math.log10 on complex, real part

Average Error: 32.1 → 17.4

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -6.35114368572189993071033166198605826169 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(\frac{-1}{re}\right) \cdot -2}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le -1.403306634450894669656271333137131311492 \cdot 10^{-200}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{elif}\;re \le -9.644694989558798059127661663723838127542 \cdot 10^{-288}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot -2\right)\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.086754484260071349249942835782837644818 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\left(\log im \cdot 2\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.488227374408068484790604899998157177197 \cdot 10^{120}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r1352224 = re;
        double r1352225 = r1352224 * r1352224;
        double r1352226 = im;
        double r1352227 = r1352226 * r1352226;
        double r1352228 = r1352225 + r1352227;
        double r1352229 = sqrt(r1352228);
        double r1352230 = log(r1352229);
        double r1352231 = 10.0;
        double r1352232 = log(r1352231);
        double r1352233 = r1352230 / r1352232;
        return r1352233;
}

↓

double f(double re, double im) {
        double r1352234 = re;
        double r1352235 = -6.3511436857219e+103;
        bool r1352236 = r1352234 <= r1352235;
        double r1352237 = 0.5;
        double r1352238 = cbrt(r1352237);
        double r1352239 = 10.0;
        double r1352240 = log(r1352239);
        double r1352241 = cbrt(r1352240);
        double r1352242 = -1.0;
        double r1352243 = r1352242 / r1352234;
        double r1352244 = log(r1352243);
        double r1352245 = -2.0;
        double r1352246 = r1352244 * r1352245;
        double r1352247 = r1352241 / r1352246;
        double r1352248 = r1352238 / r1352247;
        double r1352249 = r1352238 / r1352241;
        double r1352250 = r1352249 * r1352249;
        double r1352251 = r1352248 * r1352250;
        double r1352252 = -1.4033066344508947e-200;
        bool r1352253 = r1352234 <= r1352252;
        double r1352254 = im;
        double r1352255 = r1352254 * r1352254;
        double r1352256 = r1352234 * r1352234;
        double r1352257 = r1352255 + r1352256;
        double r1352258 = log(r1352257);
        double r1352259 = r1352241 / r1352258;
        double r1352260 = r1352238 / r1352259;
        double r1352261 = r1352250 * r1352260;
        double r1352262 = -9.644694989558798e-288;
        bool r1352263 = r1352234 <= r1352262;
        double r1352264 = sqrt(r1352237);
        double r1352265 = sqrt(r1352240);
        double r1352266 = r1352264 / r1352265;
        double r1352267 = r1352242 / r1352254;
        double r1352268 = log(r1352267);
        double r1352269 = r1352268 * r1352245;
        double r1352270 = r1352266 * r1352269;
        double r1352271 = r1352270 * r1352266;
        double r1352272 = 1.0867544842600713e-222;
        bool r1352273 = r1352234 <= r1352272;
        double r1352274 = log(r1352254);
        double r1352275 = 2.0;
        double r1352276 = r1352274 * r1352275;
        double r1352277 = r1352276 * r1352266;
        double r1352278 = r1352266 * r1352277;
        double r1352279 = 2.4882273744080685e+120;
        bool r1352280 = r1352234 <= r1352279;
        double r1352281 = r1352258 * r1352266;
        double r1352282 = r1352266 * r1352281;
        double r1352283 = 1.0;
        double r1352284 = r1352283 / r1352240;
        double r1352285 = sqrt(r1352284);
        double r1352286 = log(r1352234);
        double r1352287 = r1352286 * r1352264;
        double r1352288 = r1352285 * r1352287;
        double r1352289 = r1352288 * r1352275;
        double r1352290 = r1352266 * r1352289;
        double r1352291 = r1352280 ? r1352282 : r1352290;
        double r1352292 = r1352273 ? r1352278 : r1352291;
        double r1352293 = r1352263 ? r1352271 : r1352292;
        double r1352294 = r1352253 ? r1352261 : r1352293;
        double r1352295 = r1352236 ? r1352251 : r1352294;
        return r1352295;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 6 regimes

2. if re < -6.3511436857219e+103

   1. Initial program 52.4

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

   3. Applied pow1/2 52.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 52.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 52.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 52.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 52.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-cube-cbrt 52.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}\right) \cdot \sqrt[3]{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 52.5

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-cube-cbrt 52.4

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{2}}}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 52.4

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 52.4

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Taylor expanded around -inf 8.4

       \[\leadsto \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}}\]

   15. Simplified 8.4

       \[\leadsto \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\color{blue}{-2 \cdot \log \left(\frac{-1}{re}\right)}}}\]

   if -6.3511436857219e+103 < re < -1.4033066344508947e-200

   1. Initial program 18.7

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

   3. Applied pow1/2 18.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 18.7

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 18.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 18.8

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 18.8

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-cube-cbrt 19.4

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\left(\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}\right) \cdot \sqrt[3]{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 19.3

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-cube-cbrt 18.7

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{2}}}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1} \cdot \frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 18.6

       \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10} \cdot \sqrt[3]{\log 10}}{1}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 18.6

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   if -1.4033066344508947e-200 < re < -9.644694989558798e-288

   1. Initial program 30.4

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

   3. Applied pow1/2 30.4

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 30.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 30.5

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 30.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 30.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 30.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 30.6

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 30.4

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 30.4

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 30.4

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Simplified 30.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\]

   15. Taylor expanded around -inf 35.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\]

   16. Simplified 35.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(\log \left(\frac{-1}{im}\right) \cdot -2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\]

   if -9.644694989558798e-288 < re < 1.0867544842600713e-222

   1. Initial program 34.1

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

   3. Applied pow1/2 34.1

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 34.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 34.1

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 34.1

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 34.1

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 34.1

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 34.2

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 34.1

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 34.0

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 34.0

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Simplified 34.0

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\]

   15. Taylor expanded around inf 31.6

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(-2 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\]

   16. Simplified 31.6

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\color{blue}{\left(\log im \cdot 2\right)} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\]

   if 1.0867544842600713e-222 < re < 2.4882273744080685e+120

   1. Initial program 18.5

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

   3. Applied pow1/2 18.5

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 18.5

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 18.5

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 18.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 18.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 18.5

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 18.7

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 18.5

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 18.4

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 18.4

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Simplified 18.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\]

   if 2.4882273744080685e+120 < re

   1. Initial program 54.8

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

   3. Applied pow1/2 54.8

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 54.8

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 54.8

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 54.8

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 54.8

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 54.8

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 54.8

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 54.8

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 54.7

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 54.7

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Simplified 54.7

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)}\]

   15. Taylor expanded around 0 7.5

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log re\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
3. Recombined 6 regimes into one program.

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.35114368572189993071033166198605826169 \cdot 10^{103}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(\frac{-1}{re}\right) \cdot -2}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{elif}\;re \le -1.403306634450894669656271333137131311492 \cdot 10^{-200}:\\ \;\;\;\;\left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\frac{\sqrt[3]{\log 10}}{\log \left(im \cdot im + re \cdot re\right)}}\\ \mathbf{elif}\;re \le -9.644694989558798059127661663723838127542 \cdot 10^{-288}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(\frac{-1}{im}\right) \cdot -2\right)\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 1.086754484260071349249942835782837644818 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\left(\log im \cdot 2\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 2.488227374408068484790604899998157177197 \cdot 10^{120}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\log \left(im \cdot im + re \cdot re\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log re \cdot \sqrt{\frac{1}{2}}\right)\right) \cdot 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))