math.log/1 on complex, real part

Average Error: 31.5 → 18.0

Time: 1.5s

Precision: binary64

Math

\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.268063321609487 \cdot 10^{+32}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.17385497808198 \cdot 10^{-261}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.046877101973681 \cdot 10^{-198}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 5.756838278404119 \cdot 10^{+130}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((re <= -1.268063321609487e+32)) {
		VAR = ((double) log(((double) -(re))));
	} else {
		double VAR_1;
		if ((re <= -4.17385497808198e-261)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 2.046877101973681e-198)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 5.756838278404119e+130)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					VAR_3 = ((double) log(re));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.2680633216094869e32

   1. Initial program Error: 42.0 bits

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around -inf Error: 11.3 bits

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

   3. Simplified Error: 11.3 bits

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

   if -1.2680633216094869e32 < re < -4.1738549780819797e-261 or 2.046877101973681e-198 < re < 5.75683827840411851e130

   1. Initial program Error: 19.8 bits

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   if -4.1738549780819797e-261 < re < 2.046877101973681e-198

   1. Initial program Error: 30.6 bits

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around 0 Error: 34.7 bits

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

   if 5.75683827840411851e130 < re

   1. Initial program Error: 56.9 bits

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

   2. Taylor expanded around inf Error: 7.5 bits

      \[\leadsto \log \color{blue}{re}\]
3. Recombined 4 regimes into one program.

4. Final simplification Error: 18.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.268063321609487 \cdot 10^{+32}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -4.17385497808198 \cdot 10^{-261}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 2.046877101973681 \cdot 10^{-198}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 5.756838278404119 \cdot 10^{+130}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))