math.log10 on complex, real part

Average Error: 31.4 → 0.4

Time: 35.5s

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 r685929 = re;
        double r685930 = r685929 * r685929;
        double r685931 = im;
        double r685932 = r685931 * r685931;
        double r685933 = r685930 + r685932;
        double r685934 = sqrt(r685933);
        double r685935 = log(r685934);
        double r685936 = 10.0;
        double r685937 = log(r685936);
        double r685938 = r685935 / r685937;
        return r685938;
}

↓

double f(double re, double im) {
        double r685939 = 1.0;
        double r685940 = 10.0;
        double r685941 = log(r685940);
        double r685942 = sqrt(r685941);
        double r685943 = r685939 / r685942;
        double r685944 = re;
        double r685945 = im;
        double r685946 = hypot(r685944, r685945);
        double r685947 = log(r685946);
        double r685948 = r685943 * r685947;
        double r685949 = r685943 * r685948;
        return r685949;
}

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 pow1 0.6

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

5. Applied log-pow 0.6

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

6. Applied associate-/l* 0.6

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

8. Applied pow1 0.6

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

9. Applied log-pow 0.6

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

10. Applied add-sqr-sqrt 0.6

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

11. Applied times-frac 0.8

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

12. Applied associate-/r* 0.6

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

13. Simplified 0.6

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

15. Applied *-un-lft-identity 0.6

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

16. Applied *-un-lft-identity 0.6

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

17. Applied times-frac 0.6

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

18. Applied associate-/r* 0.6

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

19. Simplified 0.6

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

21. Applied div-inv 0.6

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

22. Simplified 0.4

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

23. 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)))