Average Error: 31.3 → 17.4
Time: 22.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.139279335366515 \cdot 10^{+91}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \le 6.605815153598046 \cdot 10^{+41}:\\ \;\;\;\;\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{\log base}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.139279335366515 \cdot 10^{+91}:\\
\;\;\;\;\frac{\log \left(-re\right)}{\log base}\\

\mathbf{elif}\;re \le 6.605815153598046 \cdot 10^{+41}:\\
\;\;\;\;\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{\log base}\right) \cdot \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log re}{\log base}\\

\end{array}
double f(double re, double im, double base) {
        double r2565963 = re;
        double r2565964 = r2565963 * r2565963;
        double r2565965 = im;
        double r2565966 = r2565965 * r2565965;
        double r2565967 = r2565964 + r2565966;
        double r2565968 = sqrt(r2565967);
        double r2565969 = log(r2565968);
        double r2565970 = base;
        double r2565971 = log(r2565970);
        double r2565972 = r2565969 * r2565971;
        double r2565973 = atan2(r2565965, r2565963);
        double r2565974 = 0.0;
        double r2565975 = r2565973 * r2565974;
        double r2565976 = r2565972 + r2565975;
        double r2565977 = r2565971 * r2565971;
        double r2565978 = r2565974 * r2565974;
        double r2565979 = r2565977 + r2565978;
        double r2565980 = r2565976 / r2565979;
        return r2565980;
}
↓
double f(double re, double im, double base) {
        double r2565981 = re;
        double r2565982 = -5.139279335366515e+91;
        bool r2565983 = r2565981 <= r2565982;
        double r2565984 = -r2565981;
        double r2565985 = log(r2565984);
        double r2565986 = base;
        double r2565987 = log(r2565986);
        double r2565988 = r2565985 / r2565987;
        double r2565989 = 6.605815153598046e+41;
        bool r2565990 = r2565981 <= r2565989;
        double r2565991 = im;
        double r2565992 = r2565991 * r2565991;
        double r2565993 = r2565981 * r2565981;
        double r2565994 = r2565992 + r2565993;
        double r2565995 = log(r2565994);
        double r2565996 = 1.0;
        double r2565997 = r2565996 / r2565987;
        double r2565998 = r2565995 * r2565997;
        double r2565999 = 0.5;
        double r2566000 = r2565998 * r2565999;
        double r2566001 = log(r2565981);
        double r2566002 = r2566001 / r2565987;
        double r2566003 = r2565990 ? r2566000 : r2566002;
        double r2566004 = r2565983 ? r2565988 : r2566003;
        return r2566004;
}

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. Split input into 3 regimes

  2. if re < -5.139279335366515e+91

    1. Initial program 47.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified47.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]

    3. Taylor expanded around -inf 9.2

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)}}{\log base}\]

    4. Simplified9.2

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)}}{\log base}\]

    if -5.139279335366515e+91 < re < 6.605815153598046e+41

    1. Initial program 21.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified21.8

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]
    3. Using strategy rm

    4. Applied pow121.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log \color{blue}{\left({base}^{1}\right)}}\]

    5. Applied log-pow21.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{1 \cdot \log base}}\]

    6. Applied pow1/221.8

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{1 \cdot \log base}\]

    7. Applied log-pow21.8

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{1 \cdot \log base}\]

    8. Applied times-frac21.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}}\]

    9. Simplified21.8

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\log base}\]
    10. Using strategy rm

    11. Applied div-inv21.9

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{1}{\log base}\right)}\]

    if 6.605815153598046e+41 < re

    1. Initial program 43.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

    2. Simplified43.3

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}}\]

    3. Taylor expanded around inf 11.8

      \[\leadsto \frac{\log \color{blue}{re}}{\log base}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.139279335366515 \cdot 10^{+91}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\log base}\\ \mathbf{elif}\;re \le 6.605815153598046 \cdot 10^{+41}:\\ \;\;\;\;\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{1}{\log base}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))