Average Error: 52.0 → 0
Time: 1.9s
Precision: 64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]
↓
\[\mathsf{fma}\left(\sqrt{9}, \sqrt{{x}^{4}}, {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9} \cdot \sqrt{{x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right)\]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\mathsf{fma}\left(\sqrt{9}, \sqrt{{x}^{4}}, {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9} \cdot \sqrt{{x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right)
double code(double x, double y) {
	return (((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y)));
}
↓
double code(double x, double y) {
	return ((fma(sqrt(9.0), sqrt(pow(x, 4.0)), pow(y, (4.0 / 2.0))) * ((sqrt(9.0) * sqrt(pow(x, 4.0))) - pow(y, (4.0 / 2.0)))) + (2.0 * (y * y)));
}

Error

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Your Program's Arguments

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. Using strategy rm

  3. Applied sqr-pow52.0

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

  4. Applied add-sqr-sqrt52.0

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

  5. Applied add-sqr-sqrt52.0

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

  6. Applied unswap-sqr52.0

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

  7. Applied difference-of-squares0

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

  8. Simplified0

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

  9. Final simplification0

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

Reproduce

herbie shell --seed 2020066 +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))))