Average Error: 30.9 → 0.4
Time: 25.0s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{1}{\log base} \cdot \tan^{-1}_* \frac{im}{re}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{1}{\log base} \cdot \tan^{-1}_* \frac{im}{re}
double f(double re, double im, double base) {
        double r29963 = im;
        double r29964 = re;
        double r29965 = atan2(r29963, r29964);
        double r29966 = base;
        double r29967 = log(r29966);
        double r29968 = r29965 * r29967;
        double r29969 = r29964 * r29964;
        double r29970 = r29963 * r29963;
        double r29971 = r29969 + r29970;
        double r29972 = sqrt(r29971);
        double r29973 = log(r29972);
        double r29974 = 0.0;
        double r29975 = r29973 * r29974;
        double r29976 = r29968 - r29975;
        double r29977 = r29967 * r29967;
        double r29978 = r29974 * r29974;
        double r29979 = r29977 + r29978;
        double r29980 = r29976 / r29979;
        return r29980;
}
↓
double f(double re, double im, double base) {
        double r29981 = 1.0;
        double r29982 = base;
        double r29983 = log(r29982);
        double r29984 = r29981 / r29983;
        double r29985 = im;
        double r29986 = re;
        double r29987 = atan2(r29985, r29986);
        double r29988 = r29984 * r29987;
        return r29988;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

    \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - 0.0 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}\]

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

    \[\leadsto \color{blue}{-\frac{\tan^{-1}_* \frac{im}{re}}{-\log base}}\]
  5. Using strategy rm

  6. Applied div-inv0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

    \[\leadsto \frac{1}{\log base} \cdot \tan^{-1}_* \frac{im}{re}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))