Average Error: 62.0 → 52.0
Time: 1.3s
Precision: binary64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[y \cdot \left(y \cdot 2\right) + \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
y \cdot \left(y \cdot 2\right) + \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}
double code(double x, double y) {
	return ((double) (((double) (9.0 * ((double) pow(x, 4.0)))) - ((double) (((double) (y * y)) * ((double) (((double) (y * y)) - 2.0))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (y * ((double) (y * 2.0)))) + ((double) cbrt(((double) pow(((double) (((double) (9.0 * ((double) pow(x, 4.0)))) - ((double) pow(y, 4.0)))), 3.0))))));
}

Error

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Results

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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

    \[\leadsto \color{blue}{y \cdot \left(y \cdot 2\right) + \left(9 \cdot {x}^{4} - {y}^{4}\right)}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube52.0

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

  6. Simplified52.0

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

  7. Final simplification52.0

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))