math.log10 on complex, real part

Average Error: 32.5 → 18.1

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.886444123297446 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.30867664875349023 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.2217527835098556 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 4.9319226435479606 \cdot 10^{132}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r44538 = re;
        double r44539 = r44538 * r44538;
        double r44540 = im;
        double r44541 = r44540 * r44540;
        double r44542 = r44539 + r44541;
        double r44543 = sqrt(r44542);
        double r44544 = log(r44543);
        double r44545 = 10.0;
        double r44546 = log(r44545);
        double r44547 = r44544 / r44546;
        return r44547;
}

↓

double f(double re, double im) {
        double r44548 = re;
        double r44549 = -1.886444123297446e+126;
        bool r44550 = r44548 <= r44549;
        double r44551 = 1.0;
        double r44552 = sqrt(r44551);
        double r44553 = r44551 / r44552;
        double r44554 = 2.0;
        double r44555 = -1.0;
        double r44556 = r44555 / r44548;
        double r44557 = log(r44556);
        double r44558 = r44554 * r44557;
        double r44559 = -r44558;
        double r44560 = 0.5;
        double r44561 = 10.0;
        double r44562 = log(r44561);
        double r44563 = sqrt(r44562);
        double r44564 = r44560 / r44563;
        double r44565 = r44564 / r44563;
        double r44566 = r44559 * r44565;
        double r44567 = r44553 * r44566;
        double r44568 = -2.3086766487534902e-209;
        bool r44569 = r44548 <= r44568;
        double r44570 = r44548 * r44548;
        double r44571 = im;
        double r44572 = r44571 * r44571;
        double r44573 = r44570 + r44572;
        double r44574 = log(r44573);
        double r44575 = r44574 * r44565;
        double r44576 = r44553 * r44575;
        double r44577 = 1.2217527835098556e-266;
        bool r44578 = r44548 <= r44577;
        double r44579 = log(r44571);
        double r44580 = r44554 * r44579;
        double r44581 = r44580 * r44565;
        double r44582 = r44553 * r44581;
        double r44583 = 4.9319226435479606e+132;
        bool r44584 = r44548 <= r44583;
        double r44585 = -2.0;
        double r44586 = r44551 / r44548;
        double r44587 = log(r44586);
        double r44588 = r44551 / r44562;
        double r44589 = sqrt(r44588);
        double r44590 = r44587 * r44589;
        double r44591 = r44585 * r44590;
        double r44592 = r44564 * r44591;
        double r44593 = r44584 ? r44576 : r44592;
        double r44594 = r44578 ? r44582 : r44593;
        double r44595 = r44569 ? r44576 : r44594;
        double r44596 = r44550 ? r44567 : r44595;
        return r44596;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.886444123297446e+126

   1. Initial program 57.6

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

   3. Applied add-sqr-sqrt 57.6

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

   4. Applied pow1/2 57.6

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

   5. Applied log-pow 57.6

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

   6. Applied times-frac 57.6

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

   8. Applied add-log-exp 57.6

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

   9. Simplified 57.6

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

   11. Applied pow1 57.6

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

   12. Applied log-pow 57.6

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

   13. Applied sqrt-prod 57.6

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

   14. Applied *-un-lft-identity 57.6

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

   15. Applied times-frac 57.6

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

   16. Applied associate-*l* 57.6

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

   17. Simplified 57.6

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

   18. Taylor expanded around -inf 9.0

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

   if -1.886444123297446e+126 < re < -2.3086766487534902e-209 or 1.2217527835098556e-266 < re < 4.9319226435479606e+132

   1. Initial program 19.9

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

   3. Applied add-sqr-sqrt 19.9

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

   4. Applied pow1/2 19.9

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

   5. Applied log-pow 19.9

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

   6. Applied times-frac 19.9

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

   8. Applied add-log-exp 19.9

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

   9. Simplified 19.7

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

   11. Applied pow1 19.7

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

   12. Applied log-pow 19.7

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

   13. Applied sqrt-prod 19.7

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

   14. Applied *-un-lft-identity 19.7

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

   15. Applied times-frac 19.7

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

   16. Applied associate-*l* 19.7

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

   17. Simplified 19.7

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

   if -2.3086766487534902e-209 < re < 1.2217527835098556e-266

   1. Initial program 31.6

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

   3. Applied add-sqr-sqrt 31.6

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

   4. Applied pow1/2 31.6

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

   5. Applied log-pow 31.6

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

   6. Applied times-frac 31.6

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

   8. Applied add-log-exp 31.6

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

   9. Simplified 31.5

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

   11. Applied pow1 31.5

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

   12. Applied log-pow 31.5

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

   13. Applied sqrt-prod 31.5

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

   14. Applied *-un-lft-identity 31.5

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

   15. Applied times-frac 31.5

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

   16. Applied associate-*l* 31.5

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

   17. Simplified 31.5

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

   18. Taylor expanded around 0 35.4

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

   if 4.9319226435479606e+132 < re

   1. Initial program 58.4

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

   3. Applied add-sqr-sqrt 58.4

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

   4. Applied pow1/2 58.4

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

   5. Applied log-pow 58.4

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

   6. Applied times-frac 58.4

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

   7. Taylor expanded around inf 6.8

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.886444123297446 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\left(-2 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le -2.30867664875349023 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 1.2217527835098556 \cdot 10^{-266}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\left(2 \cdot \log im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{elif}\;re \le 4.9319226435479606 \cdot 10^{132}:\\ \;\;\;\;\frac{1}{\sqrt{1}} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))