Average Error: 31.5 → 18.5
Time: 1.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.97733214615282287 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.6982430225033068 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.7706602894998885 \cdot 10^{-238}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6531905343411006 \cdot 10^{78}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.97733214615282287 \cdot 10^{89}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -2.6982430225033068 \cdot 10^{-174}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 3.7706602894998885 \cdot 10^{-238}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.6531905343411006 \cdot 10^{78}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
double code(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double temp;
	if ((re <= -9.977332146152823e+89)) {
		temp = (-1.0 * re);
	} else {
		double temp_1;
		if ((re <= -2.6982430225033068e-174)) {
			temp_1 = sqrt(((re * re) + (im * im)));
		} else {
			double temp_2;
			if ((re <= 3.7706602894998885e-238)) {
				temp_2 = im;
			} else {
				double temp_3;
				if ((re <= 1.6531905343411006e+78)) {
					temp_3 = sqrt(((re * re) + (im * im)));
				} else {
					temp_3 = re;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -9.977332146152823e+89

    1. Initial program 49.1

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around -inf 12.5

      \[\leadsto \color{blue}{-1 \cdot re}\]

    if -9.977332146152823e+89 < re < -2.6982430225033068e-174 or 3.7706602894998885e-238 < re < 1.6531905343411006e+78

    1. Initial program 18.3

      \[\sqrt{re \cdot re + im \cdot im}\]

    if -2.6982430225033068e-174 < re < 3.7706602894998885e-238

    1. Initial program 32.0

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around 0 33.5

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

    if 1.6531905343411006e+78 < re

    1. Initial program 48.4

      \[\sqrt{re \cdot re + im \cdot im}\]

    2. Taylor expanded around inf 11.7

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.97733214615282287 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.6982430225033068 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.7706602894998885 \cdot 10^{-238}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6531905343411006 \cdot 10^{78}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))