Average Error: 62.0 → 52.0
Time: 1.7s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r68475 = 9.0;
        double r68476 = x;
        double r68477 = 4.0;
        double r68478 = pow(r68476, r68477);
        double r68479 = r68475 * r68478;
        double r68480 = y;
        double r68481 = r68480 * r68480;
        double r68482 = 2.0;
        double r68483 = r68481 - r68482;
        double r68484 = r68481 * r68483;
        double r68485 = r68479 - r68484;
        return r68485;
}
↓
double f(double x, double y) {
        double r68486 = y;
        double r68487 = 2.0;
        double r68488 = pow(r68486, r68487);
        double r68489 = 2.0;
        double r68490 = 9.0;
        double r68491 = x;
        double r68492 = 4.0;
        double r68493 = pow(r68491, r68492);
        double r68494 = r68490 * r68493;
        double r68495 = pow(r68486, r68492);
        double r68496 = r68494 - r68495;
        double r68497 = 3.0;
        double r68498 = pow(r68496, r68497);
        double r68499 = cbrt(r68498);
        double r68500 = fma(r68488, r68489, r68499);
        return r68500;
}

Error

Derivation

  1. Initial program 62.0

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

  2. Taylor expanded around 0 62.0

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

  3. Simplified52.0

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

  5. Applied add-cbrt-cube52.0

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

  6. Simplified52.0

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

  7. Final simplification52.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (- (* 9 (pow x 4)) (* (* y y) (- (* y y) 2))))