Average Error: 0.5 → 0.5
Time: 14.0s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \sqrt[3]{{\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}^{3}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \sqrt[3]{{\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}^{3}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}
double code(double x, double y) {
	return ((double) (((double) (2.0 + ((double) (((double) (((double) (((double) sqrt(2.0)) * ((double) (((double) sin(x)) - ((double) (((double) sin(y)) / 16.0)))))) * ((double) (((double) sin(y)) - ((double) (((double) sin(x)) / 16.0)))))) * ((double) (((double) cos(x)) - ((double) cos(y)))))))) / ((double) (3.0 * ((double) (((double) (1.0 + ((double) (((double) (((double) (((double) sqrt(5.0)) - 1.0)) / 2.0)) * ((double) cos(x)))))) + ((double) (((double) (((double) (3.0 - ((double) sqrt(5.0)))) / 2.0)) * ((double) cos(y))))))))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (2.0 + ((double) cbrt(((double) pow(((double) (((double) sqrt(2.0)) * ((double) (((double) (((double) cos(x)) - ((double) cos(y)))) * ((double) (((double) (((double) sin(x)) - ((double) (((double) sin(y)) / 16.0)))) * ((double) (((double) sin(y)) - ((double) (((double) sin(x)) / 16.0)))))))))), 3.0)))))) / ((double) (3.0 * ((double) (1.0 + ((double) (((double) (((double) (((double) cos(x)) * ((double) sqrt(((double) (((double) sqrt(5.0)) - 1.0)))))) * ((double) (((double) sqrt(((double) (((double) sqrt(5.0)) - 1.0)))) / 2.0)))) + ((double) (((double) cos(y)) * ((double) (((double) (((double) (((double) (3.0 * 3.0)) - 5.0)) / ((double) (3.0 + ((double) sqrt(5.0)))))) / 2.0))))))))))));
}

Error

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Results

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Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{3 - \sqrt{5}}{2}\right)\right)}}\]
  3. Using strategy rm

  4. Applied flip--0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\color{blue}{\frac{3 \cdot 3 - \sqrt{5} \cdot \sqrt{5}}{3 + \sqrt{5}}}}{2}\right)\right)}\]

  5. Simplified0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{2} + \cos y \cdot \frac{\frac{\color{blue}{3 \cdot 3 - 5}}{3 + \sqrt{5}}}{2}\right)\right)}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\sqrt{5} - 1}{\color{blue}{1 \cdot 2}} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  8. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \frac{\color{blue}{\sqrt{\sqrt{5} - 1} \cdot \sqrt{\sqrt{5} - 1}}}{1 \cdot 2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  9. Applied times-frac0.4

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\cos x \cdot \color{blue}{\left(\frac{\sqrt{\sqrt{5} - 1}}{1} \cdot \frac{\sqrt{\sqrt{5} - 1}}{2}\right)} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  10. Applied associate-*r*0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\cos x \cdot \frac{\sqrt{\sqrt{5} - 1}}{1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2}} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  11. Simplified0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \left(\color{blue}{\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right)} \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]
  12. Using strategy rm

  13. Applied add-cbrt-cube0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)}}\right)\right)}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  14. Applied add-cbrt-cube0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}} \cdot \sqrt[3]{\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)}\right)\right)}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  15. Applied cbrt-unprod0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)\right)}}\right)}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  16. Applied add-cbrt-cube0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)}} \cdot \sqrt[3]{\left(\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)\right)}\right)}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  17. Applied cbrt-unprod0.5

    \[\leadsto \frac{2 + \sqrt{2} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  18. Applied add-cbrt-cube0.5

    \[\leadsto \frac{2 + \color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}} \cdot \sqrt[3]{\left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  19. Applied cbrt-unprod0.5

    \[\leadsto \frac{2 + \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right) \cdot \left(\left(\left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\left(\left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\left(\left(\cos x - \cos y\right) \cdot \left(\cos x - \cos y\right)\right) \cdot \left(\cos x - \cos y\right)\right)\right)\right)}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  20. Simplified0.5

    \[\leadsto \frac{2 + \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}^{3}}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

  21. Final simplification0.5

    \[\leadsto \frac{2 + \sqrt[3]{{\left(\sqrt{2} \cdot \left(\left(\cos x - \cos y\right) \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right)\right)\right)}^{3}}}{3 \cdot \left(1 + \left(\left(\cos x \cdot \sqrt{\sqrt{5} - 1}\right) \cdot \frac{\sqrt{\sqrt{5} - 1}}{2} + \cos y \cdot \frac{\frac{3 \cdot 3 - 5}{3 + \sqrt{5}}}{2}\right)\right)}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2.0 (* (* (* (sqrt 2.0) (- (sin x) (/ (sin y) 16.0))) (- (sin y) (/ (sin x) 16.0))) (- (cos x) (cos y)))) (* 3.0 (+ (+ 1.0 (* (/ (- (sqrt 5.0) 1.0) 2.0) (cos x))) (* (/ (- 3.0 (sqrt 5.0)) 2.0) (cos y))))))