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

Average Error: 36.1 → 28.0

Time: 12.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le \frac{8727737956453137}{4503599627370496}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\left(\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r423501 = x;
        double r423502 = y;
        double r423503 = 2.0;
        double r423504 = r423502 * r423503;
        double r423505 = r423501 / r423504;
        double r423506 = tan(r423505);
        double r423507 = sin(r423505);
        double r423508 = r423506 / r423507;
        return r423508;
}

↓

double f(double x, double y) {
        double r423509 = x;
        double r423510 = y;
        double r423511 = 2.0;
        double r423512 = r423510 * r423511;
        double r423513 = r423509 / r423512;
        double r423514 = tan(r423513);
        double r423515 = sin(r423513);
        double r423516 = r423514 / r423515;
        double r423517 = 8727737956453137.0;
        double r423518 = 4503599627370496.0;
        double r423519 = r423517 / r423518;
        bool r423520 = r423516 <= r423519;
        double r423521 = cos(r423513);
        double r423522 = cbrt(r423521);
        double r423523 = r423522 * r423522;
        double r423524 = r423523 * r423522;
        double r423525 = exp(r423524);
        double r423526 = log(r423525);
        double r423527 = r423515 * r423526;
        double r423528 = r423515 / r423527;
        double r423529 = 1.0;
        double r423530 = r423520 ? r423528 : r423529;
        return r423530;
}

Error

Bits error versus x

Bits error versus y

Target

Original	36.1
Target	29.3
Herbie	28.0

\[\begin{array}{l} \mathbf{if}\;y \lt -1.230369091130699363447511617672816900781 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.102852406811913849731222630299032206502 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 1.937947126429837

   1. Initial program 25.1

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

   3. Applied tan-quot 25.1

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

   4. Applied associate-/l/ 25.1

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

   6. Applied add-log-exp 25.1

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

   8. Applied add-cube-cbrt 25.1

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

   if 1.937947126429837 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 61.9

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

   2. Taylor expanded around 0 34.7

      \[\leadsto \color{blue}{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le \frac{8727737956453137}{4503599627370496}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\left(\sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{x}{y \cdot 2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< y -1.23036909113069936e114) 1 (if (< y -9.1028524068119138e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

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