math.log/2 on complex, real part

Average Error: 32.2 → 18.4

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.3745546542124324 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -4.4560413179826791 \cdot 10^{-240}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -6.00167965410244083 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.3393013779605753 \cdot 10^{-245}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 1.83729110979729275 \cdot 10^{-211}:\\ \;\;\;\;\left(\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.78842075491941293 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 2.13495784523255035 \cdot 10^{138}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

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 <= -1.3745546542124324e+144)) {
		VAR = ((double) (((double) (((double) (((double) (((double) log(((double) (-1.0 * re)))) * ((double) log(base)))) + ((double) (((double) atan2(im, re)) * 0.0)))) / ((double) sqrt(((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0)))))))) / ((double) sqrt(((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0))))))));
	} else {
		double VAR_1;
		if ((re <= -4.456041317982679e-240)) {
			VAR_1 = ((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) (((double) log(base)) * ((double) (2.0 * ((double) log(((double) cbrt(base)))))))) + ((double) (((double) log(base)) * ((double) log(((double) cbrt(base)))))))) + ((double) (0.0 * 0.0))))));
		} else {
			double VAR_2;
			if ((re <= -6.001679654102441e-286)) {
				VAR_2 = ((double) (((double) log(im)) / ((double) log(base))));
			} else {
				double VAR_3;
				if ((re <= 1.3393013779605753e-245)) {
					VAR_3 = ((double) (((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) -(((double) pow(0.0, 3.0)))) * 0.0)) + ((double) pow(((double) log(base)), 4.0)))))) * ((double) (((double) (((double) log(base)) * ((double) log(base)))) - ((double) (0.0 * 0.0))))));
				} else {
					double VAR_4;
					if ((re <= 1.8372911097972928e-211)) {
						VAR_4 = ((double) (((double) (((double) (((double) log(re)) * ((double) log(base)))) + ((double) (((double) atan2(im, re)) * 0.0)))) * ((double) (1.0 / ((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0))))))));
					} else {
						double VAR_5;
						if ((re <= 6.788420754919413e-157)) {
							VAR_5 = ((double) (((double) (((double) (((double) log(im)) * ((double) log(base)))) + ((double) (((double) atan2(im, re)) * 0.0)))) / ((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0))))));
						} else {
							double VAR_6;
							if ((re <= 2.1349578452325503e+138)) {
								VAR_6 = ((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) (1.0 / ((double) (((double) (((double) log(base)) * ((double) log(base)))) + ((double) (0.0 * 0.0))))))));
							} else {
								VAR_6 = ((double) (((double) log(((double) (1.0 / re)))) / ((double) log(((double) (1.0 / base))))));
							}
							VAR_5 = VAR_6;
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 8 regimes

2. if re < -1.3745546542124324e+144

   1. Initial program 62.0

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

   3. Applied add-sqr-sqrt 62.0

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

   4. Applied associate-/r* 62.0

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

   5. Taylor expanded around -inf 7.6

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

   if -1.3745546542124324e+144 < re < -4.456041317982679e-240

   1. Initial program 20.0

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

   3. Applied add-cube-cbrt 20.0

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

   4. Applied log-prod 20.0

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

   5. Applied distribute-lft-in 20.0

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

   6. Simplified 20.0

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

   if -4.456041317982679e-240 < re < -6.001679654102441e-286

   1. Initial program 33.4

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

   2. Taylor expanded around 0 30.9

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

   if -6.001679654102441e-286 < re < 1.3393013779605753e-245

   1. Initial program 32.1

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

   3. Applied flip-+ 32.1

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

   4. Applied associate-/r/ 32.1

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

   5. Simplified 32.1

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\]

   if 1.3393013779605753e-245 < re < 1.8372911097972928e-211

   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. Using strategy rm

   3. Applied div-inv 31.8

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

   4. Taylor expanded around inf 50.5

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

   if 1.8372911097972928e-211 < re < 6.788420754919413e-157

   1. Initial program 28.7

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

   2. Taylor expanded around 0 34.8

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

   if 6.788420754919413e-157 < re < 2.1349578452325503e+138

   1. Initial program 16.2

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

   3. Applied div-inv 16.2

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

   if 2.1349578452325503e+138 < re

   1. Initial program 60.0

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

   2. Taylor expanded around inf 8.1

      \[\leadsto \color{blue}{\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}}\]
3. Recombined 8 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3745546542124324 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -4.4560413179826791 \cdot 10^{-240}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le -6.00167965410244083 \cdot 10^{-286}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 1.3393013779605753 \cdot 10^{-245}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(-{0.0}^{3}\right) \cdot 0.0 + {\left(\log base\right)}^{4}} \cdot \left(\log base \cdot \log base - 0.0 \cdot 0.0\right)\\ \mathbf{elif}\;re \le 1.83729110979729275 \cdot 10^{-211}:\\ \;\;\;\;\left(\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 6.78842075491941293 \cdot 10^{-157}:\\ \;\;\;\;\frac{\log im \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{elif}\;re \le 2.13495784523255035 \cdot 10^{138}:\\ \;\;\;\;\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0\right) \cdot \frac{1}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(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))))