Average Error: 52.0 → 52.0
Time: 2.6s
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 r56509 = 9.0;
        double r56510 = x;
        double r56511 = 4.0;
        double r56512 = pow(r56510, r56511);
        double r56513 = r56509 * r56512;
        double r56514 = y;
        double r56515 = pow(r56514, r56511);
        double r56516 = r56513 - r56515;
        double r56517 = 2.0;
        double r56518 = r56514 * r56514;
        double r56519 = r56517 * r56518;
        double r56520 = r56516 + r56519;
        return r56520;
}
↓
double f(double x, double y) {
        double r56521 = 2.0;
        double r56522 = y;
        double r56523 = r56521 * r56522;
        double r56524 = 9.0;
        double r56525 = x;
        double r56526 = 4.0;
        double r56527 = pow(r56525, r56526);
        double r56528 = r56524 * r56527;
        double r56529 = pow(r56522, r56526);
        double r56530 = r56528 - r56529;
        double r56531 = 3.0;
        double r56532 = pow(r56530, r56531);
        double r56533 = cbrt(r56532);
        double r56534 = fma(r56523, r56522, r56533);
        return r56534;
}

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 2020081 +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))))