Average Error: 0.9 → 0.1
Time: 21.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)
double f(double re, double im) {
        double r40990 = im;
        double r40991 = re;
        double r40992 = atan2(r40990, r40991);
        double r40993 = 10.0;
        double r40994 = log(r40993);
        double r40995 = r40992 / r40994;
        return r40995;
}
↓
double f(double re, double im) {
        double r40996 = 1.0;
        double r40997 = 10.0;
        double r40998 = log(r40997);
        double r40999 = sqrt(r40998);
        double r41000 = r40996 / r40999;
        double r41001 = sqrt(r41000);
        double r41002 = cbrt(r41001);
        double r41003 = r41002 * r41002;
        double r41004 = im;
        double r41005 = re;
        double r41006 = atan2(r41004, r41005);
        double r41007 = r41006 / r40999;
        double r41008 = r41002 * r41007;
        double r41009 = r41003 * r41008;
        double r41010 = r41001 * r41009;
        return r41010;
}

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 add-sqr-sqrt0.8

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

  8. Applied associate-*l*0.9

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

  10. Applied add-cube-cbrt0.1

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

  11. Applied associate-*l*0.1

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

  12. Final simplification0.1

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

Reproduce

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