Average Error: 31.9 → 17.8
Time: 3.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\
\;\;\;\;\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 <= -6.485144449769119e+83)) {
		temp = (-1.0 * re);
	} else {
		double temp_1;
		if ((re <= 9.194803090293717e-296)) {
			temp_1 = sqrt(((re * re) + (im * im)));
		} else {
			double temp_2;
			if ((re <= 1.1600661433813666e-208)) {
				temp_2 = im;
			} else {
				double temp_3;
				if ((re <= 1.2142912145306137e+146)) {
					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 < -6.485144449769119e+83

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 11.4

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

    if -6.485144449769119e+83 < re < 9.194803090293717e-296 or 1.1600661433813666e-208 < re < 1.2142912145306137e+146

    1. Initial program 19.9

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

    if 9.194803090293717e-296 < re < 1.1600661433813666e-208

    1. Initial program 31.5

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

    2. Taylor expanded around 0 34.0

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

    if 1.2142912145306137e+146 < re

    1. Initial program 62.0

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

    2. Taylor expanded around inf 7.9

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))