math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 3.5s

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 r80125 = im;
        double r80126 = re;
        double r80127 = atan2(r80125, r80126);
        double r80128 = 10.0;
        double r80129 = log(r80128);
        double r80130 = r80127 / r80129;
        return r80130;
}

↓

double f(double re, double im) {
        double r80131 = im;
        double r80132 = re;
        double r80133 = atan2(r80131, r80132);
        double r80134 = 1.0;
        double r80135 = 10.0;
        double r80136 = log(r80135);
        double r80137 = sqrt(r80136);
        double r80138 = r80134 / r80137;
        double r80139 = cbrt(r80138);
        double r80140 = r80139 * r80139;
        double r80141 = r80133 * r80140;
        double r80142 = cbrt(r80137);
        double r80143 = r80137 * r80142;
        double r80144 = r80141 / r80143;
        return r80144;
}

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 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))