math.log10 on complex, real part

Average Error: 32.8 → 18.0

Time: 15.8s

Precision: 64

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

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \left(2 \cdot \log im\right)\right)\right)\\ \mathbf{elif}\;re \le 8.0278279982437549 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{2 \cdot \log re}}\\ \end{array}\]

C

double f(double re, double im) {
        double r42477 = re;
        double r42478 = r42477 * r42477;
        double r42479 = im;
        double r42480 = r42479 * r42479;
        double r42481 = r42478 + r42480;
        double r42482 = sqrt(r42481);
        double r42483 = log(r42482);
        double r42484 = 10.0;
        double r42485 = log(r42484);
        double r42486 = r42483 / r42485;
        return r42486;
}

↓

double f(double re, double im) {
        double r42487 = re;
        double r42488 = -4.197550803800697e+153;
        bool r42489 = r42487 <= r42488;
        double r42490 = -1.0;
        double r42491 = r42490 / r42487;
        double r42492 = log(r42491);
        double r42493 = -r42492;
        double r42494 = 10.0;
        double r42495 = log(r42494);
        double r42496 = r42493 / r42495;
        double r42497 = -4.540545293771823e-275;
        bool r42498 = r42487 <= r42497;
        double r42499 = 0.5;
        double r42500 = sqrt(r42499);
        double r42501 = sqrt(r42495);
        double r42502 = r42500 / r42501;
        double r42503 = r42487 * r42487;
        double r42504 = im;
        double r42505 = r42504 * r42504;
        double r42506 = r42503 + r42505;
        double r42507 = log(r42506);
        double r42508 = r42501 / r42507;
        double r42509 = r42500 / r42508;
        double r42510 = r42502 * r42509;
        double r42511 = 1.2766285812733717e-281;
        bool r42512 = r42487 <= r42511;
        double r42513 = sqrt(r42500);
        double r42514 = r42513 / r42501;
        double r42515 = 2.0;
        double r42516 = log(r42504);
        double r42517 = r42515 * r42516;
        double r42518 = r42513 * r42517;
        double r42519 = r42514 * r42518;
        double r42520 = r42502 * r42519;
        double r42521 = 8.027827998243755e+98;
        bool r42522 = r42487 <= r42521;
        double r42523 = log(r42487);
        double r42524 = r42515 * r42523;
        double r42525 = r42501 / r42524;
        double r42526 = r42500 / r42525;
        double r42527 = r42502 * r42526;
        double r42528 = r42522 ? r42510 : r42527;
        double r42529 = r42512 ? r42520 : r42528;
        double r42530 = r42498 ? r42510 : r42529;
        double r42531 = r42489 ? r42496 : r42530;
        return r42531;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.197550803800697e+153

   1. Initial program 63.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied pow1/2 63.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 63.9

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 63.9

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 63.9

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 63.9

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 63.9

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 63.9

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 63.9

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 63.9

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 63.9

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Taylor expanded around -inf 7.2

       \[\leadsto \color{blue}{-2 \cdot \frac{\log \left(\frac{-1}{re}\right) \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2}}{\log 10}}\]

   15. Simplified 7.1

       \[\leadsto \color{blue}{\frac{-1 \cdot \log \left(\frac{-1}{re}\right)}{\log 10}}\]

   if -4.197550803800697e+153 < re < -4.540545293771823e-275 or 1.2766285812733717e-281 < re < 8.027827998243755e+98

   1. Initial program 21.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied pow1/2 21.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 21.6

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 21.7

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 21.7

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 21.7

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 21.7

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 21.8

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 21.7

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 21.5

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 21.5

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   if -4.540545293771823e-275 < re < 1.2766285812733717e-281

   1. Initial program 33.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied pow1/2 33.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 33.3

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 33.2

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 33.2

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 33.2

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 33.2

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 33.3

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 33.3

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 33.2

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 33.2

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]
   14. Using strategy rm

   15. Applied div-inv 33.2

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\color{blue}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Applied add-sqr-sqrt 33.2

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   17. Applied sqrt-prod 33.3

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\sqrt{\sqrt{\frac{1}{2}}} \cdot \sqrt{\sqrt{\frac{1}{2}}}}}{\sqrt{\log 10} \cdot \frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\]

   18. Applied times-frac 33.2

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\sqrt{\frac{1}{2}}}}{\frac{1}{\log \left(re \cdot re + im \cdot im\right)}}\right)}\]

   19. Simplified 33.1

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)}\right)\]

   20. Taylor expanded around 0 33.5

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \color{blue}{\left(2 \cdot \log im\right)}\right)\right)\]

   if 8.027827998243755e+98 < re

   1. Initial program 51.6

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied pow1/2 51.6

      \[\leadsto \frac{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}{\log 10}\]

   4. Applied log-pow 51.6

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}{\log 10}\]

   5. Applied associate-/l* 51.6

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{\log 10}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   6. Using strategy rm

   7. Applied pow1 51.6

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{1}\right)}}}\]

   8. Applied log-pow 51.6

      \[\leadsto \frac{\frac{1}{2}}{\frac{\log 10}{\color{blue}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   9. Applied add-sqr-sqrt 51.6

      \[\leadsto \frac{\frac{1}{2}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{1 \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   10. Applied times-frac 51.7

       \[\leadsto \frac{\frac{1}{2}}{\color{blue}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 51.7

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\frac{\sqrt{\log 10}}{1} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 51.6

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{1}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Simplified 51.6

       \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   14. Taylor expanded around inf 8.0

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\color{blue}{-2 \cdot \log \left(\frac{1}{re}\right)}}}\]

   15. Simplified 8.0

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\color{blue}{2 \cdot \log re}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{-\log \left(\frac{-1}{re}\right)}{\log 10}\\ \mathbf{elif}\;re \le -4.5405452937718227 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{elif}\;re \le 1.27662858127337166 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\frac{\sqrt{\sqrt{\frac{1}{2}}}}{\sqrt{\log 10}} \cdot \left(\sqrt{\sqrt{\frac{1}{2}}} \cdot \left(2 \cdot \log im\right)\right)\right)\\ \mathbf{elif}\;re \le 8.0278279982437549 \cdot 10^{98}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{2 \cdot \log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))