Average Error: 0.8 → 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 r33103 = im;
        double r33104 = re;
        double r33105 = atan2(r33103, r33104);
        double r33106 = 10.0;
        double r33107 = log(r33106);
        double r33108 = r33105 / r33107;
        return r33108;
}
↓
double f(double re, double im) {
        double r33109 = 1.0;
        double r33110 = 10.0;
        double r33111 = log(r33110);
        double r33112 = sqrt(r33111);
        double r33113 = r33109 / r33112;
        double r33114 = im;
        double r33115 = re;
        double r33116 = atan2(r33114, r33115);
        double r33117 = r33116 / r33112;
        double r33118 = r33113 * r33117;
        double r33119 = expm1(r33118);
        double r33120 = expm1(r33119);
        double r33121 = log1p(r33120);
        double r33122 = log1p(r33121);
        return r33122;
}

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 log1p-expm1-u0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied *-un-lft-identity0.5

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

  9. Applied times-frac0.5

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

  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 2019362 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))