Average Error: 0.0 → 0.0
Time: 7.2s
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 r446595 = x;
        double r446596 = 2.0;
        double r446597 = r446595 * r446596;
        double r446598 = r446595 * r446595;
        double r446599 = r446597 + r446598;
        double r446600 = y;
        double r446601 = r446600 * r446600;
        double r446602 = r446599 + r446601;
        return r446602;
}
↓
double f(double x, double y) {
        double r446603 = y;
        double r446604 = x;
        double r446605 = 2.0;
        double r446606 = r446605 + r446604;
        double r446607 = r446604 * r446606;
        double r446608 = fma(r446603, r446603, r446607);
        return r446608;
}

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 2020047 +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)))