math.log/2 on complex, real part

Average Error: 32.0 → 17.6

Time: 12.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.2529321969344202 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le -7.7296027907141122 \cdot 10^{-226}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{elif}\;re \le 6.31099625192104387 \cdot 10^{-197}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 3.1293650531822621 \cdot 10^{107}:\\ \;\;\;\;\frac{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot {\left(\log base\right)}^{2} - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

C

double f(double re, double im, double base) {
        double r42476 = re;
        double r42477 = r42476 * r42476;
        double r42478 = im;
        double r42479 = r42478 * r42478;
        double r42480 = r42477 + r42479;
        double r42481 = sqrt(r42480);
        double r42482 = log(r42481);
        double r42483 = base;
        double r42484 = log(r42483);
        double r42485 = r42482 * r42484;
        double r42486 = atan2(r42478, r42476);
        double r42487 = 0.0;
        double r42488 = r42486 * r42487;
        double r42489 = r42485 + r42488;
        double r42490 = r42484 * r42484;
        double r42491 = r42487 * r42487;
        double r42492 = r42490 + r42491;
        double r42493 = r42489 / r42492;
        return r42493;
}

↓

double f(double re, double im, double base) {
        double r42494 = re;
        double r42495 = -8.25293219693442e+97;
        bool r42496 = r42494 <= r42495;
        double r42497 = 1.0;
        double r42498 = 0.0;
        double r42499 = r42498 * r42498;
        double r42500 = base;
        double r42501 = log(r42500);
        double r42502 = 2.0;
        double r42503 = pow(r42501, r42502);
        double r42504 = r42499 + r42503;
        double r42505 = sqrt(r42504);
        double r42506 = r42497 / r42505;
        double r42507 = im;
        double r42508 = atan2(r42507, r42494);
        double r42509 = r42508 * r42498;
        double r42510 = -r42494;
        double r42511 = log(r42510);
        double r42512 = r42501 * r42511;
        double r42513 = r42509 + r42512;
        double r42514 = r42513 / r42505;
        double r42515 = r42506 * r42514;
        double r42516 = -7.729602790714112e-226;
        bool r42517 = r42494 <= r42516;
        double r42518 = r42494 * r42494;
        double r42519 = r42507 * r42507;
        double r42520 = r42518 + r42519;
        double r42521 = sqrt(r42520);
        double r42522 = log(r42521);
        double r42523 = r42522 * r42501;
        double r42524 = r42523 + r42509;
        double r42525 = r42497 / r42504;
        double r42526 = r42524 * r42525;
        double r42527 = 6.310996251921044e-197;
        bool r42528 = r42494 <= r42527;
        double r42529 = log(r42507);
        double r42530 = r42529 / r42501;
        double r42531 = 3.129365053182262e+107;
        bool r42532 = r42494 <= r42531;
        double r42533 = r42522 * r42522;
        double r42534 = r42533 * r42503;
        double r42535 = r42509 * r42509;
        double r42536 = r42534 - r42535;
        double r42537 = r42536 / r42504;
        double r42538 = r42523 - r42509;
        double r42539 = r42537 / r42538;
        double r42540 = log(r42494);
        double r42541 = -r42540;
        double r42542 = -r42501;
        double r42543 = r42541 / r42542;
        double r42544 = r42532 ? r42539 : r42543;
        double r42545 = r42528 ? r42530 : r42544;
        double r42546 = r42517 ? r42526 : r42545;
        double r42547 = r42496 ? r42515 : r42546;
        return r42547;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -8.25293219693442e+97

   1. Initial program 51.4

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

   3. Applied add-sqr-sqrt 51.4

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

   4. Applied *-un-lft-identity 51.4

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

   5. Applied times-frac 51.4

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

   6. Simplified 51.4

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

   7. Simplified 51.4

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

   8. Taylor expanded around -inf 8.1

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

   9. Simplified 8.1

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

   if -8.25293219693442e+97 < re < -7.729602790714112e-226

   1. Initial program 20.2

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

   3. Applied div-inv 20.3

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

   4. Simplified 20.3

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

   if -7.729602790714112e-226 < re < 6.310996251921044e-197

   1. Initial program 32.3

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

   2. Taylor expanded around 0 34.5

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

   if 6.310996251921044e-197 < re < 3.129365053182262e+107

   1. Initial program 17.0

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

   3. Applied div-inv 17.1

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

   4. Simplified 17.1

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

   6. Applied flip-+ 17.1

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

   7. Applied associate-*l/ 17.1

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

   8. Simplified 17.1

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

   if 3.129365053182262e+107 < re

   1. Initial program 53.3

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

   2. Taylor expanded around inf 8.4

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

   3. Simplified 8.4

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.2529321969344202 \cdot 10^{97}:\\ \;\;\;\;\frac{1}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}} \cdot \frac{\tan^{-1}_* \frac{im}{re} \cdot 0.0 + \log base \cdot \log \left(-re\right)}{\sqrt{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}\\ \mathbf{elif}\;re \le -7.7296027907141122 \cdot 10^{-226}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}\\ \mathbf{elif}\;re \le 6.31099625192104387 \cdot 10^{-197}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 3.1293650531822621 \cdot 10^{107}:\\ \;\;\;\;\frac{\frac{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)\right) \cdot {\left(\log base\right)}^{2} - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{0.0 \cdot 0.0 + {\left(\log base\right)}^{2}}}{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base - \tan^{-1}_* \frac{im}{re} \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\log re}{-\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))