Average Error: 0.9 → 0.1
Time: 33.9s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\frac{\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\frac{\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}}{\sqrt{\log 10}} \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}
double f(double re, double im) {
        double r38174 = im;
        double r38175 = re;
        double r38176 = atan2(r38174, r38175);
        double r38177 = 10.0;
        double r38178 = log(r38177);
        double r38179 = r38176 / r38178;
        return r38179;
}
↓
double f(double re, double im) {
        double r38180 = im;
        double r38181 = re;
        double r38182 = atan2(r38180, r38181);
        double r38183 = 1.0;
        double r38184 = 10.0;
        double r38185 = log(r38184);
        double r38186 = sqrt(r38185);
        double r38187 = r38183 / r38186;
        double r38188 = sqrt(r38187);
        double r38189 = r38182 * r38188;
        double r38190 = r38189 / r38186;
        double r38191 = cbrt(r38186);
        double r38192 = r38183 / r38191;
        double r38193 = sqrt(r38192);
        double r38194 = r38190 * r38193;
        double r38195 = r38191 * r38191;
        double r38196 = r38183 / r38195;
        double r38197 = sqrt(r38196);
        double r38198 = r38194 * r38197;
        return r38198;
}

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 0.9

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

  3. Applied add-sqr-sqrt0.9

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

  4. Applied *-un-lft-identity0.9

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

  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 div-inv0.8

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

  9. Applied add-sqr-sqrt0.8

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

  10. Applied associate-*l*0.8

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

  11. Simplified0.9

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied add-sqr-sqrt0.1

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

  15. Applied times-frac0.1

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*l*0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))