math.log10 on complex, real part

Average Error: 32.3 → 18.2

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left({\left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right)}^{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right) + \log \left({\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left({\left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right)}^{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right) + \log \left({\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r111793 = re;
        double r111794 = r111793 * r111793;
        double r111795 = im;
        double r111796 = r111795 * r111795;
        double r111797 = r111794 + r111796;
        double r111798 = sqrt(r111797);
        double r111799 = log(r111798);
        double r111800 = 10.0;
        double r111801 = log(r111800);
        double r111802 = r111799 / r111801;
        return r111802;
}

↓

double f(double re, double im) {
        double r111803 = re;
        double r111804 = -2.6163751838198674e+143;
        bool r111805 = r111803 <= r111804;
        double r111806 = 1.0;
        double r111807 = 10.0;
        double r111808 = log(r111807);
        double r111809 = sqrt(r111808);
        double r111810 = r111806 / r111809;
        double r111811 = -1.0;
        double r111812 = r111811 * r111803;
        double r111813 = pow(r111812, r111810);
        double r111814 = log(r111813);
        double r111815 = r111810 * r111814;
        double r111816 = -2.0278572522938575e-184;
        bool r111817 = r111803 <= r111816;
        double r111818 = 2.0;
        double r111819 = r111803 * r111803;
        double r111820 = im;
        double r111821 = r111820 * r111820;
        double r111822 = r111819 + r111821;
        double r111823 = cbrt(r111822);
        double r111824 = fabs(r111823);
        double r111825 = cbrt(r111809);
        double r111826 = r111825 * r111825;
        double r111827 = r111806 / r111826;
        double r111828 = sqrt(r111818);
        double r111829 = r111827 / r111828;
        double r111830 = pow(r111824, r111829);
        double r111831 = sqrt(r111806);
        double r111832 = r111831 / r111825;
        double r111833 = r111832 / r111828;
        double r111834 = pow(r111830, r111833);
        double r111835 = log(r111834);
        double r111836 = r111818 * r111835;
        double r111837 = sqrt(r111823);
        double r111838 = pow(r111837, r111810);
        double r111839 = log(r111838);
        double r111840 = r111836 + r111839;
        double r111841 = r111810 * r111840;
        double r111842 = -8.804121592020497e-274;
        bool r111843 = r111803 <= r111842;
        double r111844 = log(r111820);
        double r111845 = r111844 / r111808;
        double r111846 = 1.9824297347259786e+18;
        bool r111847 = r111803 <= r111846;
        double r111848 = r111806 / r111803;
        double r111849 = r111806 / r111808;
        double r111850 = sqrt(r111849);
        double r111851 = -r111850;
        double r111852 = pow(r111848, r111851);
        double r111853 = log(r111852);
        double r111854 = r111810 * r111853;
        double r111855 = r111847 ? r111841 : r111854;
        double r111856 = r111843 ? r111845 : r111855;
        double r111857 = r111817 ? r111841 : r111856;
        double r111858 = r111805 ? r111815 : r111857;
        return r111858;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.6163751838198674e+143

   1. Initial program 60.8

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

   3. Applied add-sqr-sqrt 60.8

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

   4. Applied pow1 60.8

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

   5. Applied log-pow 60.8

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

   6. Applied times-frac 60.8

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

   8. Applied add-log-exp 60.8

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

   9. Simplified 60.8

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

   10. Taylor expanded around -inf 7.0

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

   if -2.6163751838198674e+143 < re < -2.0278572522938575e-184 or -8.804121592020497e-274 < re < 1.9824297347259786e+18

   1. Initial program 20.7

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

   3. Applied add-sqr-sqrt 20.7

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

   4. Applied pow1 20.7

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

   5. Applied log-pow 20.7

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

   6. Applied times-frac 20.6

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

   8. Applied add-log-exp 20.6

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

   9. Simplified 20.5

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

   11. Applied add-cube-cbrt 20.5

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

   12. Applied sqrt-prod 20.5

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

   13. Applied unpow-prod-down 20.5

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

   14. Applied log-prod 20.5

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

   15. Simplified 20.5

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\color{blue}{2 \cdot \log \left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right)}^{\left(\frac{\frac{1}{\sqrt{\log 10}}}{2}\right)}\right)} + \log \left({\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\]
   16. Using strategy rm

   17. Applied add-sqr-sqrt 20.8

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

   18. Applied add-cube-cbrt 20.7

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

   19. Applied add-sqr-sqrt 20.7

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

   20. Applied times-frac 20.7

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

   21. Applied times-frac 20.5

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

   22. Applied pow-unpow 20.5

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

   23. Simplified 20.5

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

   if -2.0278572522938575e-184 < re < -8.804121592020497e-274

   1. Initial program 31.4

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

   2. Taylor expanded around 0 37.0

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

   if 1.9824297347259786e+18 < re

   1. Initial program 43.0

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

   3. Applied add-sqr-sqrt 43.0

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

   4. Applied pow1 43.0

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

   5. Applied log-pow 43.0

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

   6. Applied times-frac 43.0

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

   8. Applied add-log-exp 43.0

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

   9. Simplified 42.9

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

   10. Taylor expanded around inf 13.5

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

   11. Simplified 13.4

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.61637518381986743 \cdot 10^{143}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left({\left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right)}^{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right) + \log \left({\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{elif}\;re \le -8.8041215920204974 \cdot 10^{-274}:\\ \;\;\;\;\frac{\log im}{\log 10}\\ \mathbf{elif}\;re \le 1982429734725978600:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left({\left({\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right|\right)}^{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{\sqrt{1}}{\sqrt[3]{\sqrt{\log 10}}}}{\sqrt{2}}\right)}\right) + \log \left({\left(\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\frac{1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))