Average Error: 0.9 → 0.8
Time: 2.4s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)
double f(double re, double im) {
        double r27032 = im;
        double r27033 = re;
        double r27034 = atan2(r27032, r27033);
        double r27035 = 10.0;
        double r27036 = log(r27035);
        double r27037 = r27034 / r27036;
        return r27037;
}
↓
double f(double re, double im) {
        double r27038 = 1.0;
        double r27039 = 10.0;
        double r27040 = log(r27039);
        double r27041 = sqrt(r27040);
        double r27042 = r27038 / r27041;
        double r27043 = im;
        double r27044 = re;
        double r27045 = atan2(r27043, r27044);
        double r27046 = r27045 * r27042;
        double r27047 = r27042 * r27046;
        return r27047;
}

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 add-sqr-sqrt0.9

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  4. Applied *-un-lft-identity0.9

    \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]
  6. Using strategy rm

  7. Applied div-inv0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))