math.log10 on complex, imaginary part

Average Error: 0.8 → 0.1

Time: 4.3s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r37038 = im;
        double r37039 = re;
        double r37040 = atan2(r37038, r37039);
        double r37041 = 10.0;
        double r37042 = log(r37041);
        double r37043 = r37040 / r37042;
        return r37043;
}

↓

double f(double re, double im) {
        double r37044 = 1.0;
        double r37045 = 10.0;
        double r37046 = log(r37045);
        double r37047 = sqrt(r37046);
        double r37048 = r37044 / r37047;
        double r37049 = sqrt(r37048);
        double r37050 = sqrt(r37049);
        double r37051 = im;
        double r37052 = re;
        double r37053 = atan2(r37051, r37052);
        double r37054 = r37044 / r37046;
        double r37055 = sqrt(r37054);
        double r37056 = r37053 * r37055;
        double r37057 = r37049 * r37056;
        double r37058 = r37050 * r37057;
        double r37059 = r37050 * r37058;
        return r37059;
}

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. Taylor expanded around 0 0.8

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

8. Applied add-sqr-sqrt 0.8

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

9. Applied associate-*l* 0.8

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

11. Applied add-sqr-sqrt 0.8

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

12. Applied sqrt-prod 0.1

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

13. Applied associate-*l* 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019322 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))