math.log/2 on complex, real part

Average Error: 31.5 → 17.5

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.47908061050294571 \cdot 10^{108}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\frac{0 + \log base}{-1}}\\ \mathbf{elif}\;re \le 7.02369283710352736 \cdot 10^{90}:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot \left({\left(\log base\right)}^{2} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(\frac{1}{2} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \log base\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log base \cdot \log re + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

C

double code(double re, double im, double base) {
	return (((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0)));
}

↓

double code(double re, double im, double base) {
	double VAR;
	if ((re <= -3.4790806105029457e+108)) {
		VAR = (log((-1.0 / re)) / ((0.0 + log(base)) / -1.0));
	} else {
		double VAR_1;
		if ((re <= 7.023692837103527e+90)) {
			VAR_1 = ((((0.25 * (pow(log(base), 2.0) * log(((re * re) + (im * im))))) * log(((re * re) + (im * im)))) - ((atan2(im, re) * 0.0) * (atan2(im, re) * 0.0))) / (((0.5 * (log(((re * re) + (im * im))) * log(base))) - (atan2(im, re) * 0.0)) * ((0.0 * 0.0) + pow(log(base), 2.0))));
		} else {
			VAR_1 = (((log(base) * log(re)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 3 regimes

2. if re < -3.4790806105029457e+108

   1. Initial program 52.5

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 9.1

      \[\leadsto \color{blue}{\frac{\log \left(\frac{-1}{re}\right)}{\frac{0 + \log base}{-1}}}\]

   if -3.4790806105029457e+108 < re < 7.023692837103527e+90

   1. Initial program 21.3

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

   3. Applied pow1/2 21.3

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

   4. Applied log-pow 21.3

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

   5. Applied associate-*l* 21.3

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

   7. Applied flip-+ 21.3

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

   8. Applied associate-/l/ 21.4

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

   9. Simplified 21.4

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

   11. Applied *-commutative 21.4

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

   12. Applied associate-*r* 21.4

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

   13. Applied associate-*r* 21.4

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

   14. Simplified 21.4

       \[\leadsto \frac{\color{blue}{\left(\frac{1}{4} \cdot \left({\left(\log base\right)}^{2} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right)} \cdot \log \left(re \cdot re + im \cdot im\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(\frac{1}{2} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \log base\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right)}\]

   if 7.023692837103527e+90 < re

   1. Initial program 50.9

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

   2. Taylor expanded around inf 10.4

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

   3. Simplified 10.4

      \[\leadsto \frac{\color{blue}{\log base \cdot \log re} + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.47908061050294571 \cdot 10^{108}:\\ \;\;\;\;\frac{\log \left(\frac{-1}{re}\right)}{\frac{0 + \log base}{-1}}\\ \mathbf{elif}\;re \le 7.02369283710352736 \cdot 10^{90}:\\ \;\;\;\;\frac{\left(\frac{1}{4} \cdot \left({\left(\log base\right)}^{2} \cdot \log \left(re \cdot re + im \cdot im\right)\right)\right) \cdot \log \left(re \cdot re + im \cdot im\right) - \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\left(\frac{1}{2} \cdot \left(\log \left(re \cdot re + im \cdot im\right) \cdot \log base\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log base \cdot \log re + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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))))