Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5

Average Error: 36.3 → 28.5

Time: 6.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 2 \le -1.57072877925321994 \cdot 10^{-145}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{\log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\right)}^{3}}\\ \mathbf{elif}\;y \cdot 2 \le 2.99395177104279672 \cdot 10^{-44}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{\sqrt[3]{{\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)}^{3}}}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double y) {
        double r529652 = x;
        double r529653 = y;
        double r529654 = 2.0;
        double r529655 = r529653 * r529654;
        double r529656 = r529652 / r529655;
        double r529657 = tan(r529656);
        double r529658 = sin(r529656);
        double r529659 = r529657 / r529658;
        return r529659;
}

↓

double f(double x, double y) {
        double r529660 = y;
        double r529661 = 2.0;
        double r529662 = r529660 * r529661;
        double r529663 = -1.57072877925322e-145;
        bool r529664 = r529662 <= r529663;
        double r529665 = 1.0;
        double r529666 = x;
        double r529667 = r529666 / r529662;
        double r529668 = cos(r529667);
        double r529669 = exp(r529668);
        double r529670 = log(r529669);
        double r529671 = r529665 / r529670;
        double r529672 = 3.0;
        double r529673 = pow(r529671, r529672);
        double r529674 = cbrt(r529673);
        double r529675 = 2.9939517710427967e-44;
        bool r529676 = r529662 <= r529675;
        double r529677 = 1.0;
        double r529678 = pow(r529668, r529672);
        double r529679 = cbrt(r529678);
        double r529680 = r529665 / r529679;
        double r529681 = pow(r529680, r529672);
        double r529682 = cbrt(r529681);
        double r529683 = r529676 ? r529677 : r529682;
        double r529684 = r529664 ? r529674 : r529683;
        return r529684;
}

Error

Bits error versus x

Bits error versus y

Target

Original	36.3
Target	29.5
Herbie	28.5

\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* y 2.0) < -1.57072877925322e-145

   1. Initial program 30.4

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

   3. Applied tan-quot 30.4

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

   4. Applied associate-/l/ 30.4

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

   6. Applied add-cbrt-cube 30.4

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

   7. Applied add-cbrt-cube 50.0

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

   8. Applied cbrt-unprod 50.0

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

   9. Applied add-cbrt-cube 49.8

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

   10. Applied cbrt-undiv 49.8

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

   11. Simplified 20.7

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

   13. Applied add-log-exp 20.7

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

   if -1.57072877925322e-145 < (* y 2.0) < 2.9939517710427967e-44

   1. Initial program 48.3

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

   2. Taylor expanded around 0 46.9

      \[\leadsto \color{blue}{1}\]

   if 2.9939517710427967e-44 < (* y 2.0)

   1. Initial program 28.6

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

   3. Applied tan-quot 28.6

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

   4. Applied associate-/l/ 28.6

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

   6. Applied add-cbrt-cube 28.6

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

   7. Applied add-cbrt-cube 50.8

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

   8. Applied cbrt-unprod 50.8

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

   9. Applied add-cbrt-cube 50.4

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

   10. Applied cbrt-undiv 50.4

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

   11. Simplified 15.5

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

   13. Applied add-cbrt-cube 15.5

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

   14. Simplified 15.5

       \[\leadsto \sqrt[3]{{\left(\frac{1}{\sqrt[3]{\color{blue}{{\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)}^{3}}}}\right)}^{3}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 28.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 2 \le -1.57072877925321994 \cdot 10^{-145}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{\log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\right)}^{3}}\\ \mathbf{elif}\;y \cdot 2 \le 2.99395177104279672 \cdot 10^{-44}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{1}{\sqrt[3]{{\left(\cos \left(\frac{x}{y \cdot 2}\right)\right)}^{3}}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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 (if (< y -9.102852406811914e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

  (/ (tan (/ x (* y 2))) (sin (/ x (* y 2)))))