math.log/2 on complex, real part

Average Error: 30.8 → 18.3

Time: 1.1m

Precision: 64

• could not determine a ground truth for program body

  1. re = 1.834085423193177e-220
  2. im = -1.0794532097663185e-155
  3. base = 3.5222077561159353e-203

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -1.3441617536124223 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{\log \left(-re\right)}}{\log base} \cdot \sqrt{\log \left(-re\right)}\\ \mathbf{elif}\;im \le -1541871735.4493983:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;im \le -4.4197911913014276 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \le -1.3801630952498305 \cdot 10^{-128}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;im \le 4.8135459158735706 \cdot 10^{+17}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{-re}\right)}{\log base} + \frac{\log \left(\sqrt[3]{-re}\right) + \log \left(\sqrt[3]{-re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r1675713 = re;
        double r1675714 = r1675713 * r1675713;
        double r1675715 = im;
        double r1675716 = r1675715 * r1675715;
        double r1675717 = r1675714 + r1675716;
        double r1675718 = sqrt(r1675717);
        double r1675719 = log(r1675718);
        double r1675720 = base;
        double r1675721 = log(r1675720);
        double r1675722 = r1675719 * r1675721;
        double r1675723 = atan2(r1675715, r1675713);
        double r1675724 = 0.0;
        double r1675725 = r1675723 * r1675724;
        double r1675726 = r1675722 + r1675725;
        double r1675727 = r1675721 * r1675721;
        double r1675728 = r1675724 * r1675724;
        double r1675729 = r1675727 + r1675728;
        double r1675730 = r1675726 / r1675729;
        return r1675730;
}

↓

double f(double re, double im, double base) {
        double r1675731 = im;
        double r1675732 = -1.3441617536124223e+154;
        bool r1675733 = r1675731 <= r1675732;
        double r1675734 = re;
        double r1675735 = -r1675734;
        double r1675736 = log(r1675735);
        double r1675737 = sqrt(r1675736);
        double r1675738 = base;
        double r1675739 = log(r1675738);
        double r1675740 = r1675737 / r1675739;
        double r1675741 = r1675740 * r1675737;
        double r1675742 = -1541871735.4493983;
        bool r1675743 = r1675731 <= r1675742;
        double r1675744 = r1675734 * r1675734;
        double r1675745 = r1675731 * r1675731;
        double r1675746 = r1675744 + r1675745;
        double r1675747 = sqrt(r1675746);
        double r1675748 = log(r1675747);
        double r1675749 = r1675748 / r1675739;
        double r1675750 = -4.4197911913014276e-44;
        bool r1675751 = r1675731 <= r1675750;
        double r1675752 = 1.0;
        double r1675753 = r1675752 / r1675739;
        double r1675754 = r1675753 * r1675736;
        double r1675755 = -1.3801630952498305e-128;
        bool r1675756 = r1675731 <= r1675755;
        double r1675757 = 4.8135459158735706e+17;
        bool r1675758 = r1675731 <= r1675757;
        double r1675759 = cbrt(r1675735);
        double r1675760 = log(r1675759);
        double r1675761 = r1675760 / r1675739;
        double r1675762 = r1675760 + r1675760;
        double r1675763 = r1675762 / r1675739;
        double r1675764 = r1675761 + r1675763;
        double r1675765 = log(r1675731);
        double r1675766 = r1675765 / r1675739;
        double r1675767 = r1675758 ? r1675764 : r1675766;
        double r1675768 = r1675756 ? r1675749 : r1675767;
        double r1675769 = r1675751 ? r1675754 : r1675768;
        double r1675770 = r1675743 ? r1675749 : r1675769;
        double r1675771 = r1675733 ? r1675741 : r1675770;
        return r1675771;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if im < -1.3441617536124223e+154

   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 51.4

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

   4. Simplified 51.4

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

   6. Applied clear-num 51.4

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

   7. Simplified 51.4

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

   9. Applied div-inv 51.4

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

   10. Applied *-un-lft-identity 51.4

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

   11. Applied times-frac 51.4

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

   12. Simplified 51.4

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

   14. Applied add-sqr-sqrt 51.3

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

   15. Applied associate-*r* 51.3

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

   16. Simplified 51.3

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

   if -1.3441617536124223e+154 < im < -1541871735.4493983 or -4.4197911913014276e-44 < im < -1.3801630952498305e-128

   1. Initial program 15.4

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

   2. Simplified 15.4

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

   4. Applied times-frac 15.3

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

   5. Simplified 15.3

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

   if -1541871735.4493983 < im < -4.4197911913014276e-44

   1. Initial program 16.6

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

   2. Simplified 16.6

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

   3. Taylor expanded around -inf 24.6

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

   4. Simplified 24.6

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

   6. Applied clear-num 24.6

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

   7. Simplified 24.6

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

   9. Applied div-inv 24.6

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

   10. Applied *-un-lft-identity 24.6

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

   11. Applied times-frac 24.6

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

   12. Simplified 24.6

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

   if -1.3801630952498305e-128 < im < 4.8135459158735706e+17

   1. Initial program 24.7

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

   2. Simplified 24.7

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

   3. Taylor expanded around -inf 12.3

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

   4. Simplified 12.3

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

   6. Applied clear-num 12.3

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

   7. Simplified 12.2

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

   9. Applied div-inv 12.3

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

   10. Applied *-un-lft-identity 12.3

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

   11. Applied times-frac 12.3

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

   12. Simplified 12.2

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

   14. Applied add-cube-cbrt 12.2

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

   15. Applied log-prod 12.3

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

   16. Applied distribute-lft-in 12.3

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

   17. Simplified 12.2

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

   18. Simplified 12.2

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

   if 4.8135459158735706e+17 < im

   1. Initial program 40.3

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

   2. Simplified 40.3

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

   3. Taylor expanded around 0 12.8

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

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.3441617536124223 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{\log \left(-re\right)}}{\log base} \cdot \sqrt{\log \left(-re\right)}\\ \mathbf{elif}\;im \le -1541871735.4493983:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;im \le -4.4197911913014276 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\log base} \cdot \log \left(-re\right)\\ \mathbf{elif}\;im \le -1.3801630952498305 \cdot 10^{-128}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base}\\ \mathbf{elif}\;im \le 4.8135459158735706 \cdot 10^{+17}:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{-re}\right)}{\log base} + \frac{\log \left(\sqrt[3]{-re}\right) + \log \left(\sqrt[3]{-re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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))))