Average Error: 0.0 → 0.0
Time: 1.3s
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 r431570 = x;
        double r431571 = 2.0;
        double r431572 = r431570 * r431571;
        double r431573 = r431570 * r431570;
        double r431574 = r431572 + r431573;
        double r431575 = y;
        double r431576 = r431575 * r431575;
        double r431577 = r431574 + r431576;
        return r431577;
}
↓
double f(double x, double y) {
        double r431578 = x;
        double r431579 = 2.0;
        double r431580 = y;
        double r431581 = 2.0;
        double r431582 = pow(r431580, r431581);
        double r431583 = fma(r431579, r431578, r431582);
        double r431584 = fma(r431578, r431578, r431583);
        return r431584;
}

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 2019353 +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)))