Average Error: 31.0 → 18.1
Time: 2.6s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.42682394536003885 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.74998232531126355 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8637666906269956 \cdot 10^{-113}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.1318584300381151 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.42682394536003885 \cdot 10^{82}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.74998232531126355 \cdot 10^{-281}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.8637666906269956 \cdot 10^{-113}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.1318584300381151 \cdot 10^{127}:\\
\;\;\;\;\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 <= -1.4268239453600389e+82)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= 4.7499823253112636e-281)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= 1.8637666906269956e-113)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 2.131858430038115e+127)) {
					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 < -1.4268239453600389e+82

    1. Initial program 47.7

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

    2. Taylor expanded around -inf 10.7

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

    if -1.4268239453600389e+82 < re < 4.7499823253112636e-281 or 1.8637666906269956e-113 < re < 2.131858430038115e+127

    1. Initial program 19.0

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

    if 4.7499823253112636e-281 < re < 1.8637666906269956e-113

    1. Initial program 26.9

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

    2. Taylor expanded around 0 35.3

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

    if 2.131858430038115e+127 < re

    1. Initial program 57.3

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

    2. Taylor expanded around inf 8.2

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.42682394536003885 \cdot 10^{82}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.74998232531126355 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.8637666906269956 \cdot 10^{-113}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.1318584300381151 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))