math.log10 on complex, real part

Average Error: 32.2 → 17.7

Time: 18.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.528127276764349062375093852629016219739 \cdot 10^{119}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -4.371006412922919355426993155500752563146 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.437299774034011283254659821557789091298 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.266521437225666892318788639170016686519 \cdot 10^{76}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r29765 = re;
        double r29766 = r29765 * r29765;
        double r29767 = im;
        double r29768 = r29767 * r29767;
        double r29769 = r29766 + r29768;
        double r29770 = sqrt(r29769);
        double r29771 = log(r29770);
        double r29772 = 10.0;
        double r29773 = log(r29772);
        double r29774 = r29771 / r29773;
        return r29774;
}

↓

double f(double re, double im) {
        double r29775 = re;
        double r29776 = -8.528127276764349e+119;
        bool r29777 = r29775 <= r29776;
        double r29778 = 0.5;
        double r29779 = 10.0;
        double r29780 = log(r29779);
        double r29781 = sqrt(r29780);
        double r29782 = r29778 / r29781;
        double r29783 = -2.0;
        double r29784 = -1.0;
        double r29785 = r29784 / r29775;
        double r29786 = log(r29785);
        double r29787 = r29783 * r29786;
        double r29788 = 1.0;
        double r29789 = r29788 / r29781;
        double r29790 = r29787 * r29789;
        double r29791 = r29782 * r29790;
        double r29792 = -4.371006412922919e-223;
        bool r29793 = r29775 <= r29792;
        double r29794 = cbrt(r29778);
        double r29795 = r29794 * r29794;
        double r29796 = sqrt(r29781);
        double r29797 = r29795 / r29796;
        double r29798 = r29794 / r29796;
        double r29799 = r29775 * r29775;
        double r29800 = im;
        double r29801 = r29800 * r29800;
        double r29802 = r29799 + r29801;
        double r29803 = log(r29802);
        double r29804 = r29803 / r29781;
        double r29805 = r29798 * r29804;
        double r29806 = r29797 * r29805;
        double r29807 = -2.4372997740340113e-288;
        bool r29808 = r29775 <= r29807;
        double r29809 = 2.0;
        double r29810 = log(r29800);
        double r29811 = r29809 * r29810;
        double r29812 = r29811 * r29789;
        double r29813 = r29782 * r29812;
        double r29814 = 1.2665214372256669e+76;
        bool r29815 = r29775 <= r29814;
        double r29816 = sqrt(r29782);
        double r29817 = r29816 * r29804;
        double r29818 = r29816 * r29817;
        double r29819 = log(r29775);
        double r29820 = r29809 * r29819;
        double r29821 = r29820 / r29781;
        double r29822 = r29798 * r29821;
        double r29823 = r29797 * r29822;
        double r29824 = r29815 ? r29818 : r29823;
        double r29825 = r29808 ? r29813 : r29824;
        double r29826 = r29793 ? r29806 : r29825;
        double r29827 = r29777 ? r29791 : r29826;
        return r29827;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -8.528127276764349e+119

   1. Initial program 56.2

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

   3. Applied add-sqr-sqrt 56.2

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

   4. Applied pow1/2 56.2

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

   5. Applied log-pow 56.2

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

   6. Applied times-frac 56.2

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

   8. Applied div-inv 56.2

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

   9. Taylor expanded around -inf 8.0

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

   10. Simplified 8.0

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

   if -8.528127276764349e+119 < re < -4.371006412922919e-223

   1. Initial program 19.3

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

   3. Applied add-sqr-sqrt 19.3

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

   4. Applied pow1/2 19.3

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

   5. Applied log-pow 19.3

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

   6. Applied times-frac 19.3

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

   8. Applied add-sqr-sqrt 19.3

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

   9. Applied sqrt-prod 19.8

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

   10. Applied add-cube-cbrt 19.3

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

   11. Applied times-frac 19.3

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

   12. Applied associate-*l* 19.2

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

   if -4.371006412922919e-223 < re < -2.4372997740340113e-288

   1. Initial program 30.1

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

   3. Applied add-sqr-sqrt 30.1

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

   4. Applied pow1/2 30.1

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

   5. Applied log-pow 30.1

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

   6. Applied times-frac 30.1

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

   8. Applied div-inv 30.0

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

   9. Taylor expanded around 0 32.2

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

   if -2.4372997740340113e-288 < re < 1.2665214372256669e+76

   1. Initial program 22.9

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

   3. Applied add-sqr-sqrt 22.9

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

   4. Applied pow1/2 22.9

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

   5. Applied log-pow 22.9

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

   6. Applied times-frac 22.9

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

   8. Applied add-sqr-sqrt 22.9

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

   9. Applied associate-*l* 22.8

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

   if 1.2665214372256669e+76 < re

   1. Initial program 48.3

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

   3. Applied add-sqr-sqrt 48.3

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

   4. Applied pow1/2 48.3

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

   5. Applied log-pow 48.3

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

   6. Applied times-frac 48.2

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

   8. Applied add-sqr-sqrt 48.2

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

   9. Applied sqrt-prod 48.4

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

   10. Applied add-cube-cbrt 48.2

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

   11. Applied times-frac 48.2

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

   12. Applied associate-*l* 48.2

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

   13. Taylor expanded around inf 10.5

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

   14. Simplified 10.5

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\color{blue}{2 \cdot \log re}}{\sqrt{\log 10}}\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.528127276764349062375093852629016219739 \cdot 10^{119}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -4.371006412922919355426993155500752563146 \cdot 10^{-223}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.437299774034011283254659821557789091298 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.266521437225666892318788639170016686519 \cdot 10^{76}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \left(\sqrt{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\log \left(re \cdot re + im \cdot im\right)}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{2 \cdot \log re}{\sqrt{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))