Average Error: 35.2 → 28.2
Time: 23.6s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \left(\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}\right)}\right)\right)\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \left(\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}\right)}\right)\right)
double f(double x, double y) {
        double r28300038 = x;
        double r28300039 = y;
        double r28300040 = 2.0;
        double r28300041 = r28300039 * r28300040;
        double r28300042 = r28300038 / r28300041;
        double r28300043 = tan(r28300042);
        double r28300044 = sin(r28300042);
        double r28300045 = r28300043 / r28300044;
        return r28300045;
}
↓
double f(double x, double y) {
        double r28300046 = 1.0;
        double r28300047 = x;
        double r28300048 = y;
        double r28300049 = r28300047 / r28300048;
        double r28300050 = 2.0;
        double r28300051 = r28300049 / r28300050;
        double r28300052 = cos(r28300051);
        double r28300053 = r28300046 / r28300052;
        double r28300054 = r28300053 * r28300053;
        double r28300055 = r28300053 * r28300054;
        double r28300056 = cbrt(r28300055);
        double r28300057 = expm1(r28300056);
        double r28300058 = log1p(r28300057);
        return r28300058;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original35.2
Target28.5
Herbie28.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.230369091130699363447511617672816900781 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.102852406811913849731222630299032206502 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 35.2

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

  3. Applied tan-quot35.2

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

  5. Applied log1p-expm1-u35.3

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

  6. Simplified28.2

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}\right)}\right)\]
  7. Using strategy rm

  8. Applied add-cbrt-cube28.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt[3]{\left(\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}\right) \cdot \frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}}}\right)\right)\]

  9. Final simplification28.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \left(\frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)} \cdot \frac{1}{\cos \left(\frac{\frac{x}{y}}{2}\right)}\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019172 +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)))))