math.log/2 on complex, real part

Average Error: 31.0 → 16.8

Time: 23.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -8.699527506903106763121586570753262874818 \cdot 10^{148}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 8.322808950963171226745757166064822582935 \cdot 10^{83}:\\ \;\;\;\;\frac{1}{\frac{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r36488 = re;
        double r36489 = r36488 * r36488;
        double r36490 = im;
        double r36491 = r36490 * r36490;
        double r36492 = r36489 + r36491;
        double r36493 = sqrt(r36492);
        double r36494 = log(r36493);
        double r36495 = base;
        double r36496 = log(r36495);
        double r36497 = r36494 * r36496;
        double r36498 = atan2(r36490, r36488);
        double r36499 = 0.0;
        double r36500 = r36498 * r36499;
        double r36501 = r36497 + r36500;
        double r36502 = r36496 * r36496;
        double r36503 = r36499 * r36499;
        double r36504 = r36502 + r36503;
        double r36505 = r36501 / r36504;
        return r36505;
}

↓

double f(double re, double im, double base) {
        double r36506 = im;
        double r36507 = -8.699527506903107e+148;
        bool r36508 = r36506 <= r36507;
        double r36509 = -1.0;
        double r36510 = r36509 / r36506;
        double r36511 = log(r36510);
        double r36512 = -r36511;
        double r36513 = base;
        double r36514 = log(r36513);
        double r36515 = r36512 / r36514;
        double r36516 = 8.322808950963171e+83;
        bool r36517 = r36506 <= r36516;
        double r36518 = 1.0;
        double r36519 = 2.0;
        double r36520 = pow(r36514, r36519);
        double r36521 = 0.0;
        double r36522 = r36521 * r36521;
        double r36523 = r36520 + r36522;
        double r36524 = re;
        double r36525 = r36524 * r36524;
        double r36526 = r36506 * r36506;
        double r36527 = r36525 + r36526;
        double r36528 = sqrt(r36527);
        double r36529 = log(r36528);
        double r36530 = r36529 * r36514;
        double r36531 = atan2(r36506, r36524);
        double r36532 = r36531 * r36521;
        double r36533 = r36530 + r36532;
        double r36534 = r36523 / r36533;
        double r36535 = r36518 / r36534;
        double r36536 = log(r36506);
        double r36537 = -r36536;
        double r36538 = -r36514;
        double r36539 = r36537 / r36538;
        double r36540 = r36517 ? r36535 : r36539;
        double r36541 = r36508 ? r36515 : r36540;
        return r36541;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 3 regimes

2. if im < -8.699527506903107e+148

   1. Initial program 61.8

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

   3. Applied div-inv 61.8

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

   4. Simplified 61.8

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

   6. Applied flip-+ 61.8

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

   7. Applied associate-/r/ 61.8

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

   8. Applied associate-*r* 61.8

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

   9. Simplified 61.8

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{{\left(\log base\right)}^{4} - 0.0 \cdot {0.0}^{3}}} \cdot \left({\left(\log base\right)}^{2} - 0.0 \cdot 0.0\right)\]

   10. Taylor expanded around -inf 64.0

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

   11. Simplified 7.0

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

   if -8.699527506903107e+148 < im < 8.322808950963171e+83

   1. Initial program 20.6

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

   3. Applied clear-num 20.7

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

   4. Simplified 20.7

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

   if 8.322808950963171e+83 < im

   1. Initial program 48.7

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

   3. Applied div-inv 48.7

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

   4. Simplified 48.7

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

   6. Applied flip-+ 48.7

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

   7. Applied associate-/r/ 48.7

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

   8. Applied associate-*r* 48.7

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

   9. Simplified 48.7

      \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{{\left(\log base\right)}^{4} - 0.0 \cdot {0.0}^{3}}} \cdot \left({\left(\log base\right)}^{2} - 0.0 \cdot 0.0\right)\]

   10. Taylor expanded around inf 8.9

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

   11. Simplified 8.9

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

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -8.699527506903106763121586570753262874818 \cdot 10^{148}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{im}\right)}{\log base}\\ \mathbf{elif}\;im \le 8.322808950963171226745757166064822582935 \cdot 10^{83}:\\ \;\;\;\;\frac{1}{\frac{{\left(\log base\right)}^{2} + 0.0 \cdot 0.0}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log im}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))