math.log10 on complex, real part

Average Error: 31.4 → 17.5

Time: 29.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -86.01993975715544:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)} \cdot \frac{-1}{2}}\\ \mathbf{elif}\;re \le 4.875955891928815 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\sqrt{\log 10}}{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 5.785268620035206 \cdot 10^{-190}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log im} \cdot \frac{1}{2}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 1.1977795536433536 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

C

double f(double re, double im) {
        double r761818 = re;
        double r761819 = r761818 * r761818;
        double r761820 = im;
        double r761821 = r761820 * r761820;
        double r761822 = r761819 + r761821;
        double r761823 = sqrt(r761822);
        double r761824 = log(r761823);
        double r761825 = 10.0;
        double r761826 = log(r761825);
        double r761827 = r761824 / r761826;
        return r761827;
}

↓

double f(double re, double im) {
        double r761828 = re;
        double r761829 = -86.01993975715544;
        bool r761830 = r761828 <= r761829;
        double r761831 = 0.5;
        double r761832 = sqrt(r761831);
        double r761833 = 10.0;
        double r761834 = log(r761833);
        double r761835 = -1.0;
        double r761836 = r761835 / r761828;
        double r761837 = log(r761836);
        double r761838 = r761832 * r761837;
        double r761839 = r761834 / r761838;
        double r761840 = -0.5;
        double r761841 = r761839 * r761840;
        double r761842 = r761832 / r761841;
        double r761843 = 4.875955891928815e-233;
        bool r761844 = r761828 <= r761843;
        double r761845 = sqrt(r761834);
        double r761846 = r761828 * r761828;
        double r761847 = im;
        double r761848 = r761847 * r761847;
        double r761849 = r761846 + r761848;
        double r761850 = log(r761849);
        double r761851 = r761850 / r761845;
        double r761852 = r761845 / r761851;
        double r761853 = r761852 / r761832;
        double r761854 = r761832 / r761853;
        double r761855 = 5.785268620035206e-190;
        bool r761856 = r761828 <= r761855;
        double r761857 = log(r761847);
        double r761858 = r761834 / r761857;
        double r761859 = r761858 * r761831;
        double r761860 = r761859 / r761832;
        double r761861 = r761832 / r761860;
        double r761862 = 1.1977795536433536e+113;
        bool r761863 = r761828 <= r761862;
        double r761864 = r761845 / r761850;
        double r761865 = r761832 / r761864;
        double r761866 = r761832 / r761845;
        double r761867 = r761865 * r761866;
        double r761868 = log(r761828);
        double r761869 = r761868 / r761834;
        double r761870 = r761863 ? r761867 : r761869;
        double r761871 = r761856 ? r761861 : r761870;
        double r761872 = r761844 ? r761854 : r761871;
        double r761873 = r761830 ? r761842 : r761872;
        return r761873;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -86.01993975715544

   1. Initial program 38.8

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

   3. Applied pow1 38.8

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

   4. Applied log-pow 38.8

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

   5. Applied associate-/r* 38.8

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

   6. Simplified 38.8

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

   8. Applied pow1/2 38.8

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

   9. Applied log-pow 38.8

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

   10. Applied associate-/l* 38.8

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

   12. Applied add-sqr-sqrt 38.9

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

   13. Applied associate-/l* 38.8

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

   14. Taylor expanded around -inf 12.6

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

   if -86.01993975715544 < re < 4.875955891928815e-233

   1. Initial program 24.6

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

   3. Applied pow1 24.6

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

   4. Applied log-pow 24.6

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

   5. Applied associate-/r* 24.6

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

   6. Simplified 24.6

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

   8. Applied pow1/2 24.6

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

   9. Applied log-pow 24.6

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

   10. Applied associate-/l* 24.6

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

   12. Applied add-sqr-sqrt 24.7

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

   13. Applied associate-/l* 24.5

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

   15. Applied add-sqr-sqrt 24.5

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

   16. Applied associate-/l* 24.5

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

   if 4.875955891928815e-233 < re < 5.785268620035206e-190

   1. Initial program 34.0

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

   3. Applied pow1 34.0

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

   4. Applied log-pow 34.0

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

   5. Applied associate-/r* 34.0

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

   6. Simplified 34.0

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

   8. Applied pow1/2 34.0

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

   9. Applied log-pow 34.0

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

   10. Applied associate-/l* 34.0

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

   12. Applied add-sqr-sqrt 34.1

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

   13. Applied associate-/l* 34.0

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

   14. Taylor expanded around 0 31.7

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

   if 5.785268620035206e-190 < re < 1.1977795536433536e+113

   1. Initial program 17.3

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

   3. Applied pow1 17.3

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

   4. Applied log-pow 17.3

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

   5. Applied associate-/r* 17.3

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

   6. Simplified 17.3

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

   8. Applied pow1/2 17.3

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

   9. Applied log-pow 17.3

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

   10. Applied associate-/l* 17.3

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

   12. Applied pow1 17.3

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

   13. Applied log-pow 17.3

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

   14. Applied add-sqr-sqrt 17.3

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

   15. Applied times-frac 17.5

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

   16. Applied add-sqr-sqrt 17.3

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

   17. Applied times-frac 17.2

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

   18. Simplified 17.2

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

   if 1.1977795536433536e+113 < re

   1. Initial program 53.1

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

   3. Applied pow1 53.1

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

   4. Applied log-pow 53.1

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

   5. Applied associate-/r* 53.1

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

   6. Simplified 53.1

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

   7. Taylor expanded around inf 9.4

      \[\leadsto \frac{\log \color{blue}{re}}{\log 10}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -86.01993975715544:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\log 10}{\sqrt{\frac{1}{2}} \cdot \log \left(\frac{-1}{re}\right)} \cdot \frac{-1}{2}}\\ \mathbf{elif}\;re \le 4.875955891928815 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\sqrt{\log 10}}{\frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 5.785268620035206 \cdot 10^{-190}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\frac{\log 10}{\log im} \cdot \frac{1}{2}}{\sqrt{\frac{1}{2}}}}\\ \mathbf{elif}\;re \le 1.1977795536433536 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re}{\log 10}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))