Average Error: 35.6 → 27.4
Time: 6.1s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 2.1006546564861388:\\ \;\;\;\;\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
\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 2.1006546564861388:\\
\;\;\;\;\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\\

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

\end{array}
double f(double x, double y) {
        double r746503 = x;
        double r746504 = y;
        double r746505 = 2.0;
        double r746506 = r746504 * r746505;
        double r746507 = r746503 / r746506;
        double r746508 = tan(r746507);
        double r746509 = sin(r746507);
        double r746510 = r746508 / r746509;
        return r746510;
}
↓
double f(double x, double y) {
        double r746511 = x;
        double r746512 = y;
        double r746513 = 2.0;
        double r746514 = r746512 * r746513;
        double r746515 = r746511 / r746514;
        double r746516 = tan(r746515);
        double r746517 = sin(r746515);
        double r746518 = r746516 / r746517;
        double r746519 = 2.100654656486139;
        bool r746520 = r746518 <= r746519;
        double r746521 = cbrt(r746516);
        double r746522 = r746521 * r746521;
        double r746523 = cbrt(r746517);
        double r746524 = r746523 * r746523;
        double r746525 = r746522 / r746524;
        double r746526 = r746521 / r746523;
        double r746527 = r746525 * r746526;
        double r746528 = 1.0;
        double r746529 = r746520 ? r746527 : r746528;
        return r746529;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.6
Target28.8
Herbie27.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.23036909113069936 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.1028524068119138 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))) < 2.100654656486139

    1. Initial program 24.2

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

    3. Applied add-cube-cbrt24.9

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

    4. Applied add-cube-cbrt24.2

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

    5. Applied times-frac24.2

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

    if 2.100654656486139 < (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0))))

    1. Initial program 62.3

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

    2. Taylor expanded around 0 34.8

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

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \le 2.1006546564861388:\\ \;\;\;\;\frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)} \cdot \sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}} \cdot \frac{\sqrt[3]{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sqrt[3]{\sin \left(\frac{x}{y \cdot 2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))))