Average Error: 52.0 → 52.0
Time: 2.5s
Precision: 64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]
↓
\[\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r76429 = 9.0;
        double r76430 = x;
        double r76431 = 4.0;
        double r76432 = pow(r76430, r76431);
        double r76433 = r76429 * r76432;
        double r76434 = y;
        double r76435 = pow(r76434, r76431);
        double r76436 = r76433 - r76435;
        double r76437 = 2.0;
        double r76438 = r76434 * r76434;
        double r76439 = r76437 * r76438;
        double r76440 = r76436 + r76439;
        return r76440;
}
↓
double f(double x, double y) {
        double r76441 = 2.0;
        double r76442 = y;
        double r76443 = r76441 * r76442;
        double r76444 = 9.0;
        double r76445 = x;
        double r76446 = 4.0;
        double r76447 = pow(r76445, r76446);
        double r76448 = r76444 * r76447;
        double r76449 = pow(r76442, r76446);
        double r76450 = r76448 - r76449;
        double r76451 = 3.0;
        double r76452 = pow(r76450, r76451);
        double r76453 = cbrt(r76452);
        double r76454 = fma(r76443, r76442, r76453);
        return r76454;
}

Error

Derivation

  1. Initial program 52.0

    \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]

  2. Simplified52.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot y, y, 9 \cdot {x}^{4} - {y}^{4}\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \color{blue}{\sqrt[3]{\left(\left(9 \cdot {x}^{4} - {y}^{4}\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)}}\right)\]

  5. Simplified52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{\color{blue}{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}}\right)\]

  6. Final simplification52.0

    \[\leadsto \mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (+ (- (* 9 (pow x 4)) (pow y 4)) (* 2 (* y y))))