Average Error: 62.0 → 52.0
Time: 1.9s
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 r67569 = 9.0;
        double r67570 = x;
        double r67571 = 4.0;
        double r67572 = pow(r67570, r67571);
        double r67573 = r67569 * r67572;
        double r67574 = y;
        double r67575 = r67574 * r67574;
        double r67576 = 2.0;
        double r67577 = r67575 - r67576;
        double r67578 = r67575 * r67577;
        double r67579 = r67573 - r67578;
        return r67579;
}
↓
double f(double x, double y) {
        double r67580 = y;
        double r67581 = 2.0;
        double r67582 = pow(r67580, r67581);
        double r67583 = 2.0;
        double r67584 = 9.0;
        double r67585 = x;
        double r67586 = 4.0;
        double r67587 = pow(r67585, r67586);
        double r67588 = r67584 * r67587;
        double r67589 = pow(r67580, r67586);
        double r67590 = r67588 - r67589;
        double r67591 = 3.0;
        double r67592 = pow(r67590, r67591);
        double r67593 = cbrt(r67592);
        double r67594 = fma(r67582, r67583, r67593);
        return r67594;
}

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 2020034 +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))))