math.log/2 on complex, real part

Average Error: 32.3 → 18.0

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.7113729341313481 \cdot 10^{136}:\\ \;\;\;\;\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -3.3378995032617328 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)}}\\ \mathbf{elif}\;re \le 5.49414863235609694 \cdot 10^{-172}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 8.73128570597796415 \cdot 10^{80}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log \left(\frac{1}{base}\right)}{\log \left(\frac{1}{re}\right)}}\\ \end{array}\]

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 <= -2.711372934131348e+136)) {
		VAR = ((((-1.0 * log((-1.0 / re))) * log(base)) + (atan2(im, re) * 0.0)) / (((log(base) * (2.0 * log(cbrt(base)))) + (log(base) * log(cbrt(base)))) + (0.0 * 0.0)));
	} else {
		double VAR_1;
		if ((re <= -3.3378995032617328e-307)) {
			VAR_1 = (1.0 / ((((0.0 * 0.0) * (0.0 * 0.0)) - (pow(log(base), 2.0) * pow(log(base), 2.0))) / (((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) * ((0.0 * 0.0) - pow(log(base), 2.0)))));
		} else {
			double VAR_2;
			if ((re <= 5.494148632356097e-172)) {
				VAR_2 = (log(im) / log(base));
			} else {
				double VAR_3;
				if ((re <= 8.731285705977964e+80)) {
					VAR_3 = (((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / (((log(base) * (2.0 * log(cbrt(base)))) + (log(base) * log(pow((1.0 / base), -0.3333333333333333)))) + (0.0 * 0.0)));
				} else {
					VAR_3 = (1.0 / (log((1.0 / base)) / log((1.0 / 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 < -2.711372934131348e+136

   1. Initial program 59.4

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

   3. Applied add-cube-cbrt 59.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

   4. Applied log-prod 59.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   5. Applied distribute-lft-in 59.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   6. Simplified 59.4

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Taylor expanded around -inf 64.0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(\log -1 - \log \left(\frac{-1}{base}\right)\right) \cdot \log \left(\frac{-1}{re}\right)\right)} + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   8. Simplified 8.4

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base} + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   if -2.711372934131348e+136 < re < -3.3378995032617328e-307

   1. Initial program 21.0

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

   3. Applied clear-num 21.0

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

   4. Simplified 21.0

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

   6. Applied flip-+ 21.0

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

   7. Applied associate-/l/ 21.1

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

   if -3.3378995032617328e-307 < re < 5.494148632356097e-172

   1. Initial program 31.8

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

   2. Taylor expanded around 0 34.7

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

   if 5.494148632356097e-172 < re < 8.731285705977964e+80

   1. Initial program 17.5

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

   3. Applied add-cube-cbrt 17.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log \color{blue}{\left(\left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) \cdot \sqrt[3]{base}\right)} + 0.0 \cdot 0.0}\]

   4. Applied log-prod 17.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \color{blue}{\left(\log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   5. Applied distribute-lft-in 17.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\color{blue}{\left(\log base \cdot \log \left(\sqrt[3]{base} \cdot \sqrt[3]{base}\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right)} + 0.0 \cdot 0.0}\]

   6. Simplified 17.6

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\color{blue}{\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right)} + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]

   7. Taylor expanded around inf 17.5

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \color{blue}{\left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right)}\right) + 0.0 \cdot 0.0}\]

   if 8.731285705977964e+80 < re

   1. Initial program 49.1

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

   3. Applied clear-num 49.1

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

   4. Simplified 49.1

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

   5. Taylor expanded around inf 9.9

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.7113729341313481 \cdot 10^{136}:\\ \;\;\;\;\frac{\left(-1 \cdot \log \left(\frac{-1}{re}\right)\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -3.3378995032617328 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\left(0.0 \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0\right) - {\left(\log base\right)}^{2} \cdot {\left(\log base\right)}^{2}}{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \left(0.0 \cdot 0.0 - {\left(\log base\right)}^{2}\right)}}\\ \mathbf{elif}\;re \le 5.49414863235609694 \cdot 10^{-172}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 8.73128570597796415 \cdot 10^{80}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left({\left(\frac{1}{base}\right)}^{\frac{-1}{3}}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log \left(\frac{1}{base}\right)}{\log \left(\frac{1}{re}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))