math.log10 on complex, imaginary part

Average Error: 0.9 → 0.1

Time: 3.0s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r79488 = im;
        double r79489 = re;
        double r79490 = atan2(r79488, r79489);
        double r79491 = 10.0;
        double r79492 = log(r79491);
        double r79493 = r79490 / r79492;
        return r79493;
}

↓

double f(double re, double im) {
        double r79494 = 1.0;
        double r79495 = 10.0;
        double r79496 = log(r79495);
        double r79497 = sqrt(r79496);
        double r79498 = r79494 / r79497;
        double r79499 = im;
        double r79500 = re;
        double r79501 = atan2(r79499, r79500);
        double r79502 = r79494 / r79496;
        double r79503 = sqrt(r79502);
        double r79504 = sqrt(r79503);
        double r79505 = r79501 * r79504;
        double r79506 = sqrt(r79498);
        double r79507 = sqrt(r79506);
        double r79508 = r79505 * r79507;
        double r79509 = r79508 * r79507;
        double r79510 = r79498 * r79509;
        return r79510;
}

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. Taylor expanded around 0 0.8

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

8. Applied add-sqr-sqrt 0.8

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

9. Applied sqrt-prod 0.8

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

10. Applied associate-*r* 0.8

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

12. Applied add-sqr-sqrt 0.8

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

13. Applied *-un-lft-identity 0.8

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

14. Applied times-frac 0.8

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

15. Applied sqrt-prod 0.8

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

16. Applied sqrt-prod 0.1

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

17. Applied associate-*r* 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))