Average Error: 35.9 → 27.8
Time: 5.4s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.39022469843956054:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 1.39022469843956054:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;1\\

\end{array}
double f(double x, double y) {
        double r609730 = x;
        double r609731 = y;
        double r609732 = 2.0;
        double r609733 = r609731 * r609732;
        double r609734 = r609730 / r609733;
        double r609735 = tan(r609734);
        double r609736 = sin(r609734);
        double r609737 = r609735 / r609736;
        return r609737;
}
↓
double f(double x, double y) {
        double r609738 = x;
        double r609739 = y;
        double r609740 = 2.0;
        double r609741 = r609739 * r609740;
        double r609742 = r609738 / r609741;
        double r609743 = tan(r609742);
        double r609744 = sin(r609742);
        double r609745 = r609743 / r609744;
        double r609746 = 1.3902246984395605;
        bool r609747 = r609745 <= r609746;
        double r609748 = 3.0;
        double r609749 = pow(r609745, r609748);
        double r609750 = cbrt(r609749);
        double r609751 = 1.0;
        double r609752 = r609747 ? r609750 : r609751;
        return r609752;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.9
Target29.0
Herbie27.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 1.3902246984395605

    1. Initial program 23.2

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

    3. Applied log1p-expm1-u23.3

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

    5. Applied add-cbrt-cube45.2

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

    6. Applied add-cbrt-cube44.8

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

    7. Applied cbrt-undiv44.8

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

    8. Simplified23.3

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

    10. Applied log1p-expm123.2

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

    if 1.3902246984395605 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

    1. Initial program 61.4

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

    2. Taylor expanded around 0 36.9

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

  4. Final simplification27.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.39022469843956054:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))))