math.log/2 on complex, real part

Average Error: 32.2 → 24.1

Time: 7.5s

Precision: binary64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\begin{array}{l} \mathbf{if}\;im \leq -1.3444364465238792 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log base \cdot \log \left(-re\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{elif}\;im \leq -2.3107304361754112 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{elif}\;im \leq -2.7719158935979882 \cdot 10^{-303}:\\ \;\;\;\;\frac{\log base \cdot \log \left(-re\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{{\left(\log base\right)}^{4} - {0}^{4}} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)\\ \mathbf{elif}\;im \leq 8.836668233359011 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log base}\\ \end{array}\]

C

double code(double re, double im, double base) {
	return (((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 ((im <= -1.3444364465238792e+154)) {
		VAR = ((double) ((1.0 / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0))))))) * (((double) (((double) (((double) log(base)) * ((double) log(((double) -(re)))))) + ((double) (0.0 * ((double) atan2(im, re)))))) / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0)))))))));
	} else {
		double VAR_1;
		if ((im <= -2.3107304361754112e-235)) {
			VAR_1 = ((double) ((1.0 / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0))))))) * (((double) (((double) (0.0 * ((double) atan2(im, re)))) + ((double) (((double) log(base)) * ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0)))))))));
		} else {
			double VAR_2;
			if ((im <= -2.7719158935979882e-303)) {
				VAR_2 = ((double) ((((double) (((double) (((double) log(base)) * ((double) log(((double) -(re)))))) + ((double) (0.0 * ((double) atan2(im, re)))))) / ((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_3;
				if ((im <= 8.836668233359011e+75)) {
					VAR_3 = ((double) ((1.0 / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0))))))) * (((double) (((double) (0.0 * ((double) atan2(im, re)))) + ((double) (((double) log(base)) * ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0)))))))));
				} else {
					VAR_3 = (((double) (((double) log(1.0)) + ((double) log(im)))) / ((double) log(base)));
				}
				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 4 regimes

2. if im < -1.34443644652387919e154

   1. Initial program 64.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   4. Applied *-un-lft-identity 64.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right)}}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   5. Applied times-frac 64.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0 \cdot 0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   6. Simplified 64.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   7. Simplified 64.0

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}}\]

   8. Taylor expanded around -inf 57.2

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\]

   9. Simplified 57.2

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log \color{blue}{\left(-re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\]

   if -1.34443644652387919e154 < im < -2.3107304361754112e-235 or -2.77191589359798818e-303 < im < 8.83666823335901099e75

   1. Initial program 20.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   4. Applied *-un-lft-identity 20.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0\right)}}{\sqrt{\log base \cdot \log base + 0 \cdot 0} \cdot \sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   5. Applied times-frac 20.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log base \cdot \log base + 0 \cdot 0}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}}\]

   6. Simplified 20.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{\log base \cdot \log base + 0 \cdot 0}}\]

   7. Simplified 20.8

      \[\leadsto \frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}}\]

   if -2.3107304361754112e-235 < im < -2.77191589359798818e-303

   1. Initial program 30.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
   2. Using strategy rm

   3. Applied flip-+ 30.8

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\color{blue}{\frac{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right)}{\log base \cdot \log base - 0 \cdot 0}}}\]

   4. Applied associate-/r/ 30.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\left(\log base \cdot \log base\right) \cdot \left(\log base \cdot \log base\right) - \left(0 \cdot 0\right) \cdot \left(0 \cdot 0\right)} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)}\]

   5. Simplified 30.9

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{{\left(\log base\right)}^{4} - {0}^{4}}} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)\]

   6. Taylor expanded around -inf 33.8

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{{\left(\log base\right)}^{4} - {0}^{4}} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)\]

   7. Simplified 33.8

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{{\left(\log base\right)}^{4} - {0}^{4}} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)\]

   if 8.83666823335901099e75 < im

   1. Initial program 49.2

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

   2. Taylor expanded around 0 10.2

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

   3. Simplified 10.2

      \[\leadsto \color{blue}{\frac{\log 1 + \log im}{\log base}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3444364465238792 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log base \cdot \log \left(-re\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{elif}\;im \leq -2.3107304361754112 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{elif}\;im \leq -2.7719158935979882 \cdot 10^{-303}:\\ \;\;\;\;\frac{\log base \cdot \log \left(-re\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{{\left(\log base\right)}^{4} - {0}^{4}} \cdot \left(\log base \cdot \log base - 0 \cdot 0\right)\\ \mathbf{elif}\;im \leq 8.836668233359011 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{0 \cdot \tan^{-1}_* \frac{im}{re} + \log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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))))