math.log10 on complex, imaginary part

Average Error: 0.8 → 0.8

Time: 8.6s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r35876 = im;
        double r35877 = re;
        double r35878 = atan2(r35876, r35877);
        double r35879 = 10.0;
        double r35880 = log(r35879);
        double r35881 = r35878 / r35880;
        return r35881;
}

↓

double f(double re, double im) {
        double r35882 = im;
        double r35883 = re;
        double r35884 = atan2(r35882, r35883);
        double r35885 = 1.0;
        double r35886 = 10.0;
        double r35887 = log(r35886);
        double r35888 = r35885 / r35887;
        double r35889 = sqrt(r35888);
        double r35890 = sqrt(r35887);
        double r35891 = r35889 / r35890;
        double r35892 = r35884 * r35891;
        return r35892;
}

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 associate-*l/ 0.8

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

9. Simplified 0.8

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

11. Applied *-un-lft-identity 0.8

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

12. Applied times-frac 0.8

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

13. Simplified 0.8

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

14. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019294 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))