math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 3.4s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r34702 = im;
        double r34703 = re;
        double r34704 = atan2(r34702, r34703);
        double r34705 = 10.0;
        double r34706 = log(r34705);
        double r34707 = r34704 / r34706;
        return r34707;
}

↓

double f(double re, double im) {
        double r34708 = im;
        double r34709 = re;
        double r34710 = atan2(r34708, r34709);
        double r34711 = 1.0;
        double r34712 = 10.0;
        double r34713 = log(r34712);
        double r34714 = sqrt(r34713);
        double r34715 = r34711 / r34714;
        double r34716 = cbrt(r34715);
        double r34717 = r34716 * r34716;
        double r34718 = r34710 * r34717;
        double r34719 = cbrt(r34714);
        double r34720 = r34714 * r34719;
        double r34721 = r34718 / r34720;
        return r34721;
}

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. Using strategy rm

9. Applied add-cube-cbrt 0.8

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

10. Applied associate-*r* 0.9

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

12. Applied cbrt-div 0.8

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

13. Applied associate-*r/ 0.1

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

14. Applied frac-times 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019352 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))