Average Error: 52.0 → 0
Time: 1.6s
Precision: 64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]
↓
\[\mathsf{fma}\left(2 \cdot y, y, \mathsf{fma}\left(\sqrt{9}, {x}^{\left(\frac{4}{2}\right)}, {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)\right)\]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\mathsf{fma}\left(2 \cdot y, y, \mathsf{fma}\left(\sqrt{9}, {x}^{\left(\frac{4}{2}\right)}, {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)\right)
double code(double x, double y) {
	return ((double) (((double) (((double) (9.0 * ((double) pow(x, 4.0)))) - ((double) pow(y, 4.0)))) + ((double) (2.0 * ((double) (y * y))))));
}
↓
double code(double x, double y) {
	return ((double) fma(((double) (2.0 * y)), y, ((double) (((double) fma(((double) sqrt(9.0)), ((double) pow(x, ((double) (4.0 / 2.0)))), ((double) pow(y, ((double) (4.0 / 2.0)))))) * ((double) (((double) (((double) sqrt(9.0)) * ((double) pow(x, ((double) (4.0 / 2.0)))))) - ((double) pow(y, ((double) (4.0 / 2.0))))))))));
}

Error

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Results

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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 sqr-pow52.0

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

  5. Applied sqr-pow52.0

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

  6. Applied add-sqr-sqrt52.0

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

  7. Applied unswap-sqr52.0

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

  8. Applied difference-of-squares0

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

  9. Simplified0

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

  10. Final simplification0

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

Reproduce

herbie shell --seed 2020113 +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))))