math.log10 on complex, real part

Average Error: 31.3 → 17.4

Time: 29.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -7.713582517877012 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{-2 \cdot \log \left(\frac{-1}{re}\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 4.799588986507124 \cdot 10^{+96}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r1373477 = re;
        double r1373478 = r1373477 * r1373477;
        double r1373479 = im;
        double r1373480 = r1373479 * r1373479;
        double r1373481 = r1373478 + r1373480;
        double r1373482 = sqrt(r1373481);
        double r1373483 = log(r1373482);
        double r1373484 = 10.0;
        double r1373485 = log(r1373484);
        double r1373486 = r1373483 / r1373485;
        return r1373486;
}

↓

double f(double re, double im) {
        double r1373487 = re;
        double r1373488 = -7.713582517877012e+138;
        bool r1373489 = r1373487 <= r1373488;
        double r1373490 = 0.5;
        double r1373491 = sqrt(r1373490);
        double r1373492 = 10.0;
        double r1373493 = log(r1373492);
        double r1373494 = sqrt(r1373493);
        double r1373495 = -2.0;
        double r1373496 = -1.0;
        double r1373497 = r1373496 / r1373487;
        double r1373498 = log(r1373497);
        double r1373499 = r1373495 * r1373498;
        double r1373500 = r1373494 / r1373499;
        double r1373501 = r1373491 / r1373500;
        double r1373502 = r1373491 / r1373494;
        double r1373503 = r1373501 * r1373502;
        double r1373504 = 4.799588986507124e+96;
        bool r1373505 = r1373487 <= r1373504;
        double r1373506 = cbrt(r1373491);
        double r1373507 = r1373506 * r1373506;
        double r1373508 = r1373507 * r1373502;
        double r1373509 = sqrt(r1373494);
        double r1373510 = r1373508 / r1373509;
        double r1373511 = r1373487 * r1373487;
        double r1373512 = im;
        double r1373513 = r1373512 * r1373512;
        double r1373514 = r1373511 + r1373513;
        double r1373515 = log(r1373514);
        double r1373516 = r1373509 / r1373515;
        double r1373517 = r1373506 / r1373516;
        double r1373518 = r1373510 * r1373517;
        double r1373519 = 1.0;
        double r1373520 = r1373519 / r1373487;
        double r1373521 = log(r1373520);
        double r1373522 = r1373491 * r1373521;
        double r1373523 = r1373519 / r1373493;
        double r1373524 = sqrt(r1373523);
        double r1373525 = r1373522 * r1373524;
        double r1373526 = r1373495 * r1373525;
        double r1373527 = r1373502 * r1373526;
        double r1373528 = r1373505 ? r1373518 : r1373527;
        double r1373529 = r1373489 ? r1373503 : r1373528;
        return r1373529;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -7.713582517877012e+138

   1. Initial program 57.5

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

   3. Applied pow1/2 57.5

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

   4. Applied log-pow 57.5

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

   5. Applied associate-/l* 57.5

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

   7. Applied pow1 57.5

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

   8. Applied log-pow 57.5

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

   9. Applied add-sqr-sqrt 57.5

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

   10. Applied times-frac 57.6

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

   11. Applied add-sqr-sqrt 57.5

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

   12. Applied times-frac 57.5

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

   13. Simplified 57.5

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

   14. Taylor expanded around -inf 6.9

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

   15. Simplified 6.9

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

   if -7.713582517877012e+138 < re < 4.799588986507124e+96

   1. Initial program 21.5

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

   3. Applied pow1/2 21.5

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

   4. Applied log-pow 21.5

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

   5. Applied associate-/l* 21.5

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

   7. Applied pow1 21.5

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

   8. Applied log-pow 21.5

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

   9. Applied add-sqr-sqrt 21.5

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

   10. Applied times-frac 21.6

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

   11. Applied add-sqr-sqrt 21.5

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

   12. Applied times-frac 21.4

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

   13. Simplified 21.4

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

   15. Applied *-un-lft-identity 21.4

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

   16. Applied add-sqr-sqrt 21.7

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

   17. Applied times-frac 21.7

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

   18. Applied add-cube-cbrt 21.4

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}\right) \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1} \cdot \frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   19. Applied times-frac 21.6

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1}} \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\right)}\]

   20. Applied associate-*r* 21.7

       \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{1}}\right) \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   21. Simplified 21.4

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}\right)}{\sqrt{\sqrt{\log 10}}}} \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   if 4.799588986507124e+96 < re

   1. Initial program 49.7

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

   3. Applied pow1/2 49.7

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

   4. Applied log-pow 49.7

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

   5. Applied associate-/l* 49.7

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

   7. Applied pow1 49.7

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

   8. Applied log-pow 49.7

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

   9. Applied add-sqr-sqrt 49.7

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

   10. Applied times-frac 49.7

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

   11. Applied add-sqr-sqrt 49.7

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

   12. Applied times-frac 49.7

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

   13. Simplified 49.7

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

   14. Taylor expanded around inf 10.0

       \[\leadsto \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\sqrt{\frac{1}{\log 10}} \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.713582517877012 \cdot 10^{+138}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\frac{\sqrt{\log 10}}{-2 \cdot \log \left(\frac{-1}{re}\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 4.799588986507124 \cdot 10^{+96}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{\frac{1}{2}}} \cdot \sqrt[3]{\sqrt{\frac{1}{2}}}\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}}{\sqrt{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{\sqrt{\frac{1}{2}}}}{\frac{\sqrt{\sqrt{\log 10}}}{\log \left(re \cdot re + im \cdot im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\left(\sqrt{\frac{1}{2}} \cdot \log \left(\frac{1}{re}\right)\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))