Average Error: 36.1 → 28.8
Time: 10.3s
Precision: binary64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{{\cos \left(\frac{x}{y \cdot 2}\right)}^{3}}}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{{\cos \left(\frac{x}{y \cdot 2}\right)}^{3}}}}}\right)\right) \cdot \sqrt[3]{\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}}
(FPCore (x y)
 :precision binary64
 (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))
↓
(FPCore (x y)
 :precision binary64
 (*
  (*
   (*
    (cbrt
     (/ 1.0 (* (cbrt (cos (/ x (* y 2.0)))) (cbrt (cos (/ x (* y 2.0)))))))
    (cbrt
     (/ 1.0 (* (cbrt (cos (/ x (* y 2.0)))) (cbrt (cos (/ x (* y 2.0))))))))
   (*
    (cbrt (/ 1.0 (cbrt (cos (/ x (* y 2.0))))))
    (cbrt (/ 1.0 (cbrt (cbrt (pow (cos (/ x (* y 2.0))) 3.0)))))))
  (cbrt (/ 1.0 (cos (/ x (* y 2.0)))))))
double code(double x, double y) {
	return tan(x / (y * 2.0)) / sin(x / (y * 2.0));
}
↓
double code(double x, double y) {
	return ((cbrt(1.0 / (cbrt(cos(x / (y * 2.0))) * cbrt(cos(x / (y * 2.0))))) * cbrt(1.0 / (cbrt(cos(x / (y * 2.0))) * cbrt(cos(x / (y * 2.0)))))) * (cbrt(1.0 / cbrt(cos(x / (y * 2.0)))) * cbrt(1.0 / cbrt(cbrt(pow(cos(x / (y * 2.0)), 3.0)))))) * cbrt(1.0 / cos(x / (y * 2.0)));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target29.1
Herbie28.8
\[\begin{array}{l} \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 36.1

    \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
  2. Using strategy rm

  3. Applied tan-quot_binary64_2774936.1

    \[\leadsto \frac{\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\cos \left(\frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt_binary64_2785336.1

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

  6. Simplified36.1

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

  7. Simplified28.8

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

  9. Applied add-cube-cbrt_binary64_2785328.8

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

  10. Applied add-sqr-sqrt_binary64_2786728.8

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

  11. Applied times-frac_binary64_2787728.8

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

  12. Applied cbrt-prod_binary64_2784928.8

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

  13. Applied add-cube-cbrt_binary64_2785328.8

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

  14. Applied add-sqr-sqrt_binary64_2786728.8

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

  15. Applied times-frac_binary64_2787728.8

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

  16. Applied cbrt-prod_binary64_2784928.8

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

  17. Applied swap-sqr_binary64_2790928.8

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

  18. Simplified28.8

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

  19. Simplified28.8

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

  21. Applied add-cbrt-cube_binary64_2785428.8

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

  22. Simplified28.8

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

  23. Final simplification28.8

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

Reproduce

herbie shell --seed 2020270 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.2303690911306994e+114) 1.0 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2.0))) (* (sin (/ x (* y 2.0))) (log (exp (cos (/ x (* y 2.0))))))) 1.0))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))