math.log/2 on complex, real part

Average Error: 31.9 → 18.1

Time: 8.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.7952699182275347 \cdot 10^{110}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -1.24115509350136921 \cdot 10^{-247}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 8.2237586483272037 \cdot 10^{-280}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log 1 + \log base}\\ \mathbf{elif}\;re \le 1.1064992212988812 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \end{array}\]

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 <= -4.795269918227535e+110)) {
		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 <= -1.2411550935013692e-247)) {
			VAR_1 = ((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) 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_2;
			if ((re <= 8.223758648327204e-280)) {
				VAR_2 = ((double) (((double) (((double) log(1.0)) + ((double) log(im)))) / ((double) (((double) log(1.0)) + ((double) log(base))))));
			} else {
				double VAR_3;
				if ((re <= 0.00011064992212988812)) {
					VAR_3 = ((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 {
					VAR_3 = ((double) (((double) (((double) (((double) log(re)) * ((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))))));
				}
				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 < -4.7952699182275347e110

   1. Initial program 53.2

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

   3. Applied add-sqr-sqrt 53.2

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

   4. Applied associate-/r* 53.2

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

   5. Taylor expanded around -inf 8.4

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

   if -4.7952699182275347e110 < re < -1.24115509350136921e-247

   1. Initial program 20.2

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

   3. Applied add-sqr-sqrt 20.2

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

   4. Applied associate-/r* 20.2

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

   if -1.24115509350136921e-247 < re < 8.2237586483272037e-280

   1. Initial program 31.4

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

   2. Taylor expanded around 0 32.4

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

   if 8.2237586483272037e-280 < re < 1.1064992212988812e-4

   1. Initial program 22.6

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

   3. Applied add-cube-cbrt 22.6

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

   4. Applied log-prod 22.6

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

   5. Applied distribute-lft-in 22.6

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

   6. Simplified 22.6

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

   if 1.1064992212988812e-4 < re

   1. Initial program 40.0

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

   3. Applied add-cube-cbrt 40.0

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

   4. Applied log-prod 40.0

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

   5. Applied distribute-lft-in 40.0

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

   6. Simplified 40.0

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

   7. Taylor expanded around inf 14.2

      \[\leadsto \frac{\log \color{blue}{re} \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\]
3. Recombined 5 regimes into one program.

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.7952699182275347 \cdot 10^{110}:\\ \;\;\;\;\frac{\frac{\log \left(-1 \cdot re\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le -1.24115509350136921 \cdot 10^{-247}:\\ \;\;\;\;\frac{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\\ \mathbf{elif}\;re \le 8.2237586483272037 \cdot 10^{-280}:\\ \;\;\;\;\frac{\log 1 + \log im}{\log 1 + \log base}\\ \mathbf{elif}\;re \le 1.1064992212988812 \cdot 10^{-4}:\\ \;\;\;\;\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log re \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\left(\log base \cdot \left(2 \cdot \log \left(\sqrt[3]{base}\right)\right) + \log base \cdot \log \left(\sqrt[3]{base}\right)\right) + 0.0 \cdot 0.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(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))))