math.log10 on complex, imaginary part

Average Error: 0.9 → 0.8

Time: 3.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r29191 = im;
        double r29192 = re;
        double r29193 = atan2(r29191, r29192);
        double r29194 = 10.0;
        double r29195 = log(r29194);
        double r29196 = r29193 / r29195;
        return r29196;
}

↓

double f(double re, double im) {
        double r29197 = 1.0;
        double r29198 = 10.0;
        double r29199 = log(r29198);
        double r29200 = sqrt(r29199);
        double r29201 = r29197 / r29200;
        double r29202 = im;
        double r29203 = re;
        double r29204 = atan2(r29202, r29203);
        double r29205 = sqrt(r29197);
        double r29206 = r29205 / r29200;
        double r29207 = sqrt(r29206);
        double r29208 = r29204 * r29207;
        double r29209 = sqrt(r29201);
        double r29210 = r29208 * r29209;
        double r29211 = r29201 * r29210;
        return r29211;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.9

   \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 0.9

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

4. Applied *-un-lft-identity 0.9

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

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. Using strategy rm

7. Applied div-inv 0.8

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

9. Applied add-sqr-sqrt 0.8

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

10. Applied associate-*r* 0.8

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

12. Applied pow1 0.8

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

13. Applied log-pow 0.8

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

14. Applied sqrt-prod 0.8

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

15. Applied add-sqr-sqrt 0.8

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

16. Applied times-frac 0.8

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

17. Applied sqrt-prod 0.8

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

18. Applied associate-*r* 0.8

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

19. Simplified 0.8

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

20. Final simplification 0.8

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))