math.log10 on complex, imaginary part

Average Error: 0.9 → 0.1

Time: 40.9s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r868910 = im;
        double r868911 = re;
        double r868912 = atan2(r868910, r868911);
        double r868913 = 10.0;
        double r868914 = log(r868913);
        double r868915 = r868912 / r868914;
        return r868915;
}

↓

double f(double re, double im) {
        double r868916 = 1.0;
        double r868917 = 10.0;
        double r868918 = log(r868917);
        double r868919 = sqrt(r868918);
        double r868920 = r868916 / r868919;
        double r868921 = sqrt(r868920);
        double r868922 = sqrt(r868921);
        double r868923 = im;
        double r868924 = re;
        double r868925 = atan2(r868923, r868924);
        double r868926 = r868925 * r868921;
        double r868927 = r868926 * r868922;
        double r868928 = r868922 * r868927;
        double r868929 = r868920 * r868928;
        return r868929;
}

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. Using strategy rm

9. Applied add-sqr-sqrt 0.8

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

10. Applied associate-*r* 0.8

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

12. Applied add-sqr-sqrt 0.8

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

13. Applied sqrt-prod 0.1

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

14. Applied associate-*r* 0.1

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

15. Final simplification 0.1

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

Reproduce

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