math.log10 on complex, imaginary part

Average Error: 0.8 → 0.8

Time: 3.5s

Precision: 64

Math

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

↓

\[\frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}\right)\right)}\]

C

double f(double re, double im) {
        double r36461 = im;
        double r36462 = re;
        double r36463 = atan2(r36461, r36462);
        double r36464 = 10.0;
        double r36465 = log(r36464);
        double r36466 = r36463 / r36465;
        return r36466;
}

↓

double f(double re, double im) {
        double r36467 = 1.0;
        double r36468 = 10.0;
        double r36469 = log(r36468);
        double r36470 = im;
        double r36471 = re;
        double r36472 = atan2(r36470, r36471);
        double r36473 = r36469 / r36472;
        double r36474 = cbrt(r36473);
        double r36475 = r36474 * r36474;
        double r36476 = cbrt(r36469);
        double r36477 = r36476 * r36476;
        double r36478 = r36477 / r36467;
        double r36479 = cbrt(r36478);
        double r36480 = sqrt(r36469);
        double r36481 = cbrt(r36480);
        double r36482 = r36481 / r36467;
        double r36483 = cbrt(r36482);
        double r36484 = r36481 / r36472;
        double r36485 = cbrt(r36484);
        double r36486 = r36483 * r36485;
        double r36487 = r36479 * r36486;
        double r36488 = r36475 * r36487;
        double r36489 = r36467 / r36488;
        return r36489;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.8

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

3. Applied clear-num 1.0

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

5. Applied add-cube-cbrt 0.8

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}}}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.8

   \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\log 10}{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}}}\]

8. Applied add-cube-cbrt 0.9

   \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}\right) \cdot \sqrt[3]{\log 10}}}{1 \cdot \tan^{-1}_* \frac{im}{re}}}}\]

9. Applied times-frac 0.9

   \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\color{blue}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}}\]

10. Applied cbrt-prod 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right)}}\]
11. Using strategy rm

12. Applied *-un-lft-identity 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\log 10}}{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}}\right)}\]

13. Applied add-sqr-sqrt 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}}{1 \cdot \tan^{-1}_* \frac{im}{re}}}\right)}\]

14. Applied cbrt-prod 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \sqrt[3]{\frac{\color{blue}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{1 \cdot \tan^{-1}_* \frac{im}{re}}}\right)}\]

15. Applied times-frac 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt[3]{\sqrt{\log 10}}}{1} \cdot \frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}}\right)}\]

16. Applied cbrt-prod 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}\right)}\right)}\]

17. Final simplification 0.8

    \[\leadsto \frac{1}{\left(\sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\log 10}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{1}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{\log 10}}}{\tan^{-1}_* \frac{im}{re}}}\right)\right)}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))