Average Error: 0.8 → 0.8
Time: 15.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\log 10}}}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\log 10}}}\right) \cdot \frac{1}{\sqrt{\log 10}}
double f(double re, double im) {
        double r38252 = im;
        double r38253 = re;
        double r38254 = atan2(r38252, r38253);
        double r38255 = 10.0;
        double r38256 = log(r38255);
        double r38257 = r38254 / r38256;
        return r38257;
}
↓
double f(double re, double im) {
        double r38258 = im;
        double r38259 = re;
        double r38260 = atan2(r38258, r38259);
        double r38261 = cbrt(r38260);
        double r38262 = 10.0;
        double r38263 = log(r38262);
        double r38264 = sqrt(r38263);
        double r38265 = cbrt(r38264);
        double r38266 = sqrt(r38265);
        double r38267 = r38266 * r38266;
        double r38268 = r38261 / r38267;
        double r38269 = r38261 / r38265;
        double r38270 = r38261 * r38269;
        double r38271 = r38270 / r38265;
        double r38272 = r38268 * r38271;
        double r38273 = 1.0;
        double r38274 = r38273 / r38264;
        double r38275 = r38272 * r38274;
        return r38275;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  4. Applied *-un-lft-identity0.8

    \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.5

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}}\]

  8. Applied add-cube-cbrt0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\left(\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}\]

  9. Applied times-frac1.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}\right)}\]

  10. Simplified0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\color{blue}{\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}\right)\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}}}\right)\]

  13. Final simplification0.8

    \[\leadsto \left(\frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt{\sqrt[3]{\sqrt{\log 10}}} \cdot \sqrt{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}} \cdot \frac{\sqrt[3]{\tan^{-1}_* \frac{im}{re}}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt[3]{\sqrt{\log 10}}}\right) \cdot \frac{1}{\sqrt{\log 10}}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))