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 r69477 = 9.0;
        double r69478 = x;
        double r69479 = 4.0;
        double r69480 = pow(r69478, r69479);
        double r69481 = r69477 * r69480;
        double r69482 = y;
        double r69483 = pow(r69482, r69479);
        double r69484 = r69481 - r69483;
        double r69485 = 2.0;
        double r69486 = r69482 * r69482;
        double r69487 = r69485 * r69486;
        double r69488 = r69484 + r69487;
        return r69488;
}
↓
double f(double x, double y) {
        double r69489 = 2.0;
        double r69490 = y;
        double r69491 = r69489 * r69490;
        double r69492 = 9.0;
        double r69493 = x;
        double r69494 = 4.0;
        double r69495 = pow(r69493, r69494);
        double r69496 = r69492 * r69495;
        double r69497 = pow(r69490, r69494);
        double r69498 = r69496 - r69497;
        double r69499 = 3.0;
        double r69500 = pow(r69498, r69499);
        double r69501 = cbrt(r69500);
        double r69502 = fma(r69491, r69490, r69501);
        return r69502;
}

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 2020039 +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))))