Average Error: 0.9 → 0.9
Time: 3.2s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\left(\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\left(\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}
double f(double re, double im) {
        double r28664 = im;
        double r28665 = re;
        double r28666 = atan2(r28664, r28665);
        double r28667 = 10.0;
        double r28668 = log(r28667);
        double r28669 = r28666 / r28668;
        return r28669;
}
↓
double f(double re, double im) {
        double r28670 = 1.0;
        double r28671 = 10.0;
        double r28672 = log(r28671);
        double r28673 = sqrt(r28672);
        double r28674 = r28670 / r28673;
        double r28675 = im;
        double r28676 = re;
        double r28677 = atan2(r28675, r28676);
        double r28678 = r28673 / r28677;
        double r28679 = cbrt(r28678);
        double r28680 = r28679 * r28679;
        double r28681 = r28680 * r28679;
        double r28682 = r28670 / r28681;
        double r28683 = r28674 * r28682;
        return r28683;
}

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 clear-num1.0

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

  9. Applied add-cube-cbrt0.9

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

  10. Final simplification0.9

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

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))