math.log10 on complex, real part

Average Error: 32.1 → 18.4

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log im}}\\ \mathbf{elif}\;re \le 1.13427855715340043 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log re}}\\ \end{array}\]

C

double f(double re, double im) {
        double r46760 = re;
        double r46761 = r46760 * r46760;
        double r46762 = im;
        double r46763 = r46762 * r46762;
        double r46764 = r46761 + r46763;
        double r46765 = sqrt(r46764);
        double r46766 = log(r46765);
        double r46767 = 10.0;
        double r46768 = log(r46767);
        double r46769 = r46766 / r46768;
        return r46769;
}

↓

double f(double re, double im) {
        double r46770 = re;
        double r46771 = -1.103215696956926e+72;
        bool r46772 = r46770 <= r46771;
        double r46773 = 1.0;
        double r46774 = 10.0;
        double r46775 = log(r46774);
        double r46776 = sqrt(r46775);
        double r46777 = r46773 / r46776;
        double r46778 = -1.0;
        double r46779 = r46778 * r46770;
        double r46780 = log(r46779);
        double r46781 = r46776 / r46780;
        double r46782 = r46777 / r46781;
        double r46783 = -1.3504253849915568e-194;
        bool r46784 = r46770 <= r46783;
        double r46785 = r46770 * r46770;
        double r46786 = im;
        double r46787 = r46786 * r46786;
        double r46788 = r46785 + r46787;
        double r46789 = sqrt(r46788);
        double r46790 = log(r46789);
        double r46791 = r46777 * r46790;
        double r46792 = r46777 * r46791;
        double r46793 = -2.968956980813959e-266;
        bool r46794 = r46770 <= r46793;
        double r46795 = log(r46786);
        double r46796 = r46776 / r46795;
        double r46797 = r46777 / r46796;
        double r46798 = 1.1342785571534004e-228;
        bool r46799 = r46770 <= r46798;
        double r46800 = 1.3057340609530177e-191;
        bool r46801 = r46770 <= r46800;
        double r46802 = 3.0;
        double r46803 = cbrt(r46786);
        double r46804 = log(r46803);
        double r46805 = r46775 / r46804;
        double r46806 = r46802 / r46805;
        double r46807 = 5.156219500915728e+39;
        bool r46808 = r46770 <= r46807;
        double r46809 = log(r46770);
        double r46810 = r46776 / r46809;
        double r46811 = r46777 / r46810;
        double r46812 = r46808 ? r46792 : r46811;
        double r46813 = r46801 ? r46806 : r46812;
        double r46814 = r46799 ? r46792 : r46813;
        double r46815 = r46794 ? r46797 : r46814;
        double r46816 = r46784 ? r46792 : r46815;
        double r46817 = r46772 ? r46782 : r46816;
        return r46817;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -1.103215696956926e+72

   1. Initial program 47.0

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

   3. Applied add-cube-cbrt 47.0

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

   5. Applied pow3 47.0

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

   6. Applied log-pow 47.0

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

   7. Applied associate-/l* 47.0

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

   9. Applied pow1/3 47.1

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

   10. Applied log-pow 47.0

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

   11. Applied add-sqr-sqrt 47.0

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

   12. Applied times-frac 47.2

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

   13. Applied associate-/r* 47.2

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

   14. Simplified 47.0

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

   15. Taylor expanded around -inf 9.8

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

   if -1.103215696956926e+72 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 1.1342785571534004e-228 or 1.3057340609530177e-191 < re < 5.156219500915728e+39

   1. Initial program 21.5

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

   3. Applied add-cube-cbrt 21.5

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

   5. Applied pow3 21.5

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

   6. Applied log-pow 21.5

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

   7. Applied associate-/l* 21.5

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

   9. Applied pow1/3 21.7

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

   10. Applied log-pow 21.7

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

   11. Applied add-sqr-sqrt 21.7

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

   12. Applied times-frac 22.1

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

   13. Applied associate-/r* 22.0

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

   14. Simplified 21.5

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

   16. Applied div-inv 21.5

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

   17. Applied pow1 21.5

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

   18. Applied log-pow 21.5

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

   19. Applied sqrt-prod 21.5

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

   20. Applied add-sqr-sqrt 21.5

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

   21. Applied times-frac 21.5

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

   22. Applied times-frac 21.4

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

   23. Simplified 21.4

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

   24. Simplified 21.4

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

   if -1.3504253849915568e-194 < re < -2.968956980813959e-266

   1. Initial program 32.4

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

   3. Applied add-cube-cbrt 32.4

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

   5. Applied pow3 32.4

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

   6. Applied log-pow 32.4

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

   7. Applied associate-/l* 32.4

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

   9. Applied pow1/3 32.5

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

   10. Applied log-pow 32.5

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

   11. Applied add-sqr-sqrt 32.5

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

   12. Applied times-frac 32.8

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

   13. Applied associate-/r* 32.8

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

   14. Simplified 32.4

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

   15. Taylor expanded around 0 35.6

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

   if 1.1342785571534004e-228 < re < 1.3057340609530177e-191

   1. Initial program 33.0

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

   3. Applied add-cube-cbrt 33.0

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

   5. Applied pow3 33.0

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

   6. Applied log-pow 33.0

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

   7. Applied associate-/l* 33.0

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

   8. Taylor expanded around 0 33.9

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

   if 5.156219500915728e+39 < re

   1. Initial program 44.1

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

   3. Applied add-cube-cbrt 44.0

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

   5. Applied pow3 44.0

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

   6. Applied log-pow 44.1

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

   7. Applied associate-/l* 44.1

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

   9. Applied pow1/3 44.2

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

   10. Applied log-pow 44.1

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

   11. Applied add-sqr-sqrt 44.1

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

   12. Applied times-frac 44.3

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

   13. Applied associate-/r* 44.3

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

   14. Simplified 44.0

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

   15. Taylor expanded around inf 12.1

       \[\leadsto \frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \color{blue}{re}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.10321569695692608 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log \left(-1 \cdot re\right)}}\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log im}}\\ \mathbf{elif}\;re \le 1.13427855715340043 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{elif}\;re \le 1.30573406095301773 \cdot 10^{-191}:\\ \;\;\;\;\frac{3}{\frac{\log 10}{\log \left(\sqrt[3]{im}\right)}}\\ \mathbf{elif}\;re \le 5.15621950091572796 \cdot 10^{39}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\log 10}}}{\frac{\sqrt{\log 10}}{\log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))