Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y\]
↓
\[\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, {y}^{2}\right)\right)\]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, {y}^{2}\right)\right)
double f(double x, double y) {
        double r625193 = x;
        double r625194 = 2.0;
        double r625195 = r625193 * r625194;
        double r625196 = r625193 * r625193;
        double r625197 = r625195 + r625196;
        double r625198 = y;
        double r625199 = r625198 * r625198;
        double r625200 = r625197 + r625199;
        return r625200;
}
↓
double f(double x, double y) {
        double r625201 = x;
        double r625202 = 2.0;
        double r625203 = y;
        double r625204 = 2.0;
        double r625205 = pow(r625203, r625204);
        double r625206 = fma(r625202, r625201, r625205);
        double r625207 = fma(r625201, r625201, r625206);
        return r625207;
}

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

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))