;
}

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 2019306 +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)))