math.log/2 on complex, real part

Average Error: 32.4 → 18.0

Time: 7.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -184.045576900692964:\\ \;\;\;\;\frac{\log 1 - \log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le -5.4176857200810718 \cdot 10^{-297}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base\right)}^{4} - {0.0}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 4.03283230768147503 \cdot 10^{-263}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log base}\\ \mathbf{elif}\;re \le 2.9972928323585169 \cdot 10^{140}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot {\left(\log base\right)}^{2}\right) - \tan^{-1}_* \frac{im}{re} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \left(0.0 \cdot 0.0\right)\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log re}}\\ \end{array}\]

C

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

↓

double code(double re, double im, double base) {
	double VAR;
	if ((re <= -184.04557690069296)) {
		VAR = ((double) (((double) (((double) log(1.0)) - ((double) log(((double) (-1.0 / re)))))) / ((double) log(base))));
	} else {
		double VAR_1;
		if ((re <= -5.417685720081072e-297)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) log(base)) * ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) + ((double) (((double) atan2(im, re)) * 0.0)))) / ((double) (((double) pow(((double) log(base)), 4.0)) - ((double) pow(0.0, 4.0)))))) * ((double) (((double) (((double) log(base)) * ((double) log(base)))) - ((double) (0.0 * 0.0))))));
		} else {
			double VAR_2;
			if ((re <= 4.032832307681475e-263)) {
				VAR_2 = ((double) (((double) (((double) log(1.0)) + ((double) log(im)))) / ((double) log(base))));
			} else {
				double VAR_3;
				if ((re <= 2.997292832358517e+140)) {
					VAR_3 = ((double) (((double) (((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) (((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) pow(((double) log(base)), 2.0)))))) - ((double) (((double) atan2(im, re)) * ((double) (((double) atan2(im, re)) * ((double) (0.0 * 0.0)))))))) / ((double) (((double) (((double) (0.0 * 0.0)) + ((double) pow(((double) log(base)), 2.0)))) * ((double) (((double) (((double) log(base)) * ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) - ((double) (((double) atan2(im, re)) * 0.0))))))));
				} else {
					VAR_3 = ((double) (1.0 / ((double) (((double) log(base)) / ((double) log(re))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 5 regimes

2. if re < -184.045576900692964

   1. Initial program 40.8

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 13.6

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

   if -184.045576900692964 < re < -5.4176857200810718e-297

   1. Initial program 22.4

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

   3. Applied flip-+ 22.4

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

   4. Applied associate-/r/ 22.4

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

   5. Simplified 22.4

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

   if -5.4176857200810718e-297 < re < 4.03283230768147503e-263

   1. Initial program 34.6

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

   2. Taylor expanded around 0 32.4

      \[\leadsto \color{blue}{\frac{\log 1 + \log im}{\log 1 + \log base}}\]

   3. Simplified 32.4

      \[\leadsto \color{blue}{\frac{\log 1 + \log im}{\log base}}\]

   if 4.03283230768147503e-263 < re < 2.9972928323585169e140

   1. Initial program 20.3

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

   3. Applied div-inv 20.3

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

   4. Simplified 20.3

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

   6. Applied flip-+ 20.3

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

   7. Applied frac-times 20.4

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

   8. Simplified 20.4

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

   9. Simplified 20.4

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

   if 2.9972928323585169e140 < re

   1. Initial program 61.4

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

   3. Applied clear-num 61.4

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

   4. Simplified 61.4

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

   5. Taylor expanded around inf 8.1

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

   6. Simplified 8.1

      \[\leadsto \frac{1}{\color{blue}{\frac{\log base}{\log re}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -184.045576900692964:\\ \;\;\;\;\frac{\log 1 - \log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \le -5.4176857200810718 \cdot 10^{-297}:\\ \;\;\;\;\frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{{\left(\log base\right)}^{4} - {0.0}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 4.03283230768147503 \cdot 10^{-263}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log base}\\ \mathbf{elif}\;re \le 2.9972928323585169 \cdot 10^{140}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot {\left(\log base\right)}^{2}\right) - \tan^{-1}_* \frac{im}{re} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \left(0.0 \cdot 0.0\right)\right)}{\left(0.0 \cdot 0.0 + {\left(\log base\right)}^{2}\right) \cdot \left(\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) - \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))