Average Error: 0.8 → 0.8
Time: 2.7s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)
double f(double re, double im) {
        double r75581 = im;
        double r75582 = re;
        double r75583 = atan2(r75581, r75582);
        double r75584 = 10.0;
        double r75585 = log(r75584);
        double r75586 = r75583 / r75585;
        return r75586;
}
↓
double f(double re, double im) {
        double r75587 = 1.0;
        double r75588 = 10.0;
        double r75589 = log(r75588);
        double r75590 = sqrt(r75589);
        double r75591 = r75587 / r75590;
        double r75592 = im;
        double r75593 = re;
        double r75594 = atan2(r75592, r75593);
        double r75595 = r75587 / r75589;
        double r75596 = sqrt(r75595);
        double r75597 = r75594 * r75596;
        double r75598 = r75591 * r75597;
        return r75598;
}

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 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  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. Final simplification0.8

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

Reproduce

herbie shell --seed 2020027 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))