math.log/2 on complex, real part

Average Error: 31.7 → 19.2

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -3.21907540931537393 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -7.2713402830638425 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.9780598850161828 \cdot 10^{-206}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.21416753908483653 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \mathbf{elif}\;re \le 1.492585514901269 \cdot 10^{-103}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 2.43405196972925384 \cdot 10^{101}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

C

double code(double re, double im, double base) {
	return (((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 <= -3.219075409315374e+117)) {
		VAR = ((((log((-1.0 * re)) * log(base)) + (atan2(im, re) * 0.0)) / sqrt(((log(base) * log(base)) + (0.0 * 0.0)))) / sqrt(((log(base) * log(base)) + (0.0 * 0.0))));
	} else {
		double VAR_1;
		if ((re <= -7.2713402830638425e-171)) {
			VAR_1 = ((1.0 / sqrt(((log(base) * log(base)) + (0.0 * 0.0)))) * (((log(sqrt(((re * re) + (im * im)))) * log(base)) + (atan2(im, re) * 0.0)) / sqrt(((log(base) * log(base)) + (0.0 * 0.0)))));
		} else {
			double VAR_2;
			if ((re <= 1.9780598850161828e-206)) {
				VAR_2 = (log(im) / log(base));
			} else {
				double VAR_3;
				if ((re <= 6.2141675390848365e-142)) {
					VAR_3 = (log((1.0 / re)) / log((1.0 / base)));
				} else {
					double VAR_4;
					if ((re <= 1.4925855149012688e-103)) {
						VAR_4 = (log(im) / log(base));
					} else {
						double VAR_5;
						if ((re <= 2.4340519697292538e+101)) {
							VAR_5 = (((log(((cbrt(sqrt(((re * re) + (im * im)))) * cbrt(sqrt(((re * re) + (im * im))))) * cbrt(sqrt(((re * re) + (im * im)))))) * log(base)) + (atan2(im, re) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0)));
						} else {
							VAR_5 = (log((1.0 / re)) / log((1.0 / base)));
						}
						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 5 regimes

2. if re < -3.219075409315374e+117

   1. Initial program 55.6

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

   3. Applied add-sqr-sqrt 55.6

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

   4. Applied associate-/r* 55.6

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

   5. Taylor expanded around -inf 7.9

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

   if -3.219075409315374e+117 < re < -7.2713402830638425e-171

   1. Initial program 16.1

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

   3. Applied add-sqr-sqrt 16.1

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

   4. Applied *-un-lft-identity 16.1

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

   5. Applied times-frac 16.1

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

   if -7.2713402830638425e-171 < re < 1.9780598850161828e-206 or 6.2141675390848365e-142 < re < 1.4925855149012688e-103

   1. Initial program 30.4

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

   2. Taylor expanded around 0 34.8

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

   if 1.9780598850161828e-206 < re < 6.2141675390848365e-142 or 2.4340519697292538e+101 < re

   1. Initial program 45.5

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

   2. Taylor expanded around inf 17.5

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

   if 1.4925855149012688e-103 < re < 2.4340519697292538e+101

   1. Initial program 14.5

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

   3. Applied add-cube-cbrt 14.6

      \[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
3. Recombined 5 regimes into one program.

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.21907540931537393 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -7.2713402830638425 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 1.9780598850161828 \cdot 10^{-206}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 6.21416753908483653 \cdot 10^{-142}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \mathbf{elif}\;re \le 1.492585514901269 \cdot 10^{-103}:\\ \;\;\;\;\frac{\log im}{\log base}\\ \mathbf{elif}\;re \le 2.43405196972925384 \cdot 10^{101}:\\ \;\;\;\;\frac{\log \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(\frac{1}{re}\right)}{\log \left(\frac{1}{base}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))