math.log10 on complex, real part

Average Error: 31.4 → 0.4

Time: 17.3s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)\]

C

double f(double re, double im) {
        double r624897 = re;
        double r624898 = r624897 * r624897;
        double r624899 = im;
        double r624900 = r624899 * r624899;
        double r624901 = r624898 + r624900;
        double r624902 = sqrt(r624901);
        double r624903 = log(r624902);
        double r624904 = 10.0;
        double r624905 = log(r624904);
        double r624906 = r624903 / r624905;
        return r624906;
}

↓

double f(double re, double im) {
        double r624907 = 1.0;
        double r624908 = 10.0;
        double r624909 = log(r624908);
        double r624910 = sqrt(r624909);
        double r624911 = r624907 / r624910;
        double r624912 = re;
        double r624913 = im;
        double r624914 = hypot(r624912, r624913);
        double r624915 = log(r624914);
        double r624916 = r624911 * r624915;
        double r624917 = r624911 * r624916;
        return r624917;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 31.4

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

2. Simplified 0.6

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\]
3. Using strategy rm

4. Applied clear-num 0.6

   \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]
5. Using strategy rm

6. Applied pow1 0.6

   \[\leadsto \frac{1}{\frac{\log 10}{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}}\]

7. Applied log-pow 0.6

   \[\leadsto \frac{1}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

8. Applied add-sqr-sqrt 0.6

   \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

9. Applied times-frac 0.8

   \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

10. Applied associate-/r* 0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\sqrt{\log 10}}{1}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

11. Simplified 0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\log 10}}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]
12. Using strategy rm

13. Applied pow1 0.6

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}}\]

14. Applied log-pow 0.6

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

15. Applied *-un-lft-identity 0.6

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\color{blue}{1 \cdot \sqrt{\log 10}}}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

16. Applied times-frac 0.6

    \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

17. Applied *-un-lft-identity 0.6

    \[\leadsto \frac{\frac{1}{\color{blue}{1 \cdot \sqrt{\log 10}}}}{\frac{1}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

18. Applied *-un-lft-identity 0.6

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot 1}}{1 \cdot \sqrt{\log 10}}}{\frac{1}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

19. Applied times-frac 0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{1}{\sqrt{\log 10}}}}{\frac{1}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

20. Applied times-frac 0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{1}{1}} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

21. Simplified 0.6

    \[\leadsto \color{blue}{1} \cdot \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]
22. Using strategy rm

23. Applied div-inv 0.6

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\right)}\]

24. Simplified 0.4

    \[\leadsto 1 \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)}\right)\]

25. Final simplification 0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)\right)\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))