Average Error: 0.9 → 0.2
Time: 18.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\right)
double f(double re, double im) {
        double r1484608 = im;
        double r1484609 = re;
        double r1484610 = atan2(r1484608, r1484609);
        double r1484611 = 10.0;
        double r1484612 = log(r1484611);
        double r1484613 = r1484610 / r1484612;
        return r1484613;
}
↓
double f(double re, double im) {
        double r1484614 = 1.0;
        double r1484615 = 10.0;
        double r1484616 = log(r1484615);
        double r1484617 = sqrt(r1484616);
        double r1484618 = r1484614 / r1484617;
        double r1484619 = sqrt(r1484618);
        double r1484620 = im;
        double r1484621 = re;
        double r1484622 = atan2(r1484620, r1484621);
        double r1484623 = r1484619 * r1484622;
        double r1484624 = cbrt(r1484619);
        double r1484625 = r1484623 * r1484624;
        double r1484626 = r1484624 * r1484624;
        double r1484627 = r1484625 * r1484626;
        double r1484628 = r1484618 * r1484627;
        return r1484628;
}

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. Applied associate-*r*0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied associate-*l*0.8

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied associate-*l*0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))