math.log/2 on complex, real part

Average Error: 31.6 → 17.6

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -3.181961503270047933638306263352911428766 \cdot 10^{94}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 2.041972308450815691735066150411629060859 \cdot 10^{122}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r40689 = re;
        double r40690 = r40689 * r40689;
        double r40691 = im;
        double r40692 = r40691 * r40691;
        double r40693 = r40690 + r40692;
        double r40694 = sqrt(r40693);
        double r40695 = log(r40694);
        double r40696 = base;
        double r40697 = log(r40696);
        double r40698 = r40695 * r40697;
        double r40699 = atan2(r40691, r40689);
        double r40700 = 0.0;
        double r40701 = r40699 * r40700;
        double r40702 = r40698 + r40701;
        double r40703 = r40697 * r40697;
        double r40704 = r40700 * r40700;
        double r40705 = r40703 + r40704;
        double r40706 = r40702 / r40705;
        return r40706;
}

↓

double f(double re, double im, double base) {
        double r40707 = im;
        double r40708 = -3.181961503270048e+94;
        bool r40709 = r40707 <= r40708;
        double r40710 = -1.0;
        double r40711 = r40710 / r40707;
        double r40712 = log(r40711);
        double r40713 = -r40712;
        double r40714 = base;
        double r40715 = log(r40714);
        double r40716 = r40713 / r40715;
        double r40717 = 2.0419723084508157e+122;
        bool r40718 = r40707 <= r40717;
        double r40719 = re;
        double r40720 = r40719 * r40719;
        double r40721 = r40707 * r40707;
        double r40722 = r40720 + r40721;
        double r40723 = sqrt(r40722);
        double r40724 = log(r40723);
        double r40725 = r40724 * r40715;
        double r40726 = atan2(r40707, r40719);
        double r40727 = 0.0;
        double r40728 = r40726 * r40727;
        double r40729 = r40725 + r40728;
        double r40730 = 2.0;
        double r40731 = pow(r40715, r40730);
        double r40732 = r40727 * r40727;
        double r40733 = r40731 + r40732;
        double r40734 = sqrt(r40733);
        double r40735 = r40729 / r40734;
        double r40736 = 1.0;
        double r40737 = r40732 + r40731;
        double r40738 = sqrt(r40737);
        double r40739 = r40736 / r40738;
        double r40740 = r40735 * r40739;
        double r40741 = log(r40707);
        double r40742 = r40741 * r40715;
        double r40743 = r40742 + r40728;
        double r40744 = r40715 * r40715;
        double r40745 = r40744 + r40732;
        double r40746 = r40743 / r40745;
        double r40747 = r40718 ? r40740 : r40746;
        double r40748 = r40709 ? r40716 : r40747;
        return r40748;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 3 regimes

2. if im < -3.181961503270048e+94

   1. Initial program 50.5

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

   3. Applied add-sqr-sqrt 50.5

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

   4. Applied associate-/r* 50.5

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

   5. Simplified 50.5

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

   7. Applied *-un-lft-identity 50.5

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

   8. Applied sqrt-prod 50.5

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

   9. Applied div-inv 50.5

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

   10. Applied times-frac 50.6

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

   11. Simplified 50.6

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

   12. Simplified 50.5

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

   13. Taylor expanded around -inf 64.0

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

   14. Simplified 8.9

       \[\leadsto \color{blue}{\frac{-\log \left(\frac{-1}{im}\right)}{0 + \log base}}\]

   if -3.181961503270048e+94 < im < 2.0419723084508157e+122

   1. Initial program 21.7

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

   3. Applied add-sqr-sqrt 21.7

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

   4. Applied associate-/r* 21.7

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

   5. Simplified 21.7

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

   7. Applied div-inv 21.7

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

   8. Simplified 21.7

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

   if 2.0419723084508157e+122 < im

   1. Initial program 56.6

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

   2. Taylor expanded around 0 8.2

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.181961503270047933638306263352911428766 \cdot 10^{94}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 2.041972308450815691735066150411629060859 \cdot 10^{122}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}} \cdot \frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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))))