math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 24.1s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r2499739 = im;
        double r2499740 = re;
        double r2499741 = atan2(r2499739, r2499740);
        double r2499742 = 10.0;
        double r2499743 = log(r2499742);
        double r2499744 = r2499741 / r2499743;
        return r2499744;
}

↓

double f(double re, double im) {
        double r2499745 = 1.0;
        double r2499746 = 10.0;
        double r2499747 = log(r2499746);
        double r2499748 = sqrt(r2499747);
        double r2499749 = r2499745 / r2499748;
        double r2499750 = sqrt(r2499749);
        double r2499751 = sqrt(r2499750);
        double r2499752 = im;
        double r2499753 = re;
        double r2499754 = atan2(r2499752, r2499753);
        double r2499755 = r2499750 * r2499754;
        double r2499756 = r2499751 * r2499755;
        double r2499757 = r2499751 * r2499756;
        double r2499758 = r2499757 * r2499749;
        return r2499758;
}

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-sqr-sqrt 0.8

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

14. Applied sqrt-prod 0.1

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

15. Applied associate-*l* 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))