math.log/2 on complex, real part

Average Error: 32.3 → 17.9

Time: 47.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.9762771458899918 \cdot 10^{145}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -9.7555531765393797 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -5.52667415299497418 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{0.0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log im}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {0.0}^{6}}} \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}\\ \mathbf{elif}\;re \le 5.037031638582551 \cdot 10^{101}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r47991 = re;
        double r47992 = r47991 * r47991;
        double r47993 = im;
        double r47994 = r47993 * r47993;
        double r47995 = r47992 + r47994;
        double r47996 = sqrt(r47995);
        double r47997 = log(r47996);
        double r47998 = base;
        double r47999 = log(r47998);
        double r48000 = r47997 * r47999;
        double r48001 = atan2(r47993, r47991);
        double r48002 = 0.0;
        double r48003 = r48001 * r48002;
        double r48004 = r48000 + r48003;
        double r48005 = r47999 * r47999;
        double r48006 = r48002 * r48002;
        double r48007 = r48005 + r48006;
        double r48008 = r48004 / r48007;
        return r48008;
}

↓

double f(double re, double im, double base) {
        double r48009 = re;
        double r48010 = -1.9762771458899918e+145;
        bool r48011 = r48009 <= r48010;
        double r48012 = -r48009;
        double r48013 = log(r48012);
        double r48014 = base;
        double r48015 = log(r48014);
        double r48016 = r48013 * r48015;
        double r48017 = im;
        double r48018 = atan2(r48017, r48009);
        double r48019 = 0.0;
        double r48020 = r48018 * r48019;
        double r48021 = r48016 + r48020;
        double r48022 = 2.0;
        double r48023 = cbrt(r48014);
        double r48024 = log(r48023);
        double r48025 = r48022 * r48024;
        double r48026 = r48025 * r48015;
        double r48027 = r48024 * r48015;
        double r48028 = r48026 + r48027;
        double r48029 = r48019 * r48019;
        double r48030 = r48028 + r48029;
        double r48031 = r48021 / r48030;
        double r48032 = -9.75555317653938e-240;
        bool r48033 = r48009 <= r48032;
        double r48034 = 1.0;
        double r48035 = pow(r48015, r48022);
        double r48036 = r48035 + r48029;
        double r48037 = sqrt(r48036);
        double r48038 = r48034 / r48037;
        double r48039 = r48009 * r48009;
        double r48040 = r48017 * r48017;
        double r48041 = r48039 + r48040;
        double r48042 = sqrt(r48041);
        double r48043 = log(r48042);
        double r48044 = r48043 * r48015;
        double r48045 = r48044 + r48020;
        double r48046 = r48045 / r48037;
        double r48047 = r48038 * r48046;
        double r48048 = -5.526674152994974e-299;
        bool r48049 = r48009 <= r48048;
        double r48050 = r48019 * r48018;
        double r48051 = log(r48017);
        double r48052 = r48015 * r48051;
        double r48053 = r48050 + r48052;
        double r48054 = r48053 / r48037;
        double r48055 = 3.0;
        double r48056 = pow(r48035, r48055);
        double r48057 = 6.0;
        double r48058 = pow(r48019, r48057);
        double r48059 = r48056 + r48058;
        double r48060 = sqrt(r48059);
        double r48061 = r48054 / r48060;
        double r48062 = r48035 * r48035;
        double r48063 = r48029 * r48029;
        double r48064 = r48035 * r48029;
        double r48065 = r48063 - r48064;
        double r48066 = r48062 + r48065;
        double r48067 = sqrt(r48066);
        double r48068 = r48061 * r48067;
        double r48069 = 5.037031638582551e+101;
        bool r48070 = r48009 <= r48069;
        double r48071 = r48045 / r48030;
        double r48072 = log(r48009);
        double r48073 = -r48072;
        double r48074 = -r48015;
        double r48075 = r48073 / r48074;
        double r48076 = r48070 ? r48071 : r48075;
        double r48077 = r48049 ? r48068 : r48076;
        double r48078 = r48033 ? r48047 : r48077;
        double r48079 = r48011 ? r48031 : r48078;
        return r48079;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.9762771458899918e+145

   1. Initial program 60.7

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

   3. Applied add-cube-cbrt 60.7

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

   4. Applied log-prod 60.8

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

   5. Applied distribute-lft-in 60.7

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

   6. Simplified 60.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Simplified 60.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \color{blue}{\log \left(\sqrt[3]{base}\right) \cdot \log base}\right) + 0.0 \cdot 0.0}\]

   8. Taylor expanded around -inf 7.1

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\]

   9. Simplified 7.1

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\]

   if -1.9762771458899918e+145 < re < -9.75555317653938e-240

   1. Initial program 19.7

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

   3. Applied add-sqr-sqrt 19.7

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

   4. Applied *-un-lft-identity 19.7

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

   5. Applied times-frac 19.6

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

   6. Simplified 19.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   7. Simplified 19.6

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}\]

   if -9.75555317653938e-240 < re < -5.526674152994974e-299

   1. Initial program 34.9

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

   3. Applied add-sqr-sqrt 34.9

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

   4. Applied *-un-lft-identity 34.9

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

   5. Applied times-frac 34.9

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

   6. Simplified 34.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   7. Simplified 34.9

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

   9. Applied flip3-+ 34.9

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\color{blue}{\frac{{\left({\left(\log base\right)}^{2}\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}}}}\]

   10. Applied sqrt-div 34.9

       \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\frac{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}}{\sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}}}}\]

   11. Applied associate-/r/ 35.0

       \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \color{blue}{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}\right)}\]

   12. Applied associate-*r* 34.9

       \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}}\right) \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}}\]

   13. Simplified 34.9

       \[\leadsto \color{blue}{\frac{\frac{0.0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {0.0}^{6}}}} \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}\]

   14. Taylor expanded around 0 32.1

       \[\leadsto \frac{\frac{0.0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \color{blue}{im}}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {0.0}^{6}}} \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}\]

   if -5.526674152994974e-299 < re < 5.037031638582551e+101

   1. Initial program 22.6

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

   3. Applied add-cube-cbrt 22.7

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

   4. Applied log-prod 22.7

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

   5. Applied distribute-lft-in 22.7

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

   6. Simplified 22.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Simplified 22.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \color{blue}{\log \left(\sqrt[3]{base}\right) \cdot \log base}\right) + 0.0 \cdot 0.0}\]

   if 5.037031638582551e+101 < re

   1. Initial program 53.0

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

   2. Taylor expanded around inf 9.4

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

   3. Simplified 9.4

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.9762771458899918 \cdot 10^{145}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -9.7555531765393797 \cdot 10^{-240}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -5.52667415299497418 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{0.0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log im}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{{\left({\left(\log base\right)}^{2}\right)}^{3} + {0.0}^{6}}} \cdot \sqrt{{\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2} + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot \left(0.0 \cdot 0.0\right)\right)}\\ \mathbf{elif}\;re \le 5.037031638582551 \cdot 10^{101}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))