Average Error: 0.8 → 0.2
Time: 24.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{\sqrt[3]{\frac{1}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{{\left(\sqrt[3]{\sqrt{\log 10}}\right)}^{4}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{\sqrt[3]{\frac{1}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{{\left(\sqrt[3]{\sqrt{\log 10}}\right)}^{4}}
double f(double re, double im) {
        double r39102 = im;
        double r39103 = re;
        double r39104 = atan2(r39102, r39103);
        double r39105 = 10.0;
        double r39106 = log(r39105);
        double r39107 = r39104 / r39106;
        return r39107;
}
↓
double f(double re, double im) {
        double r39108 = 1.0;
        double r39109 = 10.0;
        double r39110 = log(r39109);
        double r39111 = r39108 / r39110;
        double r39112 = cbrt(r39111);
        double r39113 = im;
        double r39114 = re;
        double r39115 = atan2(r39113, r39114);
        double r39116 = r39112 * r39115;
        double r39117 = sqrt(r39110);
        double r39118 = cbrt(r39117);
        double r39119 = 4.0;
        double r39120 = pow(r39118, r39119);
        double r39121 = r39116 / r39120;
        return r39121;
}

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. 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-cube-cbrt0.8

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

  9. Applied associate-*l*0.9

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

  11. Applied sqrt-div0.9

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

  12. Applied associate-*r/0.9

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

  13. Applied associate-*r/0.9

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

  14. Applied cbrt-div0.8

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

  15. Applied associate-*r/0.8

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

  16. Applied frac-times0.1

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

  17. Simplified1.7

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

  18. Simplified0.2

    \[\leadsto \frac{\sqrt[3]{\frac{1}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\log 10}}\right)}^{4}}}\]

  19. Final simplification0.2

    \[\leadsto \frac{\sqrt[3]{\frac{1}{\log 10}} \cdot \tan^{-1}_* \frac{im}{re}}{{\left(\sqrt[3]{\sqrt{\log 10}}\right)}^{4}}\]

Reproduce

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