math.log10 on complex, real part

Average Error: 32.2 → 18.3

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.111326592735039579788335414834132413433 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 5.246446846048960229429099302337261201176 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.178720592182588159154577176110759415934 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9.769663399544422990952065102026170637802 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\right)}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r44701 = re;
        double r44702 = r44701 * r44701;
        double r44703 = im;
        double r44704 = r44703 * r44703;
        double r44705 = r44702 + r44704;
        double r44706 = sqrt(r44705);
        double r44707 = log(r44706);
        double r44708 = 10.0;
        double r44709 = log(r44708);
        double r44710 = r44707 / r44709;
        return r44710;
}

↓

double f(double re, double im) {
        double r44711 = re;
        double r44712 = -2.1113265927350396e+117;
        bool r44713 = r44711 <= r44712;
        double r44714 = 1.0;
        double r44715 = 10.0;
        double r44716 = log(r44715);
        double r44717 = sqrt(r44716);
        double r44718 = r44714 / r44717;
        double r44719 = -1.0;
        double r44720 = r44719 * r44711;
        double r44721 = pow(r44720, r44718);
        double r44722 = log(r44721);
        double r44723 = r44718 * r44722;
        double r44724 = 5.24644684604896e-199;
        bool r44725 = r44711 <= r44724;
        double r44726 = r44711 * r44711;
        double r44727 = im;
        double r44728 = r44727 * r44727;
        double r44729 = r44726 + r44728;
        double r44730 = sqrt(r44729);
        double r44731 = pow(r44730, r44718);
        double r44732 = log(r44731);
        double r44733 = r44718 * r44732;
        double r44734 = 1.1787205921825882e-172;
        bool r44735 = r44711 <= r44734;
        double r44736 = pow(r44711, r44718);
        double r44737 = log(r44736);
        double r44738 = r44718 * r44737;
        double r44739 = 9.769663399544423e+95;
        bool r44740 = r44711 <= r44739;
        double r44741 = 2.0;
        double r44742 = cbrt(r44731);
        double r44743 = log(r44742);
        double r44744 = r44741 * r44743;
        double r44745 = r44744 * r44714;
        double r44746 = r44745 / r44717;
        double r44747 = cbrt(r44717);
        double r44748 = r44747 * r44747;
        double r44749 = r44714 / r44748;
        double r44750 = pow(r44730, r44749);
        double r44751 = r44714 / r44747;
        double r44752 = pow(r44750, r44751);
        double r44753 = cbrt(r44752);
        double r44754 = log(r44753);
        double r44755 = r44718 * r44754;
        double r44756 = r44746 + r44755;
        double r44757 = r44740 ? r44756 : r44738;
        double r44758 = r44735 ? r44738 : r44757;
        double r44759 = r44725 ? r44733 : r44758;
        double r44760 = r44713 ? r44723 : r44759;
        return r44760;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -2.1113265927350396e+117

   1. Initial program 54.6

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

   3. Applied add-sqr-sqrt 54.6

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

   4. Applied pow1 54.6

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

   5. Applied log-pow 54.6

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

   6. Applied times-frac 54.6

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

   8. Applied add-log-exp 54.6

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

   9. Simplified 54.6

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

   10. Taylor expanded around -inf 8.9

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

   if -2.1113265927350396e+117 < re < 5.24644684604896e-199

   1. Initial program 24.3

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

   3. Applied add-sqr-sqrt 24.3

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

   4. Applied pow1 24.3

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

   5. Applied log-pow 24.3

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

   6. Applied times-frac 24.3

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

   8. Applied add-log-exp 24.3

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

   9. Simplified 24.1

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

   if 5.24644684604896e-199 < re < 1.1787205921825882e-172 or 9.769663399544423e+95 < re

   1. Initial program 48.8

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

   3. Applied add-sqr-sqrt 48.8

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

   4. Applied pow1 48.8

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

   5. Applied log-pow 48.8

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

   6. Applied times-frac 48.8

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

   8. Applied add-log-exp 48.8

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

   9. Simplified 48.7

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

   10. Taylor expanded around inf 13.7

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

   if 1.1787205921825882e-172 < re < 9.769663399544423e+95

   1. Initial program 17.4

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

   3. Applied add-sqr-sqrt 17.4

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

   4. Applied pow1 17.4

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

   5. Applied log-pow 17.4

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

   6. Applied times-frac 17.4

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

   8. Applied add-log-exp 17.4

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

   9. Simplified 17.2

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

   11. Applied add-cube-cbrt 17.2

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

   12. Applied log-prod 17.3

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

   13. Applied distribute-lft-in 17.3

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

   14. Simplified 17.3

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

   16. Applied add-cube-cbrt 17.3

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

   17. Applied *-un-lft-identity 17.3

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

   18. Applied times-frac 17.3

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

   19. Applied pow-unpow 17.3

       \[\leadsto \frac{\left(2 \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{\color{blue}{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\right)}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right)}}}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.111326592735039579788335414834132413433 \cdot 10^{117}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(-1 \cdot re\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 5.246446846048960229429099302337261201176 \cdot 10^{-199}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 1.178720592182588159154577176110759415934 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \mathbf{elif}\;re \le 9.769663399544422990952065102026170637802 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(2 \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right) \cdot 1}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}} \cdot \sqrt[3]{\sqrt{\log 10}}}\right)}\right)}^{\left(\frac{1}{\sqrt[3]{\sqrt{\log 10}}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\log 10}} \cdot \log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))