math.log10 on complex, real part

1. Initial program 53.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. Step-by-step derivation

   1. *-un-lft-identity99.1%

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   2. add-sqr-sqrt99.1%

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

   3. times-frac99.2%

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

5. Applied egg-rr99.2%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}} \]
6. Step-by-step derivation

   1. clear-num99.1%

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

   2. un-div-inv99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]

   3. pow1/299.1%

      \[\leadsto \frac{\frac{1}{\color{blue}{{\log 10}^{0.5}}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   4. pow-flip99.1%

      \[\leadsto \frac{\color{blue}{{\log 10}^{\left(-0.5\right)}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

   5. metadata-eval99.1%

      \[\leadsto \frac{{\log 10}^{\color{blue}{-0.5}}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}} \]

7. Applied egg-rr99.1%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\frac{\sqrt{\log 10}}{\log \left(\mathsf{hypot}\left(re, im\right)\right)}}} \]
8. Step-by-step derivation

   1. associate-/r/99.5%

      \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

9. Simplified99.5%

   \[\leadsto \color{blue}{\frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)} \]

10. Final simplification99.5%

    \[\leadsto \frac{{\log 10}^{-0.5}}{\sqrt{\log 10}} \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right) \]

1. Initial program 53.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
4. Step-by-step derivation

   1. expm1-log1p-u67.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   2. expm1-udef67.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)} - 1} \]

   3. log1p-udef67.9%

      \[\leadsto e^{\color{blue}{\log \left(1 + \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)}} - 1 \]

   4. add-exp-log99.1%

      \[\leadsto \color{blue}{\left(1 + \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)} - 1 \]

5. Applied egg-rr99.1%

   \[\leadsto \color{blue}{\left(1 + \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right) - 1} \]

6. Final simplification99.1%

   \[\leadsto \left(1 + \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right) + -1 \]

1. Initial program 53.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

4. Final simplification99.1%

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10} \]

1. Split input into 2 regimes

2. if re < -1.07999999999999994e-44

   1. Initial program 39.0%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   5. Applied egg-rr90.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   6. Taylor expanded in re around -inf 72.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + -1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def71.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]

      2. mul-1-neg71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

      3. distribute-neg-frac71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

   8. Simplified71.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   9. Step-by-step derivation

      1. expm1-udef71.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\right)} - 1} \]

      2. log1p-udef72.0%

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

      3. add-exp-log80.5%

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

      4. neg-log80.5%

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

      5. frac-2neg80.5%

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

      6. metadata-eval80.5%

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

      7. remove-double-div80.5%

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

      8. log1p-expm1-u80.5%

         \[\leadsto \left(1 + \frac{\log \left(-re\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}}\right) - 1 \]

      9. expm1-udef80.5%

         \[\leadsto \left(1 + \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)}\right) - 1 \]

      10. add-exp-log80.5%

          \[\leadsto \left(1 + \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{10} - 1\right)}\right) - 1 \]

      11. metadata-eval80.5%

          \[\leadsto \left(1 + \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{9}\right)}\right) - 1 \]

   10. Applied egg-rr80.5%

       \[\leadsto \color{blue}{\left(1 + \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\right) - 1} \]

   if -1.07999999999999994e-44 < re

   1. Initial program 57.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

   4. Taylor expanded in re around 0 34.4%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
   5. Step-by-step derivation

      1. clear-num34.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

      2. inv-pow34.4%

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

   6. Applied egg-rr34.4%

      \[\leadsto \color{blue}{{\left(\frac{\log 10}{\log im}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-134.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   8. Simplified34.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.08 \cdot 10^{-44}:\\ \;\;\;\;\left(1 + \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

1. Split input into 2 regimes

2. if re < -1.35e-44

   1. Initial program 39.0%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   5. Applied egg-rr90.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   6. Taylor expanded in re around -inf 72.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + -1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def71.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]

      2. mul-1-neg71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

      3. distribute-neg-frac71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

   8. Simplified71.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u80.5%

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

      2. *-un-lft-identity80.5%

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

      3. neg-log80.5%

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

      4. frac-2neg80.5%

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

      5. metadata-eval80.5%

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

      6. remove-double-div80.5%

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

      7. log1p-expm1-u80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}} \]

      8. expm1-udef80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)} \]

      9. add-exp-log80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{10} - 1\right)} \]

      10. metadata-eval80.5%

          \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{9}\right)} \]

   10. Applied egg-rr80.5%

       \[\leadsto \color{blue}{1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity80.5%

          \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]

   12. Simplified80.5%

       \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]

   if -1.35e-44 < re

   1. Initial program 57.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

   4. Taylor expanded in re around 0 34.4%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
   5. Step-by-step derivation

      1. clear-num34.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

      2. inv-pow34.4%

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

   6. Applied egg-rr34.4%

      \[\leadsto \color{blue}{{\left(\frac{\log 10}{\log im}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-134.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]

   8. Simplified34.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log 10}{\log im}}\\ \end{array} \]

1. Split input into 2 regimes

2. if re < -1.2999999999999999e-44

   1. Initial program 39.0%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   5. Applied egg-rr90.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\right)\right)} \]

   6. Taylor expanded in re around -inf 72.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + -1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   7. Step-by-step derivation

      1. log1p-def71.9%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]

      2. mul-1-neg71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

      3. distribute-neg-frac71.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}}\right)\right) \]

   8. Simplified71.9%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\right)}\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u80.5%

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

      2. *-un-lft-identity80.5%

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

      3. neg-log80.5%

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

      4. frac-2neg80.5%

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

      5. metadata-eval80.5%

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

      6. remove-double-div80.5%

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

      7. log1p-expm1-u80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log 10\right)\right)}} \]

      8. expm1-udef80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{e^{\log 10} - 1}\right)} \]

      9. add-exp-log80.5%

         \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{10} - 1\right)} \]

      10. metadata-eval80.5%

          \[\leadsto 1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(\color{blue}{9}\right)} \]

   10. Applied egg-rr80.5%

       \[\leadsto \color{blue}{1 \cdot \frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity80.5%

          \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]

   12. Simplified80.5%

       \[\leadsto \color{blue}{\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}} \]

   if -1.2999999999999999e-44 < re

   1. Initial program 57.3%

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
   2. Step-by-step derivation

      1. hypot-def99.1%

         \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

   4. Taylor expanded in re around 0 34.4%

      \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
3. Recombined 2 regimes into one program.

4. Final simplification43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log \left(-re\right)}{\mathsf{log1p}\left(9\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \end{array} \]

1. Initial program 53.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

4. Taylor expanded in re around 0 29.4%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]
5. Step-by-step derivation

   1. frac-2neg29.4%

      \[\leadsto \color{blue}{\frac{-\log im}{-\log 10}} \]

   2. div-inv29.2%

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

   3. neg-log29.3%

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

   4. metadata-eval29.3%

      \[\leadsto \left(-\log im\right) \cdot \frac{1}{\log \color{blue}{0.1}} \]

6. Applied egg-rr29.3%

   \[\leadsto \color{blue}{\left(-\log im\right) \cdot \frac{1}{\log 0.1}} \]
7. Step-by-step derivation

   1. log-rec29.3%

      \[\leadsto \color{blue}{\log \left(\frac{1}{im}\right)} \cdot \frac{1}{\log 0.1} \]

   2. associate-*r/29.3%

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{im}\right) \cdot 1}{\log 0.1}} \]

   3. *-rgt-identity29.3%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1}{im}\right)}}{\log 0.1} \]

   4. log-rec29.3%

      \[\leadsto \frac{\color{blue}{-\log im}}{\log 0.1} \]

8. Simplified29.3%

   \[\leadsto \color{blue}{\frac{-\log im}{\log 0.1}} \]
9. Step-by-step derivation

   1. *-un-lft-identity29.3%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(-\log im\right)}}{\log 0.1} \]

   2. add-cube-cbrt29.0%

      \[\leadsto \frac{1 \cdot \left(-\log im\right)}{\color{blue}{\left(\sqrt[3]{\log 0.1} \cdot \sqrt[3]{\log 0.1}\right) \cdot \sqrt[3]{\log 0.1}}} \]

   3. times-frac29.0%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\log 0.1} \cdot \sqrt[3]{\log 0.1}} \cdot \frac{-\log im}{\sqrt[3]{\log 0.1}}} \]

   4. pow229.0%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}}} \cdot \frac{-\log im}{\sqrt[3]{\log 0.1}} \]

   5. add-sqr-sqrt9.2%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\color{blue}{\sqrt{-\log im} \cdot \sqrt{-\log im}}}{\sqrt[3]{\log 0.1}} \]

   6. sqrt-unprod9.6%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\color{blue}{\sqrt{\left(-\log im\right) \cdot \left(-\log im\right)}}}{\sqrt[3]{\log 0.1}} \]

   7. sqr-neg9.6%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\sqrt{\color{blue}{\log im \cdot \log im}}}{\sqrt[3]{\log 0.1}} \]

   8. sqrt-unprod0.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\color{blue}{\sqrt{\log im} \cdot \sqrt{\log im}}}{\sqrt[3]{\log 0.1}} \]

   9. add-sqr-sqrt2.3%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\color{blue}{\log im}}{\sqrt[3]{\log 0.1}} \]

10. Applied egg-rr2.3%

    \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \cdot \frac{\log im}{\sqrt[3]{\log 0.1}}} \]
11. Step-by-step derivation

    1. associate-*l/2.3%

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{\log im}{\sqrt[3]{\log 0.1}}}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}}} \]

    2. *-lft-identity2.3%

       \[\leadsto \frac{\color{blue}{\frac{\log im}{\sqrt[3]{\log 0.1}}}}{{\left(\sqrt[3]{\log 0.1}\right)}^{2}} \]

    3. associate-/r*2.3%

       \[\leadsto \color{blue}{\frac{\log im}{\sqrt[3]{\log 0.1} \cdot {\left(\sqrt[3]{\log 0.1}\right)}^{2}}} \]

    4. unpow22.3%

       \[\leadsto \frac{\log im}{\sqrt[3]{\log 0.1} \cdot \color{blue}{\left(\sqrt[3]{\log 0.1} \cdot \sqrt[3]{\log 0.1}\right)}} \]

    5. rem-3cbrt-rft2.3%

       \[\leadsto \frac{\log im}{\color{blue}{\log 0.1}} \]

12. Simplified2.3%

    \[\leadsto \color{blue}{\frac{\log im}{\log 0.1}} \]

13. Final simplification2.3%

    \[\leadsto \frac{\log im}{\log 0.1} \]

1. Initial program 53.7%

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10} \]
2. Step-by-step derivation

   1. hypot-def99.1%

      \[\leadsto \frac{\log \color{blue}{\left(\mathsf{hypot}\left(re, im\right)\right)}}{\log 10} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}} \]

4. Taylor expanded in re around 0 29.4%

   \[\leadsto \color{blue}{\frac{\log im}{\log 10}} \]

5. Final simplification29.4%

   \[\leadsto \frac{\log im}{\log 10} \]

herbie shell --seed 2023167 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))