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

Average Error: 35.5 → 28.0

Time: 12.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.012741487861675:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

C

double f(double x, double y) {
        double r3771 = x;
        double r3772 = y;
        double r3773 = 2.0;
        double r3774 = r3772 * r3773;
        double r3775 = r3771 / r3774;
        double r3776 = tan(r3775);
        double r3777 = sin(r3775);
        double r3778 = r3776 / r3777;
        return r3778;
}

↓

double f(double x, double y) {
        double r3779 = x;
        double r3780 = y;
        double r3781 = 2.0;
        double r3782 = r3780 * r3781;
        double r3783 = r3779 / r3782;
        double r3784 = tan(r3783);
        double r3785 = sin(r3783);
        double r3786 = r3784 / r3785;
        double r3787 = 1.0127414878616747;
        bool r3788 = r3786 <= r3787;
        double r3789 = expm1(r3786);
        double r3790 = expm1(r3789);
        double r3791 = log1p(r3790);
        double r3792 = log1p(r3791);
        double r3793 = 1.0;
        double r3794 = r3788 ? r3792 : r3793;
        return r3794;
}

Error

Bits error versus x

Bits error versus y

Target

Original	35.5
Target	29.0
Herbie	28.0

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

Derivation

1. Split input into 2 regimes

2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 1.0127414878616747

   1. Initial program 19.0

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

   3. Applied log1p-expm1-u 19.1

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

   5. Applied log1p-expm1-u 19.1

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

   if 1.0127414878616747 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

   1. Initial program 60.1

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

   2. Taylor expanded around 0 41.4

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

4. Final simplification 28.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.012741487861675:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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)))))