math.log/2 on complex, real part

Average Error: 32.1 → 18.0

Time: 14.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -2.941271494123403150212192649236485986394 \cdot 10^{82}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{im}\right)}{{\left(\log base\right)}^{3}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\\ \mathbf{elif}\;im \le -1.056135040441817993361662182816155280925 \cdot 10^{-243}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{{0.0}^{3} \cdot 0.0 - {\left({\left(\log base\right)}^{2}\right)}^{\left(\sqrt{4}\right)}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\\ \mathbf{elif}\;im \le 1.684512098288392462211092787468027207826 \cdot 10^{-238}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \mathbf{elif}\;im \le 3.283081512447868836362935755252953007161 \cdot 10^{107}:\\ \;\;\;\;\frac{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r44388 = re;
        double r44389 = r44388 * r44388;
        double r44390 = im;
        double r44391 = r44390 * r44390;
        double r44392 = r44389 + r44391;
        double r44393 = sqrt(r44392);
        double r44394 = log(r44393);
        double r44395 = base;
        double r44396 = log(r44395);
        double r44397 = r44394 * r44396;
        double r44398 = atan2(r44390, r44388);
        double r44399 = 0.0;
        double r44400 = r44398 * r44399;
        double r44401 = r44397 + r44400;
        double r44402 = r44396 * r44396;
        double r44403 = r44399 * r44399;
        double r44404 = r44402 + r44403;
        double r44405 = r44401 / r44404;
        return r44405;
}

↓

double f(double re, double im, double base) {
        double r44406 = im;
        double r44407 = -2.941271494123403e+82;
        bool r44408 = r44406 <= r44407;
        double r44409 = -1.0;
        double r44410 = r44409 / r44406;
        double r44411 = log(r44410);
        double r44412 = base;
        double r44413 = log(r44412);
        double r44414 = 3.0;
        double r44415 = pow(r44413, r44414);
        double r44416 = r44411 / r44415;
        double r44417 = 0.0;
        double r44418 = r44417 * r44417;
        double r44419 = 2.0;
        double r44420 = pow(r44413, r44419);
        double r44421 = r44418 - r44420;
        double r44422 = r44416 * r44421;
        double r44423 = -1.056135040441818e-243;
        bool r44424 = r44406 <= r44423;
        double r44425 = re;
        double r44426 = atan2(r44406, r44425);
        double r44427 = r44426 * r44417;
        double r44428 = r44425 * r44425;
        double r44429 = r44406 * r44406;
        double r44430 = r44428 + r44429;
        double r44431 = sqrt(r44430);
        double r44432 = log(r44431);
        double r44433 = r44413 * r44432;
        double r44434 = r44427 + r44433;
        double r44435 = pow(r44417, r44414);
        double r44436 = r44435 * r44417;
        double r44437 = 4.0;
        double r44438 = sqrt(r44437);
        double r44439 = pow(r44420, r44438);
        double r44440 = r44436 - r44439;
        double r44441 = r44434 / r44440;
        double r44442 = r44441 * r44421;
        double r44443 = 1.6845120982883925e-238;
        bool r44444 = r44406 <= r44443;
        double r44445 = log(r44425);
        double r44446 = -r44445;
        double r44447 = -r44413;
        double r44448 = r44446 / r44447;
        double r44449 = 3.283081512447869e+107;
        bool r44450 = r44406 <= r44449;
        double r44451 = r44432 * r44413;
        double r44452 = pow(r44451, r44414);
        double r44453 = pow(r44427, r44414);
        double r44454 = r44452 + r44453;
        double r44455 = r44418 + r44420;
        double r44456 = r44454 / r44455;
        double r44457 = r44451 * r44451;
        double r44458 = r44427 * r44427;
        double r44459 = r44451 * r44427;
        double r44460 = r44458 - r44459;
        double r44461 = r44457 + r44460;
        double r44462 = r44456 / r44461;
        double r44463 = log(r44406);
        double r44464 = r44463 / r44413;
        double r44465 = r44450 ? r44462 : r44464;
        double r44466 = r44444 ? r44448 : r44465;
        double r44467 = r44424 ? r44442 : r44466;
        double r44468 = r44408 ? r44422 : r44467;
        return r44468;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if im < -2.941271494123403e+82

   1. Initial program 48.6

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

   3. Applied div-inv 48.6

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

   4. Simplified 48.6

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

   6. Applied flip-+ 48.6

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

   7. Applied associate-/r/ 48.6

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

   8. Applied associate-*r* 48.6

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

   9. Simplified 48.6

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

   10. Taylor expanded around -inf 64.0

       \[\leadsto \color{blue}{\frac{\log \left(\frac{-1}{im}\right)}{{\left(\log -1 - \log \left(\frac{-1}{base}\right)\right)}^{3}}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\]

   11. Simplified 9.5

       \[\leadsto \color{blue}{\frac{\log \left(\frac{-1}{im}\right)}{{\left(0 + \log base\right)}^{3}}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\]

   if -2.941271494123403e+82 < im < -1.056135040441818e-243

   1. Initial program 21.0

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

   3. Applied div-inv 21.0

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

   4. Simplified 21.0

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

   6. Applied flip-+ 21.1

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

   7. Applied associate-/r/ 21.1

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

   8. Applied associate-*r* 21.1

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

   9. Simplified 21.1

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

   11. Applied add-sqr-sqrt 21.1

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

   12. Applied pow-unpow 21.1

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

   13. Simplified 21.1

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

   if -1.056135040441818e-243 < im < 1.6845120982883925e-238

   1. Initial program 30.8

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

   2. Taylor expanded around inf 32.7

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

   3. Simplified 32.6

      \[\leadsto \color{blue}{\frac{-\log re}{-\log base}}\]

   if 1.6845120982883925e-238 < im < 3.283081512447869e+107

   1. Initial program 20.3

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

   3. Applied div-inv 20.4

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

   4. Simplified 20.4

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

   6. Applied flip3-+ 20.4

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

   7. Applied associate-*l/ 20.4

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

   8. Simplified 20.4

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

   if 3.283081512447869e+107 < im

   1. Initial program 53.7

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

   2. Taylor expanded around 0 8.8

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.941271494123403150212192649236485986394 \cdot 10^{82}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{im}\right)}{{\left(\log base\right)}^{3}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\\ \mathbf{elif}\;im \le -1.056135040441817993361662182816155280925 \cdot 10^{-243}:\\ \;\;\;\;\frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{{0.0}^{3} \cdot 0.0 - {\left({\left(\log base\right)}^{2}\right)}^{\left(\sqrt{4}\right)}} \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)\\ \mathbf{elif}\;im \le 1.684512098288392462211092787468027207826 \cdot 10^{-238}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \mathbf{elif}\;im \le 3.283081512447868836362935755252953007161 \cdot 10^{107}:\\ \;\;\;\;\frac{\frac{{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right)}^{3} + {\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}^{3}}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) + \left(\left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) - \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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))))