math.log/1 on complex, real part

Average Error: 31.9 → 18.4

Time: 1.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.21504708687891476 \cdot 10^{144}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.3407838631968708 \cdot 10^{-187}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.6746107021451029 \cdot 10^{-294}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.3515169656148986 \cdot 10^{-256}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.676988169724194 \cdot 10^{-193}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

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 <= -1.2150470868789148e+144)) {
		VAR = ((double) log(((double) (-1.0 * re))));
	} else {
		double VAR_1;
		if ((re <= -4.340783863196871e-187)) {
			VAR_1 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
		} else {
			double VAR_2;
			if ((re <= -2.674610702145103e-294)) {
				VAR_2 = ((double) log(im));
			} else {
				double VAR_3;
				if ((re <= 4.3515169656148986e-256)) {
					VAR_3 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
				} else {
					double VAR_4;
					if ((re <= 2.676988169724194e-193)) {
						VAR_4 = ((double) log(im));
					} else {
						double VAR_5;
						if ((re <= 1.2936263172600616e+74)) {
							VAR_5 = ((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))));
						} else {
							VAR_5 = ((double) log(re));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 < -1.21504708687891476e144

   1. Initial program 61.4

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

   2. Taylor expanded around -inf 8.1

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

   if -1.21504708687891476e144 < re < -4.3407838631968708e-187 or -2.6746107021451029e-294 < re < 4.3515169656148986e-256 or 2.676988169724194e-193 < re < 1.29362631726006162e74

   1. Initial program 19.2

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

   if -4.3407838631968708e-187 < re < -2.6746107021451029e-294 or 4.3515169656148986e-256 < re < 2.676988169724194e-193

   1. Initial program 32.4

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

   2. Taylor expanded around 0 35.1

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

   if 1.29362631726006162e74 < re

   1. Initial program 47.2

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

   2. Taylor expanded around inf 11.0

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.21504708687891476 \cdot 10^{144}:\\ \;\;\;\;\log \left(-1 \cdot re\right)\\ \mathbf{elif}\;re \le -4.3407838631968708 \cdot 10^{-187}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le -2.6746107021451029 \cdot 10^{-294}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 4.3515169656148986 \cdot 10^{-256}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{elif}\;re \le 2.676988169724194 \cdot 10^{-193}:\\ \;\;\;\;\log im\\ \mathbf{elif}\;re \le 1.29362631726006162 \cdot 10^{74}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  :precision binary64
  (log (sqrt (+ (* re re) (* im im)))))