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 r76368 = 9.0;
        double r76369 = x;
        double r76370 = 4.0;
        double r76371 = pow(r76369, r76370);
        double r76372 = r76368 * r76371;
        double r76373 = y;
        double r76374 = pow(r76373, r76370);
        double r76375 = r76372 - r76374;
        double r76376 = 2.0;
        double r76377 = r76373 * r76373;
        double r76378 = r76376 * r76377;
        double r76379 = r76375 + r76378;
        return r76379;
}
↓
double f(double x, double y) {
        double r76380 = 2.0;
        double r76381 = y;
        double r76382 = r76380 * r76381;
        double r76383 = 9.0;
        double r76384 = x;
        double r76385 = 4.0;
        double r76386 = pow(r76384, r76385);
        double r76387 = r76383 * r76386;
        double r76388 = pow(r76381, r76385);
        double r76389 = r76387 - r76388;
        double r76390 = 3.0;
        double r76391 = pow(r76389, r76390);
        double r76392 = cbrt(r76391);
        double r76393 = fma(r76382, r76381, r76392);
        return r76393;
}

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))))