Average Error: 62.0 → 0
Time: 2.1s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\mathsf{fma}\left({y}^{2}, 2, \mathsf{fma}\left(\sqrt{9}, \sqrt{{x}^{4}}, \sqrt{{y}^{4}}\right) \cdot \left(\sqrt{9} \cdot \sqrt{{x}^{4}} - \sqrt{{y}^{4}}\right)\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left({y}^{2}, 2, \mathsf{fma}\left(\sqrt{9}, \sqrt{{x}^{4}}, \sqrt{{y}^{4}}\right) \cdot \left(\sqrt{9} \cdot \sqrt{{x}^{4}} - \sqrt{{y}^{4}}\right)\right)
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) fma(((double) pow(y, 2.0)), 2.0, ((double) (((double) fma(((double) sqrt(9.0)), ((double) sqrt(((double) pow(x, 4.0)))), ((double) sqrt(((double) pow(y, 4.0)))))) * ((double) (((double) (((double) sqrt(9.0)) * ((double) sqrt(((double) pow(x, 4.0)))))) - ((double) sqrt(((double) pow(y, 4.0))))))))));
}

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

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

  5. Applied add-sqr-sqrt52.0

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

  6. Applied add-sqr-sqrt52.0

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

  7. Applied add-sqr-sqrt52.0

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

  8. Applied unswap-sqr52.0

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

  9. Applied difference-of-squares0

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

  10. Simplified0

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

  11. Final simplification0

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

Reproduce

herbie shell --seed 2020114 +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))))