math.log/2 on complex, real part

Average Error: 31.9 → 17.5

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.4855642221203277 \cdot 10^{+41}:\\ \;\;\;\;\frac{\log 1 - \log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \leq 4.6156350849393567 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log re}}\\ \end{array}\]

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 ((re <= -1.4855642221203277e+41)) {
		VAR = (((double) (((double) log(1.0)) - ((double) log((-1.0 / re))))) / ((double) log(base)));
	} else {
		double VAR_1;
		if ((re <= 4.6156350849393567e+111)) {
			VAR_1 = ((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) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) + ((double) (0.0 * ((double) atan2(im, re)))))) / ((double) sqrt(((double) (((double) pow(((double) log(base)), 2.0)) + ((double) (0.0 * 0.0)))))))));
		} else {
			VAR_1 = (1.0 / (((double) log(base)) / ((double) log(re))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Split input into 3 regimes

2. if re < -1.48556422212032769e41

   1. Initial program 43.4

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

   2. Taylor expanded around -inf 64.0

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

   3. Simplified 11.7

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

   if -1.48556422212032769e41 < re < 4.61563508493935666e111

   1. Initial program 22.2

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

   3. Applied add-sqr-sqrt 22.2

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

   4. Applied *-un-lft-identity 22.2

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

   5. Applied times-frac 22.2

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

   6. Simplified 22.2

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

   7. Simplified 22.2

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

   if 4.61563508493935666e111 < re

   1. Initial program 53.1

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

   3. Applied clear-num 53.1

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

   4. Simplified 53.1

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

   5. Taylor expanded around inf 7.9

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

   6. Simplified 7.9

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

4. Final simplification 17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.4855642221203277 \cdot 10^{+41}:\\ \;\;\;\;\frac{\log 1 - \log \left(\frac{-1}{re}\right)}{\log base}\\ \mathbf{elif}\;re \leq 4.6156350849393567 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}} \cdot \frac{\log base \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right) + 0 \cdot \tan^{-1}_* \frac{im}{re}}{\sqrt{{\left(\log base\right)}^{2} + 0 \cdot 0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\log base}{\log re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))