Average Error: 32.0 → 17.9
Time: 7.8s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.151888904100587124379023402396569343398 \cdot 10^{58}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 9.450669110711002207218388065050424546591 \cdot 10^{-302}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 6.665470315389402134305579687920046870896 \cdot 10^{-268}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 5.168286053687553277469579473627394075561 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]
\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 -1.151888904100587124379023402396569343398 \cdot 10^{58}:\\
\;\;\;\;\frac{\frac{\log \left(-1 \cdot re\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 9.450669110711002207218388065050424546591 \cdot 10^{-302}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\

\mathbf{elif}\;re \le 6.665470315389402134305579687920046870896 \cdot 10^{-268}:\\
\;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\

\mathbf{elif}\;re \le 5.168286053687553277469579473627394075561 \cdot 10^{91}:\\
\;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\

\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 r42883 = re;
        double r42884 = r42883 * r42883;
        double r42885 = im;
        double r42886 = r42885 * r42885;
        double r42887 = r42884 + r42886;
        double r42888 = sqrt(r42887);
        double r42889 = log(r42888);
        double r42890 = base;
        double r42891 = log(r42890);
        double r42892 = r42889 * r42891;
        double r42893 = atan2(r42885, r42883);
        double r42894 = 0.0;
        double r42895 = r42893 * r42894;
        double r42896 = r42892 + r42895;
        double r42897 = r42891 * r42891;
        double r42898 = r42894 * r42894;
        double r42899 = r42897 + r42898;
        double r42900 = r42896 / r42899;
        return r42900;
}
↓
double f(double re, double im, double base) {
        double r42901 = re;
        double r42902 = -1.1518889041005871e+58;
        bool r42903 = r42901 <= r42902;
        double r42904 = -1.0;
        double r42905 = r42904 * r42901;
        double r42906 = log(r42905);
        double r42907 = base;
        double r42908 = log(r42907);
        double r42909 = r42906 * r42908;
        double r42910 = im;
        double r42911 = atan2(r42910, r42901);
        double r42912 = 0.0;
        double r42913 = r42911 * r42912;
        double r42914 = r42909 + r42913;
        double r42915 = r42908 * r42908;
        double r42916 = r42912 * r42912;
        double r42917 = r42915 + r42916;
        double r42918 = sqrt(r42917);
        double r42919 = r42914 / r42918;
        double r42920 = r42919 / r42918;
        double r42921 = 9.450669110711002e-302;
        bool r42922 = r42901 <= r42921;
        double r42923 = r42901 * r42901;
        double r42924 = r42910 * r42910;
        double r42925 = r42923 + r42924;
        double r42926 = sqrt(r42925);
        double r42927 = log(r42926);
        double r42928 = r42927 * r42908;
        double r42929 = r42928 + r42913;
        double r42930 = 3.0;
        double r42931 = pow(r42912, r42930);
        double r42932 = -r42931;
        double r42933 = r42932 * r42912;
        double r42934 = 4.0;
        double r42935 = pow(r42908, r42934);
        double r42936 = r42933 + r42935;
        double r42937 = r42929 / r42936;
        double r42938 = r42915 - r42916;
        double r42939 = r42937 * r42938;
        double r42940 = 6.665470315389402e-268;
        bool r42941 = r42901 <= r42940;
        double r42942 = log(r42910);
        double r42943 = r42942 * r42908;
        double r42944 = r42943 + r42913;
        double r42945 = r42944 / r42917;
        double r42946 = 5.168286053687553e+91;
        bool r42947 = r42901 <= r42946;
        double r42948 = 1.0;
        double r42949 = r42948 / r42901;
        double r42950 = log(r42949);
        double r42951 = r42948 / r42907;
        double r42952 = log(r42951);
        double r42953 = r42950 * r42952;
        double r42954 = r42953 + r42913;
        double r42955 = r42954 / r42917;
        double r42956 = r42947 ? r42939 : r42955;
        double r42957 = r42941 ? r42945 : r42956;
        double r42958 = r42922 ? r42939 : r42957;
        double r42959 = r42903 ? r42920 : r42958;
        return r42959;
}

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 < -1.1518889041005871e+58

    1. Initial program 45.0

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

    3. Applied add-sqr-sqrt45.0

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

    4. Applied associate-/r*45.0

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

    5. Taylor expanded around -inf 10.4

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

    if -1.1518889041005871e+58 < re < 9.450669110711002e-302 or 6.665470315389402e-268 < re < 5.168286053687553e+91

    1. Initial program 22.3

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

    3. Applied flip-+22.3

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

    4. Applied associate-/r/22.3

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

    5. Simplified22.3

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

    if 9.450669110711002e-302 < re < 6.665470315389402e-268

    1. Initial program 30.3

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

    2. Taylor expanded around 0 32.4

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

    if 5.168286053687553e+91 < re

    1. Initial program 49.5

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

    2. Taylor expanded around inf 9.4

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.151888904100587124379023402396569343398 \cdot 10^{58}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 9.450669110711002207218388065050424546591 \cdot 10^{-302}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 6.665470315389402134305579687920046870896 \cdot 10^{-268}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 5.168286053687553277469579473627394075561 \cdot 10^{91}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right) \cdot \log \left(\frac{1}{base}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))