math.log10 on complex, real part

Average Error: 31.0 → 0.5

Time: 4.6m

Precision: 64

Math

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

↓

\[\frac{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}}\]

C

double f(double re, double im) {
        double r6538291 = re;
        double r6538292 = r6538291 * r6538291;
        double r6538293 = im;
        double r6538294 = r6538293 * r6538293;
        double r6538295 = r6538292 + r6538294;
        double r6538296 = sqrt(r6538295);
        double r6538297 = log(r6538296);
        double r6538298 = 10.0;
        double r6538299 = log(r6538298);
        double r6538300 = r6538297 / r6538299;
        return r6538300;
}

↓

double f(double re, double im) {
        double r6538301 = re;
        double r6538302 = im;
        double r6538303 = hypot(r6538301, r6538302);
        double r6538304 = log(r6538303);
        double r6538305 = 10.0;
        double r6538306 = log(r6538305);
        double r6538307 = sqrt(r6538306);
        double r6538308 = cbrt(r6538307);
        double r6538309 = r6538304 / r6538308;
        double r6538310 = sqrt(r6538307);
        double r6538311 = r6538309 / r6538310;
        double r6538312 = 1.0;
        double r6538313 = r6538312 / r6538308;
        double r6538314 = r6538313 * r6538313;
        double r6538315 = r6538314 / r6538310;
        double r6538316 = r6538311 * r6538315;
        return r6538316;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 31.0

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

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 add-sqr-sqrt 0.6

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

5. Applied *-un-lft-identity 0.6

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

6. Applied times-frac 0.6

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

8. Applied pow1 0.6

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

9. Applied log-pow 0.6

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

10. Applied associate-/l* 0.6

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

12. Applied associate-*r/ 0.6

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

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 pow1 0.6

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

16. Applied log-pow 0.6

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

17. Applied add-sqr-sqrt 1.2

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

18. Applied times-frac 1.3

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

19. Applied add-cube-cbrt 0.7

    \[\leadsto \frac{\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt{\log 10}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

20. Applied add-sqr-sqrt 0.7

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt{\log 10}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

21. Applied times-frac 0.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt{\log 10}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

22. Applied times-frac 0.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}}\]

23. Simplified 0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}}} \cdot \frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}\]

24. Simplified 0.5

    \[\leadsto \frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \color{blue}{\frac{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}}}\]

25. Final simplification 0.5

    \[\leadsto \frac{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}}} \cdot \frac{1}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{\sqrt{\log 10}}}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))