Average Error: 62.0 → 52.0
Time: 1.8s
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 r62350 = 9.0;
        double r62351 = x;
        double r62352 = 4.0;
        double r62353 = pow(r62351, r62352);
        double r62354 = r62350 * r62353;
        double r62355 = y;
        double r62356 = r62355 * r62355;
        double r62357 = 2.0;
        double r62358 = r62356 - r62357;
        double r62359 = r62356 * r62358;
        double r62360 = r62354 - r62359;
        return r62360;
}
↓
double f(double x, double y) {
        double r62361 = y;
        double r62362 = 2.0;
        double r62363 = pow(r62361, r62362);
        double r62364 = 2.0;
        double r62365 = 9.0;
        double r62366 = x;
        double r62367 = 4.0;
        double r62368 = pow(r62366, r62367);
        double r62369 = r62365 * r62368;
        double r62370 = pow(r62361, r62367);
        double r62371 = r62369 - r62370;
        double r62372 = 3.0;
        double r62373 = pow(r62371, r62372);
        double r62374 = cbrt(r62373);
        double r62375 = fma(r62363, r62364, r62374);
        return r62375;
}

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 2020083 +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))))