math.log10 on complex, imaginary part

Average Error: 0.8 → 0.2

Time: 9.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r520561 = im;
        double r520562 = re;
        double r520563 = atan2(r520561, r520562);
        double r520564 = 10.0;
        double r520565 = log(r520564);
        double r520566 = r520563 / r520565;
        return r520566;
}

↓

double f(double re, double im) {
        double r520567 = 1.0;
        double r520568 = 10.0;
        double r520569 = log(r520568);
        double r520570 = sqrt(r520569);
        double r520571 = r520567 / r520570;
        double r520572 = sqrt(r520571);
        double r520573 = im;
        double r520574 = re;
        double r520575 = atan2(r520573, r520574);
        double r520576 = r520572 * r520575;
        double r520577 = cbrt(r520572);
        double r520578 = r520576 * r520577;
        double r520579 = r520577 * r520577;
        double r520580 = r520578 * r520579;
        double r520581 = r520571 * r520580;
        return r520581;
}

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

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

11. Applied associate-*l* 0.8

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

13. Applied add-cube-cbrt 0.1

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

14. Applied associate-*l* 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10)))