math.log10 on complex, real part

Average Error: 32.2 → 18.3

Time: 14.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -9.0150881706299043 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.28868009298563533 \cdot 10^{-244}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.6732898487300134 \cdot 10^{92}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r47245 = re;
        double r47246 = r47245 * r47245;
        double r47247 = im;
        double r47248 = r47247 * r47247;
        double r47249 = r47246 + r47248;
        double r47250 = sqrt(r47249);
        double r47251 = log(r47250);
        double r47252 = 10.0;
        double r47253 = log(r47252);
        double r47254 = r47251 / r47253;
        return r47254;
}

↓

double f(double re, double im) {
        double r47255 = re;
        double r47256 = -9.015088170629904e+107;
        bool r47257 = r47255 <= r47256;
        double r47258 = 1.0;
        double r47259 = 10.0;
        double r47260 = log(r47259);
        double r47261 = sqrt(r47260);
        double r47262 = r47258 / r47261;
        double r47263 = -r47255;
        double r47264 = log(r47263);
        double r47265 = r47264 * r47262;
        double r47266 = r47262 * r47265;
        double r47267 = -2.2886800929856353e-244;
        bool r47268 = r47255 <= r47267;
        double r47269 = r47255 * r47255;
        double r47270 = im;
        double r47271 = r47270 * r47270;
        double r47272 = r47269 + r47271;
        double r47273 = sqrt(r47272);
        double r47274 = log(r47273);
        double r47275 = r47274 * r47262;
        double r47276 = r47262 * r47275;
        double r47277 = 3.3295675416622e-138;
        bool r47278 = r47255 <= r47277;
        double r47279 = log(r47270);
        double r47280 = r47279 * r47262;
        double r47281 = r47262 * r47280;
        double r47282 = 5.673289848730013e+92;
        bool r47283 = r47255 <= r47282;
        double r47284 = r47258 / r47260;
        double r47285 = sqrt(r47284);
        double r47286 = log(r47255);
        double r47287 = r47285 * r47286;
        double r47288 = r47262 * r47287;
        double r47289 = r47283 ? r47276 : r47288;
        double r47290 = r47278 ? r47281 : r47289;
        double r47291 = r47268 ? r47276 : r47290;
        double r47292 = r47257 ? r47266 : r47291;
        return r47292;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -9.015088170629904e+107

   1. Initial program 52.8

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

   3. Applied add-sqr-sqrt 52.8

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

   4. Applied pow1 52.8

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

   5. Applied log-pow 52.8

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

   6. Applied times-frac 52.8

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

   8. Applied add-cbrt-cube 52.8

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

   9. Applied add-cbrt-cube 52.8

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

   10. Applied cbrt-undiv 52.8

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

   11. Simplified 52.8

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

   13. Applied div-inv 52.8

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

   14. Applied unpow-prod-down 52.7

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

   15. Applied cbrt-prod 52.8

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

   16. Simplified 52.7

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

   17. Simplified 52.7

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

   18. Taylor expanded around -inf 9.3

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

   19. Simplified 9.3

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

   if -9.015088170629904e+107 < re < -2.2886800929856353e-244 or 3.3295675416622e-138 < re < 5.673289848730013e+92

   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-cbrt-cube 18.7

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

   9. Applied add-cbrt-cube 18.8

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

   10. Applied cbrt-undiv 18.8

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

   11. Simplified 18.8

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

   13. Applied div-inv 18.8

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

   14. Applied unpow-prod-down 18.7

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

   15. Applied cbrt-prod 18.8

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

   16. Simplified 18.6

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

   17. Simplified 18.6

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

   if -2.2886800929856353e-244 < re < 3.3295675416622e-138

   1. Initial program 30.7

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

   3. Applied add-sqr-sqrt 30.7

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

   4. Applied pow1 30.7

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

   5. Applied log-pow 30.7

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

   6. Applied times-frac 30.7

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

   8. Applied add-cbrt-cube 30.7

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

   9. Applied add-cbrt-cube 30.8

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

   10. Applied cbrt-undiv 30.8

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

   11. Simplified 30.8

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

   13. Applied div-inv 30.8

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

   14. Applied unpow-prod-down 30.7

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

   15. Applied cbrt-prod 30.7

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

   16. Simplified 30.6

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

   17. Simplified 30.6

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

   18. Taylor expanded around 0 34.8

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

   if 5.673289848730013e+92 < re

   1. Initial program 50.1

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

   3. Applied add-sqr-sqrt 50.1

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

   4. Applied pow1 50.1

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

   5. Applied log-pow 50.1

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

   6. Applied times-frac 50.0

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

   8. Applied add-cbrt-cube 50.0

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

   9. Applied add-cbrt-cube 50.1

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

   10. Applied cbrt-undiv 50.1

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

   11. Simplified 50.1

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

   13. Applied div-inv 50.1

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

   14. Applied unpow-prod-down 50.1

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

   15. Applied cbrt-prod 50.1

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

   16. Simplified 50.0

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

   17. Simplified 50.0

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

   18. Taylor expanded around inf 9.0

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

   19. Simplified 9.0

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.0150881706299043 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(-re\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.28868009298563533 \cdot 10^{-244}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 3.3295675416621997 \cdot 10^{-138}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log im \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 5.6732898487300134 \cdot 10^{92}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \log re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))