math.log/1 on complex, real part

Average Error: 31.7 → 17.6

Time: 1.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.3699735643448252 \cdot 10^{+94}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.396105895291542 \cdot 10^{-246}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 4.97977832146901 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 33696.34187874525:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

FPCore

(FPCore (re im) :precision binary64 (log (sqrt (+ (* re re) (* im im)))))

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.3699735643448252e+94)
   (log (- re))
   (if (<= re -5.396105895291542e-246)
     (log (sqrt (+ (* re re) (* im im))))
     (if (<= re 4.97977832146901e-256)
       (log im)
       (if (<= re 33696.34187874525)
         (log (sqrt (+ (* re re) (* im im))))
         (log re))))))

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 tmp;
	if ((re <= -1.3699735643448252e+94)) {
		tmp = ((double) log(((double) -(re))));
	} else {
		double tmp_1;
		if ((re <= -5.396105895291542e-246)) {
			tmp_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double tmp_2;
			if ((re <= 4.97977832146901e-256)) {
				tmp_2 = ((double) log(im));
			} else {
				double tmp_3;
				if ((re <= 33696.34187874525)) {
					tmp_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					tmp_3 = ((double) log(re));
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1.3699735643448252e94

   1. Initial program 50.3

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

   2. Taylor expanded around -inf 10.1

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

   3. Simplified 10.1

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

   if -1.3699735643448252e94 < re < -5.39610589529154205e-246 or 4.9797783214690098e-256 < re < 33696.3418787452465

   1. Initial program 20.3

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

   if -5.39610589529154205e-246 < re < 4.9797783214690098e-256

   1. Initial program 32.7

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

   2. Taylor expanded around 0 30.0

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

   if 33696.3418787452465 < re

   1. Initial program 40.9

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

   2. Taylor expanded around inf 12.8

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3699735643448252 \cdot 10^{+94}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{elif}\;re \leq -5.396105895291542 \cdot 10^{-246}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \leq 4.97977832146901 \cdot 10^{-256}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \leq 33696.34187874525:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))