math.log10 on complex, real part

Average Error: 31.8 → 24.9

Time: 22.5s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;im \le -2.957483221806522487658889360988388662853 \cdot 10^{136}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \mathbf{elif}\;im \le -6.570997735276894967118260067026371622396 \cdot 10^{56}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{elif}\;im \le -3054555687171828544312670244831232:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;im \le -4.843689540090472204066045131134642712938 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{elif}\;im \le 9.677784630305516420633165389576243090092 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \mathbf{elif}\;im \le 3.867993196915089402122192166122080152846 \cdot 10^{-217}:\\ \;\;\;\;\sqrt[3]{{\left(-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;im \le 1.009811225965744635764018839775433289102 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;im \le 5.177363335544844238286158381799834195072 \cdot 10^{86}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r43219 = re;
        double r43220 = r43219 * r43219;
        double r43221 = im;
        double r43222 = r43221 * r43221;
        double r43223 = r43220 + r43222;
        double r43224 = sqrt(r43223);
        double r43225 = log(r43224);
        double r43226 = 10.0;
        double r43227 = log(r43226);
        double r43228 = r43225 / r43227;
        return r43228;
}

↓

double f(double re, double im) {
        double r43229 = im;
        double r43230 = -2.9574832218065225e+136;
        bool r43231 = r43229 <= r43230;
        double r43232 = 1.0;
        double r43233 = 10.0;
        double r43234 = log(r43233);
        double r43235 = sqrt(r43234);
        double r43236 = r43232 / r43235;
        double r43237 = r43232 / r43234;
        double r43238 = sqrt(r43237);
        double r43239 = re;
        double r43240 = log(r43239);
        double r43241 = r43238 * r43240;
        double r43242 = r43236 * r43241;
        double r43243 = -6.570997735276895e+56;
        bool r43244 = r43229 <= r43243;
        double r43245 = cbrt(r43235);
        double r43246 = r43232 / r43245;
        double r43247 = sqrt(r43246);
        double r43248 = sqrt(r43236);
        double r43249 = 3.0;
        double r43250 = pow(r43248, r43249);
        double r43251 = r43239 * r43239;
        double r43252 = r43229 * r43229;
        double r43253 = r43251 + r43252;
        double r43254 = sqrt(r43253);
        double r43255 = log(r43254);
        double r43256 = r43250 * r43255;
        double r43257 = r43247 * r43256;
        double r43258 = r43245 * r43245;
        double r43259 = r43232 / r43258;
        double r43260 = sqrt(r43259);
        double r43261 = r43257 * r43260;
        double r43262 = -3.0545556871718285e+33;
        bool r43263 = r43229 <= r43262;
        double r43264 = -1.0;
        double r43265 = r43264 / r43235;
        double r43266 = r43264 / r43239;
        double r43267 = log(r43266);
        double r43268 = r43267 * r43238;
        double r43269 = r43265 * r43268;
        double r43270 = -4.843689540090472e-262;
        bool r43271 = r43229 <= r43270;
        double r43272 = 9.677784630305516e-234;
        bool r43273 = r43229 <= r43272;
        double r43274 = 3.867993196915089e-217;
        bool r43275 = r43229 <= r43274;
        double r43276 = r43267 / r43234;
        double r43277 = -r43276;
        double r43278 = pow(r43277, r43249);
        double r43279 = cbrt(r43278);
        double r43280 = 1.0098112259657446e-185;
        bool r43281 = r43229 <= r43280;
        double r43282 = r43240 / r43234;
        double r43283 = pow(r43282, r43249);
        double r43284 = cbrt(r43283);
        double r43285 = 5.177363335544844e+86;
        bool r43286 = r43229 <= r43285;
        double r43287 = log(r43229);
        double r43288 = r43287 * r43238;
        double r43289 = r43248 * r43288;
        double r43290 = r43248 * r43289;
        double r43291 = r43286 ? r43261 : r43290;
        double r43292 = r43281 ? r43284 : r43291;
        double r43293 = r43275 ? r43279 : r43292;
        double r43294 = r43273 ? r43242 : r43293;
        double r43295 = r43271 ? r43261 : r43294;
        double r43296 = r43263 ? r43269 : r43295;
        double r43297 = r43244 ? r43261 : r43296;
        double r43298 = r43231 ? r43242 : r43297;
        return r43298;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 6 regimes

2. if im < -2.9574832218065225e+136 or -4.843689540090472e-262 < im < 9.677784630305516e-234

   1. Initial program 48.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 48.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 48.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 48.3

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 48.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]

   7. Taylor expanded around inf 46.9

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-1 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

   8. Simplified 46.9

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)}\]

   if -2.9574832218065225e+136 < im < -6.570997735276895e+56 or -3.0545556871718285e+33 < im < -4.843689540090472e-262 or 1.0098112259657446e-185 < im < 5.177363335544844e+86

   1. Initial program 18.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 18.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 18.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 18.7

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 18.7

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

   8. Applied add-sqr-sqrt 18.7

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\]

   9. Applied associate-*l* 18.8

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

   11. Applied add-cube-cbrt 18.6

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)\]

   12. Applied add-cube-cbrt 18.6

       \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}\right) \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)\]

   13. Applied times-frac 18.6

       \[\leadsto \sqrt{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)\]

   14. Applied sqrt-prod 18.6

       \[\leadsto \color{blue}{\left(\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}}\right)} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)\]

   15. Applied associate-*l* 18.7

       \[\leadsto \color{blue}{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)\right)}\]

   16. Simplified 18.6

       \[\leadsto \sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right)}\]

   if -6.570997735276895e+56 < im < -3.0545556871718285e+33

   1. Initial program 24.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 24.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 24.8

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 24.8

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 24.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]

   7. Taylor expanded around -inf 44.1

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-1 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

   8. Simplified 44.1

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(-\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

   if 9.677784630305516e-234 < im < 3.867993196915089e-217

   1. Initial program 30.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 30.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

   4. Applied add-cbrt-cube 30.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

   5. Applied cbrt-undiv 30.2

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

   6. Simplified 30.2

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]

   7. Taylor expanded around -inf 33.4

      \[\leadsto \sqrt[3]{{\color{blue}{\left(-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}}^{3}}\]

   8. Simplified 33.4

      \[\leadsto \sqrt[3]{{\color{blue}{\left(-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}}^{3}}\]

   if 3.867993196915089e-217 < im < 1.0098112259657446e-185

   1. Initial program 35.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 35.9

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

   4. Applied add-cbrt-cube 35.8

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}{\sqrt[3]{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}\]

   5. Applied cbrt-undiv 35.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\left(\log 10 \cdot \log 10\right) \cdot \log 10}}}\]

   6. Simplified 35.5

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\right)}^{3}}}\]

   7. Taylor expanded around inf 34.9

      \[\leadsto \sqrt[3]{{\left(\frac{\log \color{blue}{re}}{\log 10}\right)}^{3}}\]

   if 5.177363335544844e+86 < im

   1. Initial program 48.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 48.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 48.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 48.4

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 48.4

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

   8. Applied add-sqr-sqrt 48.4

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\]

   9. Applied associate-*l* 48.4

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}\right)}\]

   10. Taylor expanded around 0 9.8

       \[\leadsto \sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \color{blue}{\left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)\]
3. Recombined 6 regimes into one program.

4. Final simplification 24.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.957483221806522487658889360988388662853 \cdot 10^{136}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \mathbf{elif}\;im \le -6.570997735276894967118260067026371622396 \cdot 10^{56}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{elif}\;im \le -3054555687171828544312670244831232:\\ \;\;\;\;\frac{-1}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\\ \mathbf{elif}\;im \le -4.843689540090472204066045131134642712938 \cdot 10^{-262}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{elif}\;im \le 9.677784630305516420633165389576243090092 \cdot 10^{-234}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \mathbf{elif}\;im \le 3.867993196915089402122192166122080152846 \cdot 10^{-217}:\\ \;\;\;\;\sqrt[3]{{\left(-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;im \le 1.009811225965744635764018839775433289102 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log re}{\log 10}\right)}^{3}}\\ \mathbf{elif}\;im \le 5.177363335544844238286158381799834195072 \cdot 10^{86}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}}}} \cdot \left({\left(\sqrt{\frac{1}{\sqrt{\log 10}}}\right)}^{3} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))