math.log/2 on complex, real part

Average Error: 32.5 → 18.4

Time: 7.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.03956723925942709 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -6.37440255965590493 \cdot 10^{-279}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 8.0032969337860817 \cdot 10^{-168}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.0657693485277471 \cdot 10^{114}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r47419 = re;
        double r47420 = r47419 * r47419;
        double r47421 = im;
        double r47422 = r47421 * r47421;
        double r47423 = r47420 + r47422;
        double r47424 = sqrt(r47423);
        double r47425 = log(r47424);
        double r47426 = base;
        double r47427 = log(r47426);
        double r47428 = r47425 * r47427;
        double r47429 = atan2(r47421, r47419);
        double r47430 = 0.0;
        double r47431 = r47429 * r47430;
        double r47432 = r47428 + r47431;
        double r47433 = r47427 * r47427;
        double r47434 = r47430 * r47430;
        double r47435 = r47433 + r47434;
        double r47436 = r47432 / r47435;
        return r47436;
}

↓

double f(double re, double im, double base) {
        double r47437 = re;
        double r47438 = -1.039567239259427e+103;
        bool r47439 = r47437 <= r47438;
        double r47440 = 1.0;
        double r47441 = base;
        double r47442 = log(r47441);
        double r47443 = 6.0;
        double r47444 = pow(r47442, r47443);
        double r47445 = cbrt(r47444);
        double r47446 = 0.0;
        double r47447 = r47446 * r47446;
        double r47448 = r47445 + r47447;
        double r47449 = sqrt(r47448);
        double r47450 = r47440 / r47449;
        double r47451 = -1.0;
        double r47452 = r47451 * r47437;
        double r47453 = log(r47452);
        double r47454 = r47453 * r47442;
        double r47455 = im;
        double r47456 = atan2(r47455, r47437);
        double r47457 = r47456 * r47446;
        double r47458 = r47454 + r47457;
        double r47459 = r47442 * r47442;
        double r47460 = r47459 + r47447;
        double r47461 = sqrt(r47460);
        double r47462 = r47458 / r47461;
        double r47463 = r47450 * r47462;
        double r47464 = -6.374402559655905e-279;
        bool r47465 = r47437 <= r47464;
        double r47466 = r47437 * r47437;
        double r47467 = r47455 * r47455;
        double r47468 = r47466 + r47467;
        double r47469 = sqrt(r47468);
        double r47470 = log(r47469);
        double r47471 = r47470 * r47442;
        double r47472 = r47471 + r47457;
        double r47473 = 2.0;
        double r47474 = cbrt(r47441);
        double r47475 = log(r47474);
        double r47476 = r47473 * r47475;
        double r47477 = r47442 * r47476;
        double r47478 = r47442 * r47475;
        double r47479 = r47477 + r47478;
        double r47480 = r47479 + r47447;
        double r47481 = r47472 / r47480;
        double r47482 = 8.003296933786082e-168;
        bool r47483 = r47437 <= r47482;
        double r47484 = log(r47455);
        double r47485 = r47484 / r47442;
        double r47486 = 1.0657693485277471e+114;
        bool r47487 = r47437 <= r47486;
        double r47488 = r47472 / r47461;
        double r47489 = r47450 * r47488;
        double r47490 = r47440 / r47437;
        double r47491 = log(r47490);
        double r47492 = r47440 / r47441;
        double r47493 = log(r47492);
        double r47494 = r47491 / r47493;
        double r47495 = r47487 ? r47489 : r47494;
        double r47496 = r47483 ? r47485 : r47495;
        double r47497 = r47465 ? r47481 : r47496;
        double r47498 = r47439 ? r47463 : r47497;
        return r47498;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -1.039567239259427e+103

   1. Initial program 52.6

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

   3. Applied add-sqr-sqrt 52.6

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

   4. Applied *-un-lft-identity 52.6

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

   5. Applied times-frac 52.6

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

   7. Applied add-cbrt-cube 52.6

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

   8. Applied add-cbrt-cube 52.6

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

   9. Applied cbrt-unprod 52.6

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

   10. Simplified 52.6

       \[\leadsto \frac{1}{\sqrt{\sqrt[3]{\color{blue}{{\left(\log base\right)}^{6}}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   11. Taylor expanded around -inf 9.5

       \[\leadsto \frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   if -1.039567239259427e+103 < re < -6.374402559655905e-279

   1. Initial program 21.1

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

   3. Applied add-cube-cbrt 21.1

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

   4. Applied log-prod 21.1

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

   5. Applied distribute-lft-in 21.1

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

   6. Simplified 21.1

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

   if -6.374402559655905e-279 < re < 8.003296933786082e-168

   1. Initial program 31.9

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

   2. Taylor expanded around 0 36.3

      \[\leadsto \color{blue}{\frac{\log im}{\log base}}\]

   if 8.003296933786082e-168 < re < 1.0657693485277471e+114

   1. Initial program 17.1

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

   3. Applied add-sqr-sqrt 17.1

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

   4. Applied *-un-lft-identity 17.1

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

   5. Applied times-frac 17.1

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

   7. Applied add-cbrt-cube 17.2

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

   8. Applied add-cbrt-cube 17.3

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

   9. Applied cbrt-unprod 17.2

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

   10. Simplified 17.2

       \[\leadsto \frac{1}{\sqrt{\sqrt[3]{\color{blue}{{\left(\log base\right)}^{6}}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]

   if 1.0657693485277471e+114 < re

   1. Initial program 55.7

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

   2. Taylor expanded around inf 9.1

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.03956723925942709 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -6.37440255965590493 \cdot 10^{-279}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 8.0032969337860817 \cdot 10^{-168}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.0657693485277471 \cdot 10^{114}:\\ \;\;\;\;\frac{1}{\sqrt{\sqrt[3]{{\left(\log base\right)}^{6}} + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))