Average Error: 0.5 → 0.5
Time: 20.0s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \sqrt{2} \cdot \left(\left(\sin x - \frac{\sin y}{16}\right) \cdot \left(\left(\sin y - \frac{\sin x}{16}\right) \cdot \left(\cos x - \cos y\right)\right)\right)}{3 \cdot \left(1 + \frac{\frac{\cos x \cdot \left(\left(3 + \sqrt{5}\right) \cdot \left(5 \cdot \sqrt{5} - {1}^{3}\right)\right) + \cos y \cdot \left(\left(3 \cdot 3 - 5\right) \cdot \left(5 + 1 \cdot \left(1 + \sqrt{5}\right)\right)\right)}{e^{\log \left(\left(3 + \sqrt{5}\right) \cdot \left(5 + 1 \cdot \left(1 + \sqrt{5}\right)\right)\right)}}}{2}\right)}\]
\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{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 + \frac{\frac{\cos x \cdot \left(\left(3 + \sqrt{5}\right) \cdot \left(5 \cdot \sqrt{5} - {1}^{3}\right)\right) + \cos y \cdot \left(\left(3 \cdot 3 - 5\right) \cdot \left(5 + 1 \cdot \left(1 + \sqrt{5}\right)\right)\right)}{e^{\log \left(\left(3 + \sqrt{5}\right) \cdot \left(5 + 1 \cdot \left(1 + \sqrt{5}\right)\right)\right)}}}{2}\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) (((double) sqrt(2.0)) * ((double) (((double) (((double) sin(x)) - ((double) (((double) sin(y)) / 16.0)))) * ((double) (((double) (((double) sin(y)) - ((double) (((double) sin(x)) / 16.0)))) * ((double) (((double) cos(x)) - ((double) cos(y)))))))))))) / ((double) (3.0 * ((double) (1.0 + ((double) (((double) (((double) (((double) (((double) cos(x)) * ((double) (((double) (3.0 + ((double) sqrt(5.0)))) * ((double) (((double) (5.0 * ((double) sqrt(5.0)))) - ((double) pow(1.0, 3.0)))))))) + ((double) (((double) cos(y)) * ((double) (((double) (((double) (3.0 * 3.0)) - 5.0)) * ((double) (5.0 + ((double) (1.0 * ((double) (1.0 + ((double) sqrt(5.0)))))))))))))) / ((double) exp(((double) log(((double) (((double) (3.0 + ((double) sqrt(5.0)))) * ((double) (5.0 + ((double) (1.0 * ((double) (1.0 + ((double) sqrt(5.0)))))))))))))))) / 2.0))))))));
}

Error

Bits error versus x

Bits error versus y

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Results

Enter valid numbers for all inputs

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

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

  4. Applied flip--0.5

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

  5. Applied associate-*r/0.5

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

  6. Applied flip3--0.7

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

  7. Applied associate-*r/0.6

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

  8. Applied frac-add0.6

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

  9. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-exp-log0.4

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

  13. Applied add-exp-log0.5

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

  14. Applied prod-exp0.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020181 
(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))))))