math.log/2 on complex, real part

Average Error: 31.9 → 18.1

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.678553877806604 \cdot 10^{41}:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.69081280904729871 \cdot 10^{-246}:\\ \;\;\;\;\frac{1}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.4235180232042555 \cdot 10^{-227}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.589229227697289 \cdot 10^{74}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(1 \cdot {base}^{\frac{1}{3}}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r42690 = re;
        double r42691 = r42690 * r42690;
        double r42692 = im;
        double r42693 = r42692 * r42692;
        double r42694 = r42691 + r42693;
        double r42695 = sqrt(r42694);
        double r42696 = log(r42695);
        double r42697 = base;
        double r42698 = log(r42697);
        double r42699 = r42696 * r42698;
        double r42700 = atan2(r42692, r42690);
        double r42701 = 0.0;
        double r42702 = r42700 * r42701;
        double r42703 = r42699 + r42702;
        double r42704 = r42698 * r42698;
        double r42705 = r42701 * r42701;
        double r42706 = r42704 + r42705;
        double r42707 = r42703 / r42706;
        return r42707;
}

↓

double f(double re, double im, double base) {
        double r42708 = re;
        double r42709 = -3.6785538778066045e+41;
        bool r42710 = r42708 <= r42709;
        double r42711 = -1.0;
        double r42712 = r42711 / r42708;
        double r42713 = log(r42712);
        double r42714 = r42711 * r42713;
        double r42715 = base;
        double r42716 = log(r42715);
        double r42717 = r42714 * r42716;
        double r42718 = im;
        double r42719 = atan2(r42718, r42708);
        double r42720 = 0.0;
        double r42721 = r42719 * r42720;
        double r42722 = r42717 + r42721;
        double r42723 = r42716 * r42716;
        double r42724 = r42720 * r42720;
        double r42725 = r42723 + r42724;
        double r42726 = sqrt(r42725);
        double r42727 = r42722 / r42726;
        double r42728 = r42727 / r42726;
        double r42729 = 4.690812809047299e-246;
        bool r42730 = r42708 <= r42729;
        double r42731 = 1.0;
        double r42732 = 2.0;
        double r42733 = pow(r42716, r42732);
        double r42734 = r42724 + r42733;
        double r42735 = r42708 * r42708;
        double r42736 = r42718 * r42718;
        double r42737 = r42735 + r42736;
        double r42738 = sqrt(r42737);
        double r42739 = log(r42738);
        double r42740 = r42739 * r42716;
        double r42741 = r42740 + r42721;
        double r42742 = r42734 / r42741;
        double r42743 = r42731 / r42742;
        double r42744 = 4.423518023204255e-227;
        bool r42745 = r42708 <= r42744;
        double r42746 = log(r42708);
        double r42747 = r42746 * r42716;
        double r42748 = r42747 + r42721;
        double r42749 = r42748 / r42725;
        double r42750 = 6.589229227697289e+74;
        bool r42751 = r42708 <= r42750;
        double r42752 = 0.3333333333333333;
        double r42753 = pow(r42715, r42752);
        double r42754 = r42731 * r42753;
        double r42755 = log(r42754);
        double r42756 = r42732 * r42755;
        double r42757 = r42716 * r42756;
        double r42758 = cbrt(r42715);
        double r42759 = log(r42758);
        double r42760 = r42716 * r42759;
        double r42761 = r42757 + r42760;
        double r42762 = r42761 + r42724;
        double r42763 = r42741 / r42762;
        double r42764 = r42732 * r42759;
        double r42765 = r42716 * r42764;
        double r42766 = r42765 + r42760;
        double r42767 = r42766 + r42724;
        double r42768 = r42748 / r42767;
        double r42769 = r42751 ? r42763 : r42768;
        double r42770 = r42745 ? r42749 : r42769;
        double r42771 = r42730 ? r42743 : r42770;
        double r42772 = r42710 ? r42728 : r42771;
        return r42772;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -3.6785538778066045e+41

   1. Initial program 43.6

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

   3. Applied add-sqr-sqrt 43.6

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

   4. Applied associate-/r* 43.5

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

   5. Taylor expanded around -inf 64.0

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

   6. Simplified 11.3

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

   if -3.6785538778066045e+41 < re < 4.690812809047299e-246

   1. Initial program 23.8

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

   3. Applied clear-num 23.8

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

   4. Simplified 23.8

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

   if 4.690812809047299e-246 < re < 4.423518023204255e-227

   1. Initial program 33.3

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

   2. Taylor expanded around inf 51.4

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

   if 4.423518023204255e-227 < re < 6.589229227697289e+74

   1. Initial program 20.4

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

   3. Applied add-cube-cbrt 20.4

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

   4. Applied log-prod 20.4

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

   5. Applied distribute-lft-in 20.4

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

   6. Simplified 20.4

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

   8. Applied *-un-lft-identity 20.4

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

   9. Applied cbrt-prod 20.4

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

   10. Simplified 20.4

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

   11. Simplified 20.4

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

   if 6.589229227697289e+74 < re

   1. Initial program 47.7

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

   3. Applied add-cube-cbrt 47.7

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

   4. Applied log-prod 47.7

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

   5. Applied distribute-lft-in 47.7

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

   6. Simplified 47.7

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

   7. Taylor expanded around inf 10.3

      \[\leadsto \frac{\log \color{blue}{re} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.678553877806604 \cdot 10^{41}:\\ \;\;\;\;\frac{\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.69081280904729871 \cdot 10^{-246}:\\ \;\;\;\;\frac{1}{\frac{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{elif}\;re \le 4.4235180232042555 \cdot 10^{-227}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.589229227697289 \cdot 10^{74}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(1 \cdot {base}^{\frac{1}{3}}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))