math.log/1 on complex, real part

Average Error: 31.4 → 18.0

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;-\log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -2.3437866053113895 \cdot 10^{-235}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.2168114165841524 \cdot 10^{-238}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.1593645025980273 \cdot 10^{82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 <= -2.3080563981284436e+37)) {
		VAR = ((double) -(((double) log(((double) (-1.0 / re))))));
	} else {
		double VAR_1;
		if ((re <= -2.3437866053113895e-235)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= 7.216811416584152e-238)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 4.159364502598027e+82)) {
					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 < -2.3080563981284436e+37

   1. Initial program 42.2

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

   2. Taylor expanded around -inf 11.5

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

   if -2.3080563981284436e+37 < re < -2.3437866053113895e-235 or 7.216811416584152e-238 < re < 4.159364502598027e+82

   1. Initial program 20.3

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

   if -2.3437866053113895e-235 < re < 7.216811416584152e-238

   1. Initial program 31.7

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

   2. Taylor expanded around 0 33.8

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

   if 4.159364502598027e+82 < re

   1. Initial program 48.4

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

   2. Taylor expanded around inf 9.8

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.30805639812844361 \cdot 10^{37}:\\ \;\;\;\;-\log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -2.3437866053113895 \cdot 10^{-235}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 7.2168114165841524 \cdot 10^{-238}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.1593645025980273 \cdot 10^{82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))