Average Error: 35.5 → 28.0
Time: 5.5s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 1.012741487861675:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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}
double f(double x, double y) {
        double r1192901 = x;
        double r1192902 = y;
        double r1192903 = 2.0;
        double r1192904 = r1192902 * r1192903;
        double r1192905 = r1192901 / r1192904;
        double r1192906 = tan(r1192905);
        double r1192907 = sin(r1192905);
        double r1192908 = r1192906 / r1192907;
        return r1192908;
}
↓
double f(double x, double y) {
        double r1192909 = x;
        double r1192910 = y;
        double r1192911 = 2.0;
        double r1192912 = r1192910 * r1192911;
        double r1192913 = r1192909 / r1192912;
        double r1192914 = tan(r1192913);
        double r1192915 = sin(r1192913);
        double r1192916 = r1192914 / r1192915;
        double r1192917 = 1.0127414878616747;
        bool r1192918 = r1192916 <= r1192917;
        double r1192919 = expm1(r1192916);
        double r1192920 = expm1(r1192919);
        double r1192921 = log1p(r1192920);
        double r1192922 = log1p(r1192921);
        double r1192923 = 1.0;
        double r1192924 = r1192918 ? r1192922 : r1192923;
        return r1192924;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.5
Target29.0
Herbie28.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

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-u19.1

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

    5. Applied log1p-expm1-u19.1

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

    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 simplification28.0

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

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)))))