Average Error: 36.1 → 28.9
Time: 26.2s
Precision: 64
\[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}\]
↓
\[\log \left(1 + \mathsf{expm1}\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\]
\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)}
↓
\log \left(1 + \mathsf{expm1}\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)
double f(double x, double y) {
        double r412621 = x;
        double r412622 = y;
        double r412623 = 2.0;
        double r412624 = r412622 * r412623;
        double r412625 = r412621 / r412624;
        double r412626 = tan(r412625);
        double r412627 = sin(r412625);
        double r412628 = r412626 / r412627;
        return r412628;
}
↓
double f(double x, double y) {
        double r412629 = 1.0;
        double r412630 = x;
        double r412631 = y;
        double r412632 = 2.0;
        double r412633 = r412631 * r412632;
        double r412634 = r412630 / r412633;
        double r412635 = cos(r412634);
        double r412636 = r412629 / r412635;
        double r412637 = expm1(r412636);
        double r412638 = r412629 + r412637;
        double r412639 = log(r412638);
        return r412639;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.1
Target29.3
Herbie28.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.230369091130699363447511617672816900781 \cdot 10^{114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \lt -9.102852406811913849731222630299032206502 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sin \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right) \cdot \log \left(e^{\cos \left(\frac{x}{y \cdot 2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Derivation

  1. Initial program 36.1

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

  3. Applied tan-quot36.1

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

  4. Applied associate-/l/36.1

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

  6. Applied log1p-expm1-u36.1

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

  7. Simplified28.9

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

  9. Applied log1p-udef28.9

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

  10. Final simplification28.9

    \[\leadsto \log \left(1 + \mathsf{expm1}\left(\frac{1}{\cos \left(\frac{x}{y \cdot 2}\right)}\right)\right)\]

Reproduce

herbie shell --seed 2019303 +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 -1.23036909113069936e114) 1 (if (< y -9.1028524068119138e-222) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1))

  (/ (tan (/ x (* y 2))) (sin (/ x (* y 2)))))