Average Error: 0.8 → 0.1
Time: 17.8s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right|\right) \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left|\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right|\right) \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)
double f(double re, double im) {
        double r1267509 = im;
        double r1267510 = re;
        double r1267511 = atan2(r1267509, r1267510);
        double r1267512 = 10.0;
        double r1267513 = log(r1267512);
        double r1267514 = r1267511 / r1267513;
        return r1267514;
}
↓
double f(double re, double im) {
        double r1267515 = 1.0;
        double r1267516 = 10.0;
        double r1267517 = log(r1267516);
        double r1267518 = sqrt(r1267517);
        double r1267519 = cbrt(r1267518);
        double r1267520 = r1267515 / r1267519;
        double r1267521 = sqrt(r1267520);
        double r1267522 = r1267515 / r1267518;
        double r1267523 = sqrt(r1267522);
        double r1267524 = fabs(r1267520);
        double r1267525 = r1267523 * r1267524;
        double r1267526 = im;
        double r1267527 = re;
        double r1267528 = atan2(r1267526, r1267527);
        double r1267529 = r1267528 / r1267518;
        double r1267530 = r1267525 * r1267529;
        double r1267531 = r1267521 * r1267530;
        return r1267531;
}

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. Applied associate-*r*0.8

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

  10. Applied add-sqr-sqrt0.8

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

  11. Applied associate-*r*0.8

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied *-un-lft-identity0.1

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

  15. Applied times-frac0.1

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*r*0.2

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))