math.log/2 on complex, real part

Average Error: 31.9 → 18.0

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.4702290859562653754013000213245692038 \cdot 10^{113}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -1.607695456247534001972433141424386115394 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.791830782571059225299768583227875412425 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \left(\sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)} \cdot \frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)\\ \mathbf{elif}\;re \le 1.230336623445155882199636404066337824284 \cdot 10^{145}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right)} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \sqrt{\log base \cdot \log base - 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r43122 = re;
        double r43123 = r43122 * r43122;
        double r43124 = im;
        double r43125 = r43124 * r43124;
        double r43126 = r43123 + r43125;
        double r43127 = sqrt(r43126);
        double r43128 = log(r43127);
        double r43129 = base;
        double r43130 = log(r43129);
        double r43131 = r43128 * r43130;
        double r43132 = atan2(r43124, r43122);
        double r43133 = 0.0;
        double r43134 = r43132 * r43133;
        double r43135 = r43131 + r43134;
        double r43136 = r43130 * r43130;
        double r43137 = r43133 * r43133;
        double r43138 = r43136 + r43137;
        double r43139 = r43135 / r43138;
        return r43139;
}

↓

double f(double re, double im, double base) {
        double r43140 = re;
        double r43141 = -1.4702290859562654e+113;
        bool r43142 = r43140 <= r43141;
        double r43143 = -1.0;
        double r43144 = r43143 * r43140;
        double r43145 = log(r43144);
        double r43146 = base;
        double r43147 = log(r43146);
        double r43148 = r43145 * r43147;
        double r43149 = im;
        double r43150 = atan2(r43149, r43140);
        double r43151 = 0.0;
        double r43152 = r43150 * r43151;
        double r43153 = r43148 + r43152;
        double r43154 = r43147 * r43147;
        double r43155 = r43151 * r43151;
        double r43156 = r43154 + r43155;
        double r43157 = r43153 / r43156;
        double r43158 = -1.607695456247534e-186;
        bool r43159 = r43140 <= r43158;
        double r43160 = 1.0;
        double r43161 = 2.0;
        double r43162 = pow(r43147, r43161);
        double r43163 = r43162 + r43155;
        double r43164 = sqrt(r43163);
        double r43165 = r43160 / r43164;
        double r43166 = r43140 * r43140;
        double r43167 = r43149 * r43149;
        double r43168 = r43166 + r43167;
        double r43169 = sqrt(r43168);
        double r43170 = log(r43169);
        double r43171 = r43170 * r43147;
        double r43172 = r43171 + r43152;
        double r43173 = sqrt(r43156);
        double r43174 = r43172 / r43173;
        double r43175 = r43165 * r43174;
        double r43176 = 1.7918307825710592e-180;
        bool r43177 = r43140 <= r43176;
        double r43178 = 3.0;
        double r43179 = pow(r43154, r43178);
        double r43180 = pow(r43155, r43178);
        double r43181 = r43179 + r43180;
        double r43182 = sqrt(r43181);
        double r43183 = r43160 / r43182;
        double r43184 = r43154 * r43154;
        double r43185 = r43155 * r43155;
        double r43186 = r43154 * r43155;
        double r43187 = r43185 - r43186;
        double r43188 = r43184 + r43187;
        double r43189 = sqrt(r43188);
        double r43190 = log(r43149);
        double r43191 = r43190 * r43147;
        double r43192 = r43191 + r43152;
        double r43193 = r43192 / r43173;
        double r43194 = r43189 * r43193;
        double r43195 = r43183 * r43194;
        double r43196 = 1.2303366234451559e+145;
        bool r43197 = r43140 <= r43196;
        double r43198 = r43184 - r43185;
        double r43199 = sqrt(r43198);
        double r43200 = r43199 * r43173;
        double r43201 = r43172 / r43200;
        double r43202 = r43154 - r43155;
        double r43203 = sqrt(r43202);
        double r43204 = r43201 * r43203;
        double r43205 = r43160 / r43140;
        double r43206 = log(r43205);
        double r43207 = r43160 / r43146;
        double r43208 = log(r43207);
        double r43209 = r43206 / r43208;
        double r43210 = r43197 ? r43204 : r43209;
        double r43211 = r43177 ? r43195 : r43210;
        double r43212 = r43159 ? r43175 : r43211;
        double r43213 = r43142 ? r43157 : r43212;
        return r43213;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.4702290859562654e+113

   1. Initial program 54.9

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

   2. Taylor expanded around -inf 9.1

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

   if -1.4702290859562654e+113 < re < -1.607695456247534e-186

   1. Initial program 18.3

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

   3. Applied add-sqr-sqrt 18.3

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

   4. Applied *-un-lft-identity 18.3

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

   5. Applied times-frac 18.3

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

   7. Applied pow1 18.3

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

   8. Applied pow1 18.3

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

   9. Applied pow-prod-up 18.3

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

   10. Simplified 18.3

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

   if -1.607695456247534e-186 < re < 1.7918307825710592e-180

   1. Initial program 31.6

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

   3. Applied add-sqr-sqrt 31.6

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

   4. Applied *-un-lft-identity 31.6

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

   5. Applied times-frac 31.6

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

   7. Applied flip3-+ 31.7

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

   8. Applied sqrt-div 31.7

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

   9. Applied associate-/r/ 31.7

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

   10. Applied associate-*l* 31.7

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

   11. Taylor expanded around 0 35.0

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

   if 1.7918307825710592e-180 < re < 1.2303366234451559e+145

   1. Initial program 16.5

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

   3. Applied add-sqr-sqrt 16.5

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

   4. Applied *-un-lft-identity 16.5

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

   5. Applied times-frac 16.5

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

   7. Applied flip-+ 16.6

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

   8. Applied sqrt-div 16.6

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

   9. Applied associate-/r/ 16.6

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

   10. Applied associate-*r* 16.6

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

   11. Simplified 16.6

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

   if 1.2303366234451559e+145 < re

   1. Initial program 60.5

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

   2. Taylor expanded around inf 6.6

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4702290859562653754013000213245692038 \cdot 10^{113}:\\ \;\;\;\;\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -1.607695456247534001972433141424386115394 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.791830782571059225299768583227875412425 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base \cdot \log base\right)}^{3} + {\left(0.0 \cdot 0.0\right)}^{3}}} \cdot \left(\sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) + \left(\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - \left(\log base \cdot \log base\right) \cdot \left(0.0 \cdot 0.0\right)\right)} \cdot \frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\right)\\ \mathbf{elif}\;re \le 1.230336623445155882199636404066337824284 \cdot 10^{145}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right)} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \sqrt{\log base \cdot \log base - 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))