math.log10 on complex, real part

Average Error: 31.5 → 0.3

Time: 21.5s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r33858 = re;
        double r33859 = r33858 * r33858;
        double r33860 = im;
        double r33861 = r33860 * r33860;
        double r33862 = r33859 + r33861;
        double r33863 = sqrt(r33862);
        double r33864 = log(r33863);
        double r33865 = 10.0;
        double r33866 = log(r33865);
        double r33867 = r33864 / r33866;
        return r33867;
}

↓

double f(double re, double im) {
        double r33868 = 1.0;
        double r33869 = 10.0;
        double r33870 = log(r33869);
        double r33871 = sqrt(r33870);
        double r33872 = r33868 / r33871;
        double r33873 = r33872 / r33871;
        double r33874 = re;
        double r33875 = im;
        double r33876 = hypot(r33874, r33875);
        double r33877 = log(r33876);
        double r33878 = r33873 * r33877;
        return r33878;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 31.5

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

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 pow1 0.6

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

6. Applied log-pow 0.6

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

7. Applied times-frac 0.6

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

9. Applied div-inv 0.4

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

11. Applied add-log-exp 0.4

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

12. Simplified 0.3

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

14. Applied log-pow 0.4

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

15. Applied associate-*r* 0.3

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))