Average Error: 0.8 → 0.7
Time: 3.6s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)\right)\right)
double f(double re, double im) {
        double r46939 = im;
        double r46940 = re;
        double r46941 = atan2(r46939, r46940);
        double r46942 = 10.0;
        double r46943 = log(r46942);
        double r46944 = r46941 / r46943;
        return r46944;
}
↓
double f(double re, double im) {
        double r46945 = 1.0;
        double r46946 = 10.0;
        double r46947 = log(r46946);
        double r46948 = sqrt(r46947);
        double r46949 = r46945 / r46948;
        double r46950 = im;
        double r46951 = re;
        double r46952 = atan2(r46950, r46951);
        double r46953 = r46952 * r46949;
        double r46954 = r46949 * r46953;
        double r46955 = expm1(r46954);
        double r46956 = log1p(r46955);
        return r46956;
}

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 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  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 div-inv0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))