Average Error: 31.8 → 18.5
Time: 2.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.25443431135417537 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.40477998933113187 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.02637373854441201 \cdot 10^{-303}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.032966617252637 \cdot 10^{102}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.25443431135417537 \cdot 10^{129}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.40477998933113187 \cdot 10^{-158}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -1.02637373854441201 \cdot 10^{-303}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.032966617252637 \cdot 10^{102}:\\
\;\;\;\;\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 <= -6.254434311354175e+129)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= -1.4047799893311319e-158)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= -1.026373738544412e-303)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 2.032966617252637e+102)) {
					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 < -6.254434311354175e+129

    1. Initial program 58.3

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

    2. Taylor expanded around -inf 9.4

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

    if -6.254434311354175e+129 < re < -1.4047799893311319e-158 or -1.026373738544412e-303 < re < 2.032966617252637e+102

    1. Initial program 19.4

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

    if -1.4047799893311319e-158 < re < -1.026373738544412e-303

    1. Initial program 30.5

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

    2. Taylor expanded around 0 35.2

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

    if 2.032966617252637e+102 < re

    1. Initial program 52.3

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.25443431135417537 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.40477998933113187 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.02637373854441201 \cdot 10^{-303}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.032966617252637 \cdot 10^{102}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))