Average Error: 0.8 → 0.7
Time: 3.5s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(1 \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}}{\sqrt{\log 10}}\right)\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(1 \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}}{\sqrt{\log 10}}\right)\right)
double f(double re, double im) {
        double r28004 = im;
        double r28005 = re;
        double r28006 = atan2(r28004, r28005);
        double r28007 = 10.0;
        double r28008 = log(r28007);
        double r28009 = r28006 / r28008;
        return r28009;
}
↓
double f(double re, double im) {
        double r28010 = 1.0;
        double r28011 = im;
        double r28012 = re;
        double r28013 = atan2(r28011, r28012);
        double r28014 = 10.0;
        double r28015 = log(r28014);
        double r28016 = r28010 / r28015;
        double r28017 = sqrt(r28016);
        double r28018 = r28013 * r28017;
        double r28019 = sqrt(r28015);
        double r28020 = r28018 / r28019;
        double r28021 = r28010 * r28020;
        double r28022 = expm1(r28021);
        double r28023 = log1p(r28022);
        return r28023;
}

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. Taylor expanded around 0 0.7

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

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

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

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

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

  12. Applied times-frac0.7

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

  13. Applied associate-*l*0.7

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

  14. Simplified0.7

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

  15. Final simplification0.7

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))