Average Error: 0.9 → 0.9
Time: 3.3s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\left(\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \frac{1}{\left(\sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}} \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}\right) \cdot \sqrt[3]{\frac{\sqrt{\log 10}}{\tan^{-1}_* \frac{im}{re}}}}
double f(double re, double im) {
        double r27373 = im;
        double r27374 = re;
        double r27375 = atan2(r27373, r27374);
        double r27376 = 10.0;
        double r27377 = log(r27376);
        double r27378 = r27375 / r27377;
        return r27378;
}
↓
double f(double re, double im) {
        double r27379 = 1.0;
        double r27380 = 10.0;
        double r27381 = log(r27380);
        double r27382 = sqrt(r27381);
        double r27383 = r27379 / r27382;
        double r27384 = im;
        double r27385 = re;
        double r27386 = atan2(r27384, r27385);
        double r27387 = r27382 / r27386;
        double r27388 = cbrt(r27387);
        double r27389 = r27388 * r27388;
        double r27390 = r27389 * r27388;
        double r27391 = r27379 / r27390;
        double r27392 = r27383 * r27391;
        return r27392;
}

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. Using strategy rm

  7. Applied clear-num1.0

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

  9. Applied add-cube-cbrt0.9

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

  10. Final simplification0.9

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))