Average Error: 32.3 → 18.3
Time: 7.6s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1225739876249368 \cdot 10^{21}:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -4.27711132562022042 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 3.510677278861231 \cdot 10^{-284}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 5.9451169291533458 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1225739876249368 \cdot 10^{21}:\\
\;\;\;\;\frac{\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le -4.27711132562022042 \cdot 10^{-185}:\\
\;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{elif}\;re \le 3.510677278861231 \cdot 10^{-284}:\\
\;\;\;\;\frac{\log im}{\log base}\\

\mathbf{elif}\;re \le 5.9451169291533458 \cdot 10^{55}:\\
\;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\end{array}
double f(double re, double im, double base) {
        double r50925 = re;
        double r50926 = r50925 * r50925;
        double r50927 = im;
        double r50928 = r50927 * r50927;
        double r50929 = r50926 + r50928;
        double r50930 = sqrt(r50929);
        double r50931 = log(r50930);
        double r50932 = base;
        double r50933 = log(r50932);
        double r50934 = r50931 * r50933;
        double r50935 = atan2(r50927, r50925);
        double r50936 = 0.0;
        double r50937 = r50935 * r50936;
        double r50938 = r50934 + r50937;
        double r50939 = r50933 * r50933;
        double r50940 = r50936 * r50936;
        double r50941 = r50939 + r50940;
        double r50942 = r50938 / r50941;
        return r50942;
}
↓
double f(double re, double im, double base) {
        double r50943 = re;
        double r50944 = -1.1225739876249368e+21;
        bool r50945 = r50943 <= r50944;
        double r50946 = -1.0;
        double r50947 = r50946 / r50943;
        double r50948 = log(r50947);
        double r50949 = r50946 * r50948;
        double r50950 = base;
        double r50951 = log(r50950);
        double r50952 = r50949 * r50951;
        double r50953 = im;
        double r50954 = atan2(r50953, r50943);
        double r50955 = 0.0;
        double r50956 = r50954 * r50955;
        double r50957 = r50952 + r50956;
        double r50958 = r50951 * r50951;
        double r50959 = r50955 * r50955;
        double r50960 = r50958 + r50959;
        double r50961 = sqrt(r50960);
        double r50962 = r50957 / r50961;
        double r50963 = r50962 / r50961;
        double r50964 = -4.2771113256202204e-185;
        bool r50965 = r50943 <= r50964;
        double r50966 = r50943 * r50943;
        double r50967 = r50953 * r50953;
        double r50968 = r50966 + r50967;
        double r50969 = cbrt(r50968);
        double r50970 = fabs(r50969);
        double r50971 = sqrt(r50969);
        double r50972 = r50970 * r50971;
        double r50973 = log(r50972);
        double r50974 = r50973 * r50951;
        double r50975 = r50974 + r50956;
        double r50976 = r50975 / r50961;
        double r50977 = r50976 / r50961;
        double r50978 = 3.510677278861231e-284;
        bool r50979 = r50943 <= r50978;
        double r50980 = log(r50953);
        double r50981 = r50980 / r50951;
        double r50982 = 5.945116929153346e+55;
        bool r50983 = r50943 <= r50982;
        double r50984 = 1.0;
        double r50985 = r50984 / r50943;
        double r50986 = log(r50985);
        double r50987 = r50984 / r50950;
        double r50988 = log(r50987);
        double r50989 = r50986 * r50988;
        double r50990 = r50989 + r50956;
        double r50991 = r50990 / r50960;
        double r50992 = r50983 ? r50977 : r50991;
        double r50993 = r50979 ? r50981 : r50992;
        double r50994 = r50965 ? r50977 : r50993;
        double r50995 = r50945 ? r50963 : r50994;
        return r50995;
}

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 4 regimes

  2. if re < -1.1225739876249368e+21

    1. Initial program 43.2

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

    3. Applied add-sqr-sqrt43.2

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

    4. Applied associate-/r*43.2

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

    5. Taylor expanded around -inf 64.0

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

    6. Simplified12.5

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

    if -1.1225739876249368e+21 < re < -4.2771113256202204e-185 or 3.510677278861231e-284 < re < 5.945116929153346e+55

    1. Initial program 20.3

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

    3. Applied add-sqr-sqrt20.3

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

    4. Applied associate-/r*20.2

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt20.2

      \[\leadsto \frac{\frac{\log \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    7. Applied sqrt-prod20.2

      \[\leadsto \frac{\frac{\log \color{blue}{\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    8. Simplified20.2

      \[\leadsto \frac{\frac{\log \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

    if -4.2771113256202204e-185 < re < 3.510677278861231e-284

    1. Initial program 32.3

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

    2. Taylor expanded around 0 35.1

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

    if 5.945116929153346e+55 < re

    1. Initial program 45.5

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

    2. Taylor expanded around inf 10.8

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right)} + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1225739876249368 \cdot 10^{21}:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -4.27711132562022042 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 3.510677278861231 \cdot 10^{-284}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 5.9451169291533458 \cdot 10^{55}:\\ \;\;\;\;\frac{\frac{\log \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))