Average Error: 31.1 → 17.3
Time: 1.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.4585822033812435 \cdot 10^{103}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.4585822033812435 \cdot 10^{103}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\
\;\;\;\;\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 <= -7.458582203381243e+103)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= 2.5643987853721874e-244)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= 2.9654757473955426e-173)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.5563902308845874e+144)) {
					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 < -7.458582203381243e+103

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 10.5

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

    if -7.458582203381243e+103 < re < 2.5643987853721874e-244 or 2.9654757473955426e-173 < re < 1.5563902308845874e+144

    1. Initial program 19.7

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

    if 2.5643987853721874e-244 < re < 2.9654757473955426e-173

    1. Initial program 31.4

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

    2. Taylor expanded around 0 35.6

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

    if 1.5563902308845874e+144 < re

    1. Initial program 61.2

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

    2. Taylor expanded around inf 6.2

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.4585822033812435 \cdot 10^{103}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.56439878537218745 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.9654757473955426 \cdot 10^{-173}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.556390230884587 \cdot 10^{144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))