Average Error: 0.8 → 0.1
Time: 22.5s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}
double f(double re, double im) {
        double r23407 = im;
        double r23408 = re;
        double r23409 = atan2(r23407, r23408);
        double r23410 = 10.0;
        double r23411 = log(r23410);
        double r23412 = r23409 / r23411;
        return r23412;
}
↓
double f(double re, double im) {
        double r23413 = 1.0;
        double r23414 = 10.0;
        double r23415 = log(r23414);
        double r23416 = sqrt(r23415);
        double r23417 = cbrt(r23416);
        double r23418 = r23413 / r23417;
        double r23419 = sqrt(r23418);
        double r23420 = im;
        double r23421 = re;
        double r23422 = atan2(r23420, r23421);
        double r23423 = r23422 / r23416;
        double r23424 = r23419 * r23423;
        double r23425 = r23417 * r23417;
        double r23426 = r23413 / r23425;
        double r23427 = sqrt(r23426);
        double r23428 = r23424 * r23427;
        double r23429 = r23413 / r23416;
        double r23430 = sqrt(r23429);
        double r23431 = r23428 * r23430;
        return r23431;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.8

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

  3. Applied add-sqr-sqrt0.8

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

  4. Applied *-un-lft-identity0.8

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

  5. Applied times-frac0.8

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

  7. Applied div-inv0.8

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

  9. Applied add-sqr-sqrt0.8

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

  10. Applied associate-*l*0.8

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

  11. Simplified0.9

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied add-sqr-sqrt0.1

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

  15. Applied times-frac0.1

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*l*0.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))