Average Error: 62.0 → 52.0
Time: 20.6s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\mathsf{fma}\left(y, 2 \cdot y, \sqrt[3]{{\left({x}^{4} \cdot 9 - {y}^{4}\right)}^{3}}\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left(y, 2 \cdot y, \sqrt[3]{{\left({x}^{4} \cdot 9 - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r98263 = 9.0;
        double r98264 = x;
        double r98265 = 4.0;
        double r98266 = pow(r98264, r98265);
        double r98267 = r98263 * r98266;
        double r98268 = y;
        double r98269 = r98268 * r98268;
        double r98270 = 2.0;
        double r98271 = r98269 - r98270;
        double r98272 = r98269 * r98271;
        double r98273 = r98267 - r98272;
        return r98273;
}
↓
double f(double x, double y) {
        double r98274 = y;
        double r98275 = 2.0;
        double r98276 = r98275 * r98274;
        double r98277 = x;
        double r98278 = 4.0;
        double r98279 = pow(r98277, r98278);
        double r98280 = 9.0;
        double r98281 = r98279 * r98280;
        double r98282 = 4.0;
        double r98283 = pow(r98274, r98282);
        double r98284 = r98281 - r98283;
        double r98285 = 3.0;
        double r98286 = pow(r98284, r98285);
        double r98287 = cbrt(r98286);
        double r98288 = fma(r98274, r98276, r98287);
        return r98288;
}

Error

Derivation

  1. Initial program 62.0

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

  2. Simplified62.0

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

  3. Taylor expanded around -inf 64.0

    \[\leadsto \color{blue}{\left(9 \cdot e^{4 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + 2 \cdot {y}^{2}\right) - {y}^{4}}\]

  4. Simplified55.0

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

  6. Applied add-cbrt-cube55.0

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

  7. Simplified52.0

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

  8. Final simplification52.0

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

Reproduce

herbie shell --seed 2019323 +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))))