Average Error: 0.9 → 0.4
Time: 13.2s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{\sqrt{\frac{1}{\log 10}}}{\sqrt{\log 10}}\right)\right)\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{\sqrt{\frac{1}{\log 10}}}{\sqrt{\log 10}}\right)\right)\right)\right)
double f(double re, double im) {
        double r21186 = im;
        double r21187 = re;
        double r21188 = atan2(r21186, r21187);
        double r21189 = 10.0;
        double r21190 = log(r21189);
        double r21191 = r21188 / r21190;
        return r21191;
}
↓
double f(double re, double im) {
        double r21192 = im;
        double r21193 = re;
        double r21194 = atan2(r21192, r21193);
        double r21195 = 1.0;
        double r21196 = 10.0;
        double r21197 = log(r21196);
        double r21198 = r21195 / r21197;
        double r21199 = sqrt(r21198);
        double r21200 = sqrt(r21197);
        double r21201 = r21199 / r21200;
        double r21202 = r21194 * r21201;
        double r21203 = expm1(r21202);
        double r21204 = expm1(r21203);
        double r21205 = log1p(r21204);
        double r21206 = log1p(r21205);
        return r21206;
}

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 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. Taylor expanded around 0 0.7

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

  10. Applied log1p-expm1-u0.5

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  (/ (atan2 im re) (log 10.0)))