math.log10 on complex, real part

Average Error: 32.3 → 18.2

Time: 7.3s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r57650 = re;
        double r57651 = r57650 * r57650;
        double r57652 = im;
        double r57653 = r57652 * r57652;
        double r57654 = r57651 + r57653;
        double r57655 = sqrt(r57654);
        double r57656 = log(r57655);
        double r57657 = 10.0;
        double r57658 = log(r57657);
        double r57659 = r57656 / r57658;
        return r57659;
}

↓

double f(double re, double im) {
        double r57660 = re;
        double r57661 = -2.6163751838198674e+143;
        bool r57662 = r57660 <= r57661;
        double r57663 = 1.0;
        double r57664 = 10.0;
        double r57665 = log(r57664);
        double r57666 = sqrt(r57665);
        double r57667 = r57663 / r57666;
        double r57668 = -1.0;
        double r57669 = r57668 / r57660;
        double r57670 = r57663 / r57665;
        double r57671 = sqrt(r57670);
        double r57672 = -r57671;
        double r57673 = pow(r57669, r57672);
        double r57674 = log(r57673);
        double r57675 = r57667 * r57674;
        double r57676 = -2.0278572522938575e-184;
        bool r57677 = r57660 <= r57676;
        double r57678 = 2.0;
        double r57679 = r57660 * r57660;
        double r57680 = im;
        double r57681 = r57680 * r57680;
        double r57682 = r57679 + r57681;
        double r57683 = cbrt(r57682);
        double r57684 = fabs(r57683);
        double r57685 = cbrt(r57666);
        double r57686 = r57685 * r57685;
        double r57687 = r57663 / r57686;
        double r57688 = sqrt(r57678);
        double r57689 = r57687 / r57688;
        double r57690 = pow(r57684, r57689);
        double r57691 = sqrt(r57663);
        double r57692 = r57691 / r57685;
        double r57693 = r57692 / r57688;
        double r57694 = pow(r57690, r57693);
        double r57695 = log(r57694);
        double r57696 = r57678 * r57695;
        double r57697 = sqrt(r57683);
        double r57698 = pow(r57697, r57667);
        double r57699 = log(r57698);
        double r57700 = r57696 + r57699;
        double r57701 = r57667 * r57700;
        double r57702 = -8.804121592020497e-274;
        bool r57703 = r57660 <= r57702;
        double r57704 = log(r57680);
        double r57705 = r57704 / r57665;
        double r57706 = 1.9824297347259786e+18;
        bool r57707 = r57660 <= r57706;
        double r57708 = r57663 / r57660;
        double r57709 = pow(r57708, r57672);
        double r57710 = log(r57709);
        double r57711 = r57667 * r57710;
        double r57712 = r57707 ? r57701 : r57711;
        double r57713 = r57703 ? r57705 : r57712;
        double r57714 = r57677 ? r57701 : r57713;
        double r57715 = r57662 ? r57675 : r57714;
        return r57715;
}

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 7.1

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

   11. Simplified 7.0

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

   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

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

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)))