math.log/2 on complex, real part

Average Error: 31.8 → 17.6

Time: 25.0s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.860564632808157146272715482474331974986 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{-1}{3} \cdot \log \left(\frac{-1}{re}\right)}{\log \left(\sqrt[3]{-base} \cdot \sqrt[3]{-1}\right)}\\ \mathbf{elif}\;re \le -2.550928749919678760355455544426569849248 \cdot 10^{-162}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 4.352011132147789479254882523295138287074 \cdot 10^{-295}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 5.983180677799418047864441563914340286047 \cdot 10^{93}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r100781 = re;
        double r100782 = r100781 * r100781;
        double r100783 = im;
        double r100784 = r100783 * r100783;
        double r100785 = r100782 + r100784;
        double r100786 = sqrt(r100785);
        double r100787 = log(r100786);
        double r100788 = base;
        double r100789 = log(r100788);
        double r100790 = r100787 * r100789;
        double r100791 = atan2(r100783, r100781);
        double r100792 = 0.0;
        double r100793 = r100791 * r100792;
        double r100794 = r100790 + r100793;
        double r100795 = r100789 * r100789;
        double r100796 = r100792 * r100792;
        double r100797 = r100795 + r100796;
        double r100798 = r100794 / r100797;
        return r100798;
}

↓

double f(double re, double im, double base) {
        double r100799 = re;
        double r100800 = -1.860564632808157e+70;
        bool r100801 = r100799 <= r100800;
        double r100802 = -0.3333333333333333;
        double r100803 = -1.0;
        double r100804 = r100803 / r100799;
        double r100805 = log(r100804);
        double r100806 = r100802 * r100805;
        double r100807 = base;
        double r100808 = -r100807;
        double r100809 = cbrt(r100808);
        double r100810 = cbrt(r100803);
        double r100811 = r100809 * r100810;
        double r100812 = log(r100811);
        double r100813 = r100806 / r100812;
        double r100814 = -2.550928749919679e-162;
        bool r100815 = r100799 <= r100814;
        double r100816 = r100799 * r100799;
        double r100817 = im;
        double r100818 = r100817 * r100817;
        double r100819 = r100816 + r100818;
        double r100820 = sqrt(r100819);
        double r100821 = log(r100820);
        double r100822 = log(r100807);
        double r100823 = r100821 * r100822;
        double r100824 = atan2(r100817, r100799);
        double r100825 = 0.0;
        double r100826 = r100824 * r100825;
        double r100827 = r100823 + r100826;
        double r100828 = 2.0;
        double r100829 = pow(r100822, r100828);
        double r100830 = r100825 * r100825;
        double r100831 = r100829 + r100830;
        double r100832 = sqrt(r100831);
        double r100833 = r100827 / r100832;
        double r100834 = 1.0;
        double r100835 = r100830 + r100829;
        double r100836 = sqrt(r100835);
        double r100837 = r100834 / r100836;
        double r100838 = r100833 * r100837;
        double r100839 = 4.3520111321477895e-295;
        bool r100840 = r100799 <= r100839;
        double r100841 = log(r100817);
        double r100842 = r100841 * r100822;
        double r100843 = r100842 + r100826;
        double r100844 = cbrt(r100807);
        double r100845 = log(r100844);
        double r100846 = r100828 * r100845;
        double r100847 = r100846 * r100822;
        double r100848 = r100845 * r100822;
        double r100849 = r100847 + r100848;
        double r100850 = r100849 + r100830;
        double r100851 = r100843 / r100850;
        double r100852 = 5.983180677799418e+93;
        bool r100853 = r100799 <= r100852;
        double r100854 = r100822 * r100822;
        double r100855 = r100854 + r100830;
        double r100856 = sqrt(r100855);
        double r100857 = r100833 / r100856;
        double r100858 = log(r100799);
        double r100859 = -r100858;
        double r100860 = -r100822;
        double r100861 = r100859 / r100860;
        double r100862 = r100853 ? r100857 : r100861;
        double r100863 = r100840 ? r100851 : r100862;
        double r100864 = r100815 ? r100838 : r100863;
        double r100865 = r100801 ? r100813 : r100864;
        return r100865;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.860564632808157e+70

   1. Initial program 46.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 46.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

   4. Applied log-prod 46.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   5. Applied distribute-lft-in 46.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   6. Simplified 46.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Simplified 46.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \color{blue}{\log \left(\sqrt[3]{base}\right) \cdot \log base}\right) + 0.0 \cdot 0.0}\]

   8. Taylor expanded around -inf 64.0

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

   9. Simplified 10.2

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

   if -1.860564632808157e+70 < re < -2.550928749919679e-162

   1. Initial program 15.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 15.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied associate-/r* 15.5

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   5. Simplified 15.5

      \[\leadsto \frac{\color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
   6. Using strategy rm

   7. Applied div-inv 15.6

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   8. Simplified 15.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \color{blue}{\frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}}\]

   if -2.550928749919679e-162 < re < 4.3520111321477895e-295

   1. Initial program 32.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

   4. Applied log-prod 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   5. Applied distribute-lft-in 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   6. Simplified 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Simplified 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \color{blue}{\log \left(\sqrt[3]{base}\right) \cdot \log base}\right) + 0.0 \cdot 0.0}\]

   8. Taylor expanded around 0 34.9

      \[\leadsto \frac{\log \color{blue}{im} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\]

   if 4.3520111321477895e-295 < re < 5.983180677799418e+93

   1. Initial program 21.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 21.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0} \cdot \sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   4. Applied associate-/r* 21.5

      \[\leadsto \color{blue}{\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}\]

   5. Simplified 21.5

      \[\leadsto \frac{\color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   if 5.983180677799418e+93 < re

   1. Initial program 50.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

   2. Taylor expanded around inf 8.2

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]

   3. Simplified 8.2

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.860564632808157146272715482474331974986 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{-1}{3} \cdot \log \left(\frac{-1}{re}\right)}{\log \left(\sqrt[3]{-base} \cdot \sqrt[3]{-1}\right)}\\ \mathbf{elif}\;re \le -2.550928749919678760355455544426569849248 \cdot 10^{-162}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le 4.352011132147789479254882523295138287074 \cdot 10^{-295}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 5.983180677799418047864441563914340286047 \cdot 10^{93}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))