Average Error: 35.2 → 28.2
Time: 22.9s
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 r25041226 = x;
        double r25041227 = y;
        double r25041228 = 2.0;
        double r25041229 = r25041227 * r25041228;
        double r25041230 = r25041226 / r25041229;
        double r25041231 = tan(r25041230);
        double r25041232 = sin(r25041230);
        double r25041233 = r25041231 / r25041232;
        return r25041233;
}
↓
double f(double x, double y) {
        double r25041234 = 1.0;
        double r25041235 = x;
        double r25041236 = y;
        double r25041237 = r25041235 / r25041236;
        double r25041238 = 2.0;
        double r25041239 = r25041237 / r25041238;
        double r25041240 = cos(r25041239);
        double r25041241 = r25041234 / r25041240;
        double r25041242 = r25041241 * r25041241;
        double r25041243 = r25041241 * r25041242;
        double r25041244 = cbrt(r25041243);
        double r25041245 = expm1(r25041244);
        double r25041246 = log1p(r25041245);
        return r25041246;
}

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)))))