math.log10 on complex, imaginary part

Average Error: 0.8 → 0.2

Time: 17.7s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r1606732 = im;
        double r1606733 = re;
        double r1606734 = atan2(r1606732, r1606733);
        double r1606735 = 10.0;
        double r1606736 = log(r1606735);
        double r1606737 = r1606734 / r1606736;
        return r1606737;
}

↓

double f(double re, double im) {
        double r1606738 = 1.0;
        double r1606739 = 10.0;
        double r1606740 = log(r1606739);
        double r1606741 = sqrt(r1606740);
        double r1606742 = r1606738 / r1606741;
        double r1606743 = sqrt(r1606742);
        double r1606744 = cbrt(r1606743);
        double r1606745 = r1606744 * r1606744;
        double r1606746 = im;
        double r1606747 = re;
        double r1606748 = atan2(r1606746, r1606747);
        double r1606749 = r1606748 * r1606742;
        double r1606750 = r1606743 * r1606749;
        double r1606751 = r1606745 * r1606750;
        double r1606752 = r1606744 * r1606751;
        return r1606752;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.8

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

3. Applied add-sqr-sqrt 0.8

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

4. Applied *-un-lft-identity 0.8

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

5. Applied times-frac 0.8

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

7. Applied div-inv 0.8

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

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-sqrt 0.8

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

11. Applied associate-*r* 0.8

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

13. Applied add-cube-cbrt 0.1

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

14. Applied associate-*r* 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019163 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))