math.log/2 on complex, real part

Average Error: 30.9 → 17.5

Time: 1.2m

Precision: 64

• could not determine a ground truth for program body

  1. re = 6.141686545302404e-199
  2. im = 17491141879405.176
  3. base = 2.330187764053037e+289

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\begin{array}{l} \mathbf{if}\;im \le -6.319225497138611 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\log \left(\frac{-1}{re}\right)}{\left(\log base \cdot \log base\right) \cdot \log base} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right)\right)}{-\left(\log base \cdot \log base - \log base \cdot 0\right)}\\ \mathbf{elif}\;im \le -9.784975811452057 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le -3.7539404597618837 \cdot 10^{+52}:\\ \;\;\;\;\left(\log base \cdot \log base - \log base \cdot 0\right) \cdot \frac{-\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}\\ \mathbf{elif}\;im \le -1.5129456424795354 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le 2.125432684072948 \cdot 10^{-114}:\\ \;\;\;\;\left(\log base \cdot \log base - \log base \cdot 0\right) \cdot \frac{-\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}\\ \mathbf{elif}\;im \le 1.731637496866189 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le 1.3029063758199714 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\log \left(\frac{-1}{re}\right)}{\left(\log \left(\sqrt[3]{base}\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + \log \left(\sqrt[3]{base}\right) \cdot \log base}}{\log base} \cdot \left(-\left(\log base \cdot \log base - \log base \cdot 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r1644661 = re;
        double r1644662 = r1644661 * r1644661;
        double r1644663 = im;
        double r1644664 = r1644663 * r1644663;
        double r1644665 = r1644662 + r1644664;
        double r1644666 = sqrt(r1644665);
        double r1644667 = log(r1644666);
        double r1644668 = base;
        double r1644669 = log(r1644668);
        double r1644670 = r1644667 * r1644669;
        double r1644671 = atan2(r1644663, r1644661);
        double r1644672 = 0.0;
        double r1644673 = r1644671 * r1644672;
        double r1644674 = r1644670 + r1644673;
        double r1644675 = r1644669 * r1644669;
        double r1644676 = r1644672 * r1644672;
        double r1644677 = r1644675 + r1644676;
        double r1644678 = r1644674 / r1644677;
        return r1644678;
}

↓

double f(double re, double im, double base) {
        double r1644679 = im;
        double r1644680 = -6.319225497138611e+155;
        bool r1644681 = r1644679 <= r1644680;
        double r1644682 = -1.0;
        double r1644683 = re;
        double r1644684 = r1644682 / r1644683;
        double r1644685 = log(r1644684);
        double r1644686 = base;
        double r1644687 = log(r1644686);
        double r1644688 = r1644687 * r1644687;
        double r1644689 = r1644688 * r1644687;
        double r1644690 = r1644685 / r1644689;
        double r1644691 = r1644688 * r1644688;
        double r1644692 = r1644690 * r1644691;
        double r1644693 = 0.0;
        double r1644694 = r1644687 * r1644693;
        double r1644695 = r1644688 - r1644694;
        double r1644696 = -r1644695;
        double r1644697 = r1644692 / r1644696;
        double r1644698 = -9.784975811452057e+64;
        bool r1644699 = r1644679 <= r1644698;
        double r1644700 = 1.0;
        double r1644701 = r1644683 * r1644683;
        double r1644702 = r1644679 * r1644679;
        double r1644703 = r1644701 + r1644702;
        double r1644704 = sqrt(r1644703);
        double r1644705 = log(r1644704);
        double r1644706 = r1644687 / r1644705;
        double r1644707 = r1644700 / r1644706;
        double r1644708 = -3.7539404597618837e+52;
        bool r1644709 = r1644679 <= r1644708;
        double r1644710 = r1644685 / r1644688;
        double r1644711 = -r1644710;
        double r1644712 = r1644711 / r1644687;
        double r1644713 = r1644695 * r1644712;
        double r1644714 = -1.5129456424795354e-88;
        bool r1644715 = r1644679 <= r1644714;
        double r1644716 = 2.125432684072948e-114;
        bool r1644717 = r1644679 <= r1644716;
        double r1644718 = 1.731637496866189e-54;
        bool r1644719 = r1644679 <= r1644718;
        double r1644720 = 1.3029063758199714e-09;
        bool r1644721 = r1644679 <= r1644720;
        double r1644722 = cbrt(r1644686);
        double r1644723 = log(r1644722);
        double r1644724 = r1644723 * r1644687;
        double r1644725 = r1644724 + r1644724;
        double r1644726 = r1644725 + r1644724;
        double r1644727 = r1644685 / r1644726;
        double r1644728 = r1644727 / r1644687;
        double r1644729 = r1644728 * r1644696;
        double r1644730 = log(r1644679);
        double r1644731 = r1644730 / r1644687;
        double r1644732 = r1644721 ? r1644729 : r1644731;
        double r1644733 = r1644719 ? r1644707 : r1644732;
        double r1644734 = r1644717 ? r1644713 : r1644733;
        double r1644735 = r1644715 ? r1644707 : r1644734;
        double r1644736 = r1644709 ? r1644713 : r1644735;
        double r1644737 = r1644699 ? r1644707 : r1644736;
        double r1644738 = r1644681 ? r1644697 : r1644737;
        return r1644738;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if im < -6.319225497138611e+155

   1. Initial program 62.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 62.0

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified 50.8

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{0 + \log base}}\]
   5. Using strategy rm

   6. Applied flip3-+ 50.8

      \[\leadsto -\frac{\log \left(\frac{-1}{re}\right)}{\color{blue}{\frac{{0}^{3} + {\left(\log base\right)}^{3}}{0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)}}}\]

   7. Applied associate-/r/ 50.8

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{-1}{re}\right)}{{0}^{3} + {\left(\log base\right)}^{3}} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)}\]

   8. Simplified 50.8

      \[\leadsto -\color{blue}{\frac{\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]
   9. Using strategy rm

   10. Applied flip-+ 50.8

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

   11. Applied associate-*r/ 50.8

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

   12. Simplified 50.8

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

   if -6.319225497138611e+155 < im < -9.784975811452057e+64 or -3.7539404597618837e+52 < im < -1.5129456424795354e-88 or 2.125432684072948e-114 < im < 1.731637496866189e-54

   1. Initial program 16.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 16.3

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

   4. Applied clear-num 16.4

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

   5. Simplified 16.3

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

   if -9.784975811452057e+64 < im < -3.7539404597618837e+52 or -1.5129456424795354e-88 < im < 2.125432684072948e-114

   1. Initial program 26.4

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 26.4

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified 9.1

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{0 + \log base}}\]
   5. Using strategy rm

   6. Applied flip3-+ 9.2

      \[\leadsto -\frac{\log \left(\frac{-1}{re}\right)}{\color{blue}{\frac{{0}^{3} + {\left(\log base\right)}^{3}}{0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)}}}\]

   7. Applied associate-/r/ 9.3

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{-1}{re}\right)}{{0}^{3} + {\left(\log base\right)}^{3}} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)}\]

   8. Simplified 9.2

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

   if 1.731637496866189e-54 < im < 1.3029063758199714e-09

   1. Initial program 18.1

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 18.1

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

   3. Taylor expanded around -inf 62.8

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

   4. Simplified 22.0

      \[\leadsto \color{blue}{-\frac{\log \left(\frac{-1}{re}\right)}{0 + \log base}}\]
   5. Using strategy rm

   6. Applied flip3-+ 22.0

      \[\leadsto -\frac{\log \left(\frac{-1}{re}\right)}{\color{blue}{\frac{{0}^{3} + {\left(\log base\right)}^{3}}{0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)}}}\]

   7. Applied associate-/r/ 22.1

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{-1}{re}\right)}{{0}^{3} + {\left(\log base\right)}^{3}} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)}\]

   8. Simplified 22.0

      \[\leadsto -\color{blue}{\frac{\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 22.0

       \[\leadsto -\frac{\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)}}}{\log base} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]

   11. Applied log-prod 22.1

       \[\leadsto -\frac{\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)}}}{\log base} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]

   12. Applied distribute-lft-in 22.1

       \[\leadsto -\frac{\frac{\log \left(\frac{-1}{re}\right)}{\color{blue}{\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)}}}{\log base} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]

   13. Simplified 22.1

       \[\leadsto -\frac{\frac{\log \left(\frac{-1}{re}\right)}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)}}{\log base} \cdot \left(0 \cdot 0 + \left(\log base \cdot \log base - 0 \cdot \log base\right)\right)\]

   if 1.3029063758199714e-09 < im

   1. Initial program 37.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Simplified 37.8

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

   3. Taylor expanded around 0 13.3

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -6.319225497138611 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{\log \left(\frac{-1}{re}\right)}{\left(\log base \cdot \log base\right) \cdot \log base} \cdot \left(\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right)\right)}{-\left(\log base \cdot \log base - \log base \cdot 0\right)}\\ \mathbf{elif}\;im \le -9.784975811452057 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le -3.7539404597618837 \cdot 10^{+52}:\\ \;\;\;\;\left(\log base \cdot \log base - \log base \cdot 0\right) \cdot \frac{-\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}\\ \mathbf{elif}\;im \le -1.5129456424795354 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le 2.125432684072948 \cdot 10^{-114}:\\ \;\;\;\;\left(\log base \cdot \log base - \log base \cdot 0\right) \cdot \frac{-\frac{\log \left(\frac{-1}{re}\right)}{\log base \cdot \log base}}{\log base}\\ \mathbf{elif}\;im \le 1.731637496866189 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\\ \mathbf{elif}\;im \le 1.3029063758199714 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\log \left(\frac{-1}{re}\right)}{\left(\log \left(\sqrt[3]{base}\right) \cdot \log base + \log \left(\sqrt[3]{base}\right) \cdot \log base\right) + \log \left(\sqrt[3]{base}\right) \cdot \log base}}{\log base} \cdot \left(-\left(\log base \cdot \log base - \log base \cdot 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0))))