Average Error: 0.8 → 0.8
Time: 13.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\sqrt{\frac{\sqrt{1}}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{\sqrt{\frac{\sqrt{1}}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}
double f(double re, double im) {
        double r41693 = im;
        double r41694 = re;
        double r41695 = atan2(r41693, r41694);
        double r41696 = 10.0;
        double r41697 = log(r41696);
        double r41698 = r41695 / r41697;
        return r41698;
}
↓
double f(double re, double im) {
        double r41699 = 1.0;
        double r41700 = sqrt(r41699);
        double r41701 = 10.0;
        double r41702 = log(r41701);
        double r41703 = r41700 / r41702;
        double r41704 = sqrt(r41703);
        double r41705 = im;
        double r41706 = re;
        double r41707 = atan2(r41705, r41706);
        double r41708 = r41704 * r41707;
        double r41709 = sqrt(r41702);
        double r41710 = r41708 / r41709;
        return r41710;
}

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. Taylor expanded around 0 0.8

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

  8. Applied pow10.8

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

  9. Applied log-pow0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied times-frac0.8

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

  12. Applied sqrt-prod0.8

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

  13. Applied associate-*r*0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2019291 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))