Average Error: 31.5 → 18.0
Time: 896.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.88499570758548 \cdot 10^{124}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.0386532336525431 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8902523850390375 \cdot 10^{-167}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.30263506617115279 \cdot 10^{154}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.88499570758548 \cdot 10^{124}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -8.0386532336525431 \cdot 10^{-303}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.8902523850390375 \cdot 10^{-167}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.30263506617115279 \cdot 10^{154}:\\
\;\;\;\;\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 VAR;
	if ((re <= -2.88499570758548e+124)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= -8.038653233652543e-303)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= 1.8902523850390375e-167)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.3026350661711528e+154)) {
					VAR_3 = sqrt(((re * re) + (im * im)));
				} else {
					VAR_3 = re;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -2.88499570758548e+124

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 9.6

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

    if -2.88499570758548e+124 < re < -8.038653233652543e-303 or 1.8902523850390375e-167 < re < 1.3026350661711528e+154

    1. Initial program 18.8

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

    if -8.038653233652543e-303 < re < 1.8902523850390375e-167

    1. Initial program 31.0

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

    2. Taylor expanded around 0 34.5

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

    if 1.3026350661711528e+154 < re

    1. Initial program 64.0

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.88499570758548 \cdot 10^{124}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.0386532336525431 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8902523850390375 \cdot 10^{-167}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.30263506617115279 \cdot 10^{154}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))