Average Error: 0.9 → 0.1
Time: 22.2s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}
double f(double re, double im) {
        double r34877 = im;
        double r34878 = re;
        double r34879 = atan2(r34877, r34878);
        double r34880 = 10.0;
        double r34881 = log(r34880);
        double r34882 = r34879 / r34881;
        return r34882;
}
↓
double f(double re, double im) {
        double r34883 = 1.0;
        double r34884 = 10.0;
        double r34885 = log(r34884);
        double r34886 = sqrt(r34885);
        double r34887 = r34883 / r34886;
        double r34888 = sqrt(r34887);
        double r34889 = sqrt(r34888);
        double r34890 = im;
        double r34891 = re;
        double r34892 = atan2(r34890, r34891);
        double r34893 = r34883 / r34885;
        double r34894 = sqrt(r34893);
        double r34895 = sqrt(r34894);
        double r34896 = r34892 * r34895;
        double r34897 = r34896 * r34889;
        double r34898 = r34889 * r34897;
        double r34899 = r34898 * r34887;
        return r34899;
}

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 0.9

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

  3. Applied add-sqr-sqrt0.9

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

  4. Applied *-un-lft-identity0.9

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

  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-sqr-sqrt0.8

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

  9. Applied sqrt-prod0.8

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

  10. Applied associate-*r*0.8

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

  12. Applied add-sqr-sqrt0.8

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

  13. Applied add-sqr-sqrt0.8

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

  14. Applied times-frac0.8

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

  15. Applied sqrt-prod0.8

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*r*0.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))