Average Error: 0.9 → 0.5
Time: 3.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\right)\right)
double f(double re, double im) {
        double r33822 = im;
        double r33823 = re;
        double r33824 = atan2(r33822, r33823);
        double r33825 = 10.0;
        double r33826 = log(r33825);
        double r33827 = r33824 / r33826;
        return r33827;
}
↓
double f(double re, double im) {
        double r33828 = 1.0;
        double r33829 = 10.0;
        double r33830 = log(r33829);
        double r33831 = sqrt(r33830);
        double r33832 = r33828 / r33831;
        double r33833 = im;
        double r33834 = re;
        double r33835 = atan2(r33833, r33834);
        double r33836 = r33835 / r33831;
        double r33837 = r33832 * r33836;
        double r33838 = expm1(r33837);
        double r33839 = expm1(r33838);
        double r33840 = log1p(r33839);
        double r33841 = log1p(r33840);
        return r33841;
}

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 log1p-expm1-u0.7

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\right)\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\right)\right)\]

  6. Applied *-un-lft-identity0.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\right)\right)\]

  7. Applied times-frac0.7

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\right)\right)\]
  8. Using strategy rm

  9. Applied log1p-expm1-u0.5

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\right)}\right)\]

  10. Final simplification0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))