Average Error: 0.9 → 0.1
Time: 22.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right)\right)\right) \cdot \frac{1}{\sqrt{\log 10}}
double f(double re, double im) {
        double r19001 = im;
        double r19002 = re;
        double r19003 = atan2(r19001, r19002);
        double r19004 = 10.0;
        double r19005 = log(r19004);
        double r19006 = r19003 / r19005;
        return r19006;
}
↓
double f(double re, double im) {
        double r19007 = 1.0;
        double r19008 = 10.0;
        double r19009 = log(r19008);
        double r19010 = sqrt(r19009);
        double r19011 = r19007 / r19010;
        double r19012 = sqrt(r19011);
        double r19013 = sqrt(r19012);
        double r19014 = im;
        double r19015 = re;
        double r19016 = atan2(r19014, r19015);
        double r19017 = r19007 / r19009;
        double r19018 = sqrt(r19017);
        double r19019 = sqrt(r19018);
        double r19020 = r19016 * r19019;
        double r19021 = r19013 * r19020;
        double r19022 = r19013 * r19021;
        double r19023 = r19022 * r19011;
        return r19023;
}

Error

Bits error versus re

Bits error versus im

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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-cube-cbrt0.8

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

  14. Applied times-frac0.8

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

  15. Applied sqrt-prod0.8

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

  16. Applied sqrt-prod0.1

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

  17. Applied associate-*r*0.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2019310 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))