math.log10 on complex, imaginary part

Average Error: 0.9 → 0.2

Time: 47.7s

Precision: 64

Math

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10.0}\]

↓

\[\frac{1}{\sqrt{\log 10.0}} \cdot \left(\left(\left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right)\right)\]

C

double f(double re, double im) {
        double r989863 = im;
        double r989864 = re;
        double r989865 = atan2(r989863, r989864);
        double r989866 = 10.0;
        double r989867 = log(r989866);
        double r989868 = r989865 / r989867;
        return r989868;
}

↓

double f(double re, double im) {
        double r989869 = 1.0;
        double r989870 = 10.0;
        double r989871 = log(r989870);
        double r989872 = sqrt(r989871);
        double r989873 = r989869 / r989872;
        double r989874 = sqrt(r989873);
        double r989875 = im;
        double r989876 = re;
        double r989877 = atan2(r989875, r989876);
        double r989878 = r989874 * r989877;
        double r989879 = cbrt(r989874);
        double r989880 = r989878 * r989879;
        double r989881 = r989879 * r989879;
        double r989882 = r989880 * r989881;
        double r989883 = r989873 * r989882;
        return r989883;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.9

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10.0}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.9

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10.0} \cdot \sqrt{\log 10.0}}}\]

4. Applied *-un-lft-identity 0.9

   \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10.0} \cdot \sqrt{\log 10.0}}\]

5. Applied times-frac 0.8

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10.0}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10.0}}}\]
6. Using strategy rm

7. Applied div-inv 0.8

   \[\leadsto \frac{1}{\sqrt{\log 10.0}} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10.0}}\right)}\]

8. Applied associate-*r* 0.8

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{\log 10.0}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\sqrt{\log 10.0}}}\]
9. Using strategy rm

10. Applied add-sqr-sqrt 0.8

    \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10.0}}}\right)} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \frac{1}{\sqrt{\log 10.0}}\]

11. Applied associate-*l* 0.8

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \tan^{-1}_* \frac{im}{re}\right)\right)} \cdot \frac{1}{\sqrt{\log 10.0}}\]
12. Using strategy rm

13. Applied add-cube-cbrt 0.1

    \[\leadsto \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right)} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \tan^{-1}_* \frac{im}{re}\right)\right) \cdot \frac{1}{\sqrt{\log 10.0}}\]

14. Applied associate-*l* 0.2

    \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \tan^{-1}_* \frac{im}{re}\right)\right)\right)} \cdot \frac{1}{\sqrt{\log 10.0}}\]

15. Final simplification 0.2

    \[\leadsto \frac{1}{\sqrt{\log 10.0}} \cdot \left(\left(\left(\sqrt{\frac{1}{\sqrt{\log 10.0}}} \cdot \tan^{-1}_* \frac{im}{re}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\sqrt{\log 10.0}}}}\right)\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))