math.log10 on complex, real part

Average Error: 32.4 → 18.0

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -44144913.1003666148:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le 5.7520550571681611 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log re \cdot 2}}\\ \end{array}\]

C

double f(double re, double im) {
        double r31640 = re;
        double r31641 = r31640 * r31640;
        double r31642 = im;
        double r31643 = r31642 * r31642;
        double r31644 = r31641 + r31643;
        double r31645 = sqrt(r31644);
        double r31646 = log(r31645);
        double r31647 = 10.0;
        double r31648 = log(r31647);
        double r31649 = r31646 / r31648;
        return r31649;
}

↓

double f(double re, double im) {
        double r31650 = re;
        double r31651 = -44144913.100366615;
        bool r31652 = r31650 <= r31651;
        double r31653 = 0.5;
        double r31654 = 10.0;
        double r31655 = log(r31654);
        double r31656 = sqrt(r31655);
        double r31657 = r31653 / r31656;
        double r31658 = sqrt(r31657);
        double r31659 = cbrt(r31655);
        double r31660 = fabs(r31659);
        double r31661 = r31658 / r31660;
        double r31662 = sqrt(r31659);
        double r31663 = -2.0;
        double r31664 = -1.0;
        double r31665 = r31664 / r31650;
        double r31666 = log(r31665);
        double r31667 = r31663 * r31666;
        double r31668 = r31662 / r31667;
        double r31669 = r31658 / r31668;
        double r31670 = r31661 * r31669;
        double r31671 = 5.752055057168161e+125;
        bool r31672 = r31650 <= r31671;
        double r31673 = r31650 * r31650;
        double r31674 = im;
        double r31675 = r31674 * r31674;
        double r31676 = r31673 + r31675;
        double r31677 = log(r31676);
        double r31678 = r31662 / r31677;
        double r31679 = r31658 / r31678;
        double r31680 = r31661 * r31679;
        double r31681 = log(r31650);
        double r31682 = 2.0;
        double r31683 = r31681 * r31682;
        double r31684 = r31662 / r31683;
        double r31685 = r31658 / r31684;
        double r31686 = r31661 * r31685;
        double r31687 = r31672 ? r31680 : r31686;
        double r31688 = r31652 ? r31670 : r31687;
        return r31688;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -44144913.100366615

   1. Initial program 41.8

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

   3. Applied pow1/2 41.8

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

   4. Applied log-pow 41.8

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

   5. Applied associate-/l* 41.8

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

   7. Applied pow1 41.8

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

   8. Applied log-pow 41.8

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

   9. Applied add-sqr-sqrt 41.8

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

   10. Applied times-frac 41.9

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

   11. Applied associate-/r* 41.8

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

   12. Simplified 41.8

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

   14. Applied pow1 41.8

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

   15. Applied log-pow 41.8

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

   16. Applied add-cube-cbrt 42.0

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

   17. Applied sqrt-prod 42.0

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

   18. Applied times-frac 42.0

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

   19. Applied add-sqr-sqrt 41.8

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

   20. Applied times-frac 41.8

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

   21. Simplified 41.8

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

   22. Taylor expanded around -inf 12.9

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

   23. Simplified 12.9

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

   if -44144913.100366615 < re < 5.752055057168161e+125

   1. Initial program 22.7

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

   3. Applied pow1/2 22.7

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

   4. Applied log-pow 22.7

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

   5. Applied associate-/l* 22.7

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

   7. Applied pow1 22.7

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

   8. Applied log-pow 22.7

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

   9. Applied add-sqr-sqrt 22.7

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

   10. Applied times-frac 22.8

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

   11. Applied associate-/r* 22.7

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

   12. Simplified 22.7

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

   14. Applied pow1 22.7

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

   15. Applied log-pow 22.7

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

   16. Applied add-cube-cbrt 23.1

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

   17. Applied sqrt-prod 23.1

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

   18. Applied times-frac 23.0

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

   19. Applied add-sqr-sqrt 22.6

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

   20. Applied times-frac 22.6

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

   21. Simplified 22.6

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

   if 5.752055057168161e+125 < re

   1. Initial program 56.1

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

   3. Applied pow1/2 56.1

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

   4. Applied log-pow 56.1

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

   5. Applied associate-/l* 56.1

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

   7. Applied pow1 56.1

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

   8. Applied log-pow 56.1

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

   9. Applied add-sqr-sqrt 56.1

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

   10. Applied times-frac 56.1

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

   11. Applied associate-/r* 56.1

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

   12. Simplified 56.1

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

   14. Applied pow1 56.1

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

   15. Applied log-pow 56.1

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

   16. Applied add-cube-cbrt 56.2

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

   17. Applied sqrt-prod 56.2

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

   18. Applied times-frac 56.1

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

   19. Applied add-sqr-sqrt 56.1

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

   20. Applied times-frac 56.1

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

   21. Simplified 56.1

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

   22. Taylor expanded around inf 8.1

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

   23. Simplified 8.1

       \[\leadsto \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\color{blue}{\log re \cdot 2}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -44144913.1003666148:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{-2 \cdot \log \left(\frac{-1}{re}\right)}}\\ \mathbf{elif}\;re \le 5.7520550571681611 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\left|\sqrt[3]{\log 10}\right|} \cdot \frac{\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}}}{\frac{\sqrt{\sqrt[3]{\log 10}}}{\log re \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10.0)))