Average Error: 31.5 → 17.8
Time: 1.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.1469124341159173 \cdot 10^{94}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.19651783577238613 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1868703169839283 \cdot 10^{-289}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.2975351703027165 \cdot 10^{62}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.1469124341159173 \cdot 10^{94}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -3.19651783577238613 \cdot 10^{-203}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.1868703169839283 \cdot 10^{-289}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 9.2975351703027165 \cdot 10^{62}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
double code(double re, double im) {
	return ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
}
↓
double code(double re, double im) {
	double VAR;
	if ((re <= -2.1469124341159173e+94)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -3.196517835772386e-203)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 1.1868703169839283e-289)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 9.297535170302716e+62)) {
					VAR_3 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (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 < -2.1469124341159173e+94

    1. Initial program 50.8

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

    2. Taylor expanded around -inf 9.6

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

    if -2.1469124341159173e+94 < re < -3.196517835772386e-203 or 1.1868703169839283e-289 < re < 9.297535170302716e+62

    1. Initial program 19.4

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

    if -3.196517835772386e-203 < re < 1.1868703169839283e-289

    1. Initial program 30.5

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

    2. Taylor expanded around 0 34.6

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

    if 9.297535170302716e+62 < re

    1. Initial program 45.9

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

    2. Taylor expanded around inf 12.6

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.1469124341159173 \cdot 10^{94}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.19651783577238613 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1868703169839283 \cdot 10^{-289}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.2975351703027165 \cdot 10^{62}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))