Average Error: 0.9 → 0.8
Time: 3.2s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\log 10}\right)}^{\frac{1}{4}}\right) \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot {\left(\frac{1}{\log 10}\right)}^{\frac{1}{4}}\right) \cdot \sqrt{\sqrt{\frac{1}{\log 10}}}\right)
double f(double re, double im) {
        double r26762 = im;
        double r26763 = re;
        double r26764 = atan2(r26762, r26763);
        double r26765 = 10.0;
        double r26766 = log(r26765);
        double r26767 = r26764 / r26766;
        return r26767;
}
↓
double f(double re, double im) {
        double r26768 = 1.0;
        double r26769 = 10.0;
        double r26770 = log(r26769);
        double r26771 = sqrt(r26770);
        double r26772 = r26768 / r26771;
        double r26773 = im;
        double r26774 = re;
        double r26775 = atan2(r26773, r26774);
        double r26776 = r26768 / r26770;
        double r26777 = 0.25;
        double r26778 = pow(r26776, r26777);
        double r26779 = r26775 * r26778;
        double r26780 = sqrt(r26776);
        double r26781 = sqrt(r26780);
        double r26782 = r26779 * r26781;
        double r26783 = r26772 * r26782;
        return r26783;
}

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. Taylor expanded around 0 0.8

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

  12. Final simplification0.8

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

Reproduce

herbie shell --seed 2020024 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))