math.log10 on complex, real part

Average Error: 30.6 → 17.1

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -7.993069603181868 \cdot 10^{+120}:\\ \;\;\;\;\left(\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 3.263648024588438 \cdot 10^{+109}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r2727474 = re;
        double r2727475 = r2727474 * r2727474;
        double r2727476 = im;
        double r2727477 = r2727476 * r2727476;
        double r2727478 = r2727475 + r2727477;
        double r2727479 = sqrt(r2727478);
        double r2727480 = log(r2727479);
        double r2727481 = 10.0;
        double r2727482 = log(r2727481);
        double r2727483 = r2727480 / r2727482;
        return r2727483;
}

↓

double f(double re, double im) {
        double r2727484 = im;
        double r2727485 = -7.993069603181868e+120;
        bool r2727486 = r2727484 <= r2727485;
        double r2727487 = -2.0;
        double r2727488 = -1.0;
        double r2727489 = r2727488 / r2727484;
        double r2727490 = log(r2727489);
        double r2727491 = r2727487 * r2727490;
        double r2727492 = 0.5;
        double r2727493 = sqrt(r2727492);
        double r2727494 = 10.0;
        double r2727495 = log(r2727494);
        double r2727496 = sqrt(r2727495);
        double r2727497 = r2727493 / r2727496;
        double r2727498 = r2727491 * r2727497;
        double r2727499 = r2727498 * r2727497;
        double r2727500 = 3.263648024588438e+109;
        bool r2727501 = r2727484 <= r2727500;
        double r2727502 = re;
        double r2727503 = r2727502 * r2727502;
        double r2727504 = r2727484 * r2727484;
        double r2727505 = r2727503 + r2727504;
        double r2727506 = log(r2727505);
        double r2727507 = cbrt(r2727492);
        double r2727508 = cbrt(r2727495);
        double r2727509 = r2727507 / r2727508;
        double r2727510 = r2727506 * r2727509;
        double r2727511 = r2727509 * r2727509;
        double r2727512 = r2727510 * r2727511;
        double r2727513 = 2.0;
        double r2727514 = log(r2727484);
        double r2727515 = r2727513 * r2727514;
        double r2727516 = r2727515 * r2727509;
        double r2727517 = r2727516 * r2727511;
        double r2727518 = r2727501 ? r2727512 : r2727517;
        double r2727519 = r2727486 ? r2727499 : r2727518;
        return r2727519;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if im < -7.993069603181868e+120

   1. Initial program 54.1

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

   3. Applied pow1/2 54.1

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

   4. Applied log-pow 54.1

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

   5. Applied associate-/l* 54.1

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

   7. Applied *-un-lft-identity 54.1

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

   8. Applied add-sqr-sqrt 54.1

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

   9. Applied times-frac 54.1

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

   10. Applied add-sqr-sqrt 54.0

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

   11. Applied times-frac 54.0

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

   12. Simplified 54.0

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

   13. Simplified 54.0

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

   14. Taylor expanded around -inf 7.1

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

   15. Simplified 7.1

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

   if -7.993069603181868e+120 < im < 3.263648024588438e+109

   1. Initial program 21.2

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

   3. Applied pow1/2 21.2

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

   4. Applied log-pow 21.2

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

   5. Applied associate-/l* 21.2

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

   7. Applied pow1 21.2

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

   8. Applied log-pow 21.2

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

   9. Applied add-cube-cbrt 21.7

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

   10. Applied times-frac 21.7

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

   11. Applied add-cube-cbrt 21.1

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

   12. Applied times-frac 21.1

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

   13. Simplified 21.1

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

   14. Simplified 21.0

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

   if 3.263648024588438e+109 < im

   1. Initial program 51.7

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

   3. Applied pow1/2 51.7

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

   4. Applied log-pow 51.7

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

   5. Applied associate-/l* 51.7

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

   7. Applied pow1 51.7

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

   8. Applied log-pow 51.7

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

   9. Applied add-cube-cbrt 51.8

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

   10. Applied times-frac 51.8

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

   11. Applied add-cube-cbrt 51.7

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

   12. Applied times-frac 51.7

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

   13. Simplified 51.7

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

   14. Simplified 51.6

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

   15. Taylor expanded around inf 8.8

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

   16. Simplified 8.8

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

4. Final simplification 17.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -7.993069603181868 \cdot 10^{+120}:\\ \;\;\;\;\left(\left(-2 \cdot \log \left(\frac{-1}{im}\right)\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\right) \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{\log 10}}\\ \mathbf{elif}\;im \le 3.263648024588438 \cdot 10^{+109}:\\ \;\;\;\;\left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \log im\right) \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{\log 10}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))