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 r1267534 = im;
        double r1267535 = re;
        double r1267536 = atan2(r1267534, r1267535);
        double r1267537 = 10.0;
        double r1267538 = log(r1267537);
        double r1267539 = r1267536 / r1267538;
        return r1267539;
}
↓
double f(double re, double im) {
        double r1267540 = 1.0;
        double r1267541 = 10.0;
        double r1267542 = log(r1267541);
        double r1267543 = sqrt(r1267542);
        double r1267544 = cbrt(r1267543);
        double r1267545 = r1267540 / r1267544;
        double r1267546 = sqrt(r1267545);
        double r1267547 = r1267540 / r1267543;
        double r1267548 = sqrt(r1267547);
        double r1267549 = fabs(r1267545);
        double r1267550 = r1267548 * r1267549;
        double r1267551 = im;
        double r1267552 = re;
        double r1267553 = atan2(r1267551, r1267552);
        double r1267554 = r1267553 / r1267543;
        double r1267555 = r1267550 * r1267554;
        double r1267556 = r1267546 * r1267555;
        return r1267556;
}

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)))