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

Average Error: 35.4 → 27.5

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \le 2.69286171955532038779779213655274361372:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{\frac{x}{y}}{2}\right)}{\sin \left(\frac{\frac{x}{y}}{2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r436220 = x;
        double r436221 = y;
        double r436222 = 2.0;
        double r436223 = r436221 * r436222;
        double r436224 = r436220 / r436223;
        double r436225 = tan(r436224);
        double r436226 = sin(r436224);
        double r436227 = r436225 / r436226;
        return r436227;
}

↓

double f(double x, double y) {
        double r436228 = x;
        double r436229 = 2.0;
        double r436230 = y;
        double r436231 = r436229 * r436230;
        double r436232 = r436228 / r436231;
        double r436233 = tan(r436232);
        double r436234 = sin(r436232);
        double r436235 = r436233 / r436234;
        double r436236 = 2.6928617195553204;
        bool r436237 = r436235 <= r436236;
        double r436238 = r436228 / r436230;
        double r436239 = r436238 / r436229;
        double r436240 = tan(r436239);
        double r436241 = sin(r436239);
        double r436242 = r436240 / r436241;
        double r436243 = expm1(r436242);
        double r436244 = expm1(r436243);
        double r436245 = log1p(r436244);
        double r436246 = log1p(r436245);
        double r436247 = 1.0;
        double r436248 = r436237 ? r436246 : r436247;
        return r436248;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.4
Target	28.9
Herbie	27.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 2.6928617195553204

   1. Initial program 24.9

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

   2. Simplified 24.9

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

   4. Applied log1p-expm1-u 24.9

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)}\right)\right)}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 45.7

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sqrt[3]{\left(\sin \left(\frac{x}{2 \cdot y}\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)}}}\right)\right)\]

   7. Applied add-cbrt-cube 45.2

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\sqrt[3]{\left(\tan \left(\frac{x}{2 \cdot y}\right) \cdot \tan \left(\frac{x}{2 \cdot y}\right)\right) \cdot \tan \left(\frac{x}{2 \cdot y}\right)}}}{\sqrt[3]{\left(\sin \left(\frac{x}{2 \cdot y}\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)}}\right)\right)\]

   8. Applied cbrt-undiv 45.2

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt[3]{\frac{\left(\tan \left(\frac{x}{2 \cdot y}\right) \cdot \tan \left(\frac{x}{2 \cdot y}\right)\right) \cdot \tan \left(\frac{x}{2 \cdot y}\right)}{\left(\sin \left(\frac{x}{2 \cdot y}\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)\right) \cdot \sin \left(\frac{x}{2 \cdot y}\right)}}}\right)\right)\]

   9. Simplified 24.9

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\color{blue}{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}}\right)\right)\]
   10. Using strategy rm

   11. Applied log1p-expm1-u 24.9

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\right)\right)\right)}\right)\]

   12. Simplified 24.9

       \[\leadsto \mathsf{log1p}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{\frac{x}{y}}{2}\right)}{\sin \left(\frac{\frac{x}{y}}{2}\right)}\right)\right)}\right)\right)\]

   if 2.6928617195553204 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 62.7

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

   2. Simplified 62.7

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

   3. Taylor expanded around 0 34.1

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

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\frac{x}{2 \cdot y}\right)} \le 2.69286171955532038779779213655274361372:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{\frac{x}{y}}{2}\right)}{\sin \left(\frac{\frac{x}{y}}{2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"

  :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)))))