Average Error: 31.5 → 17.5
Time: 3.0s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.07621210558784363 \cdot 10^{93}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.24837089590244688 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.46573922954344998 \cdot 10^{-184}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.0706681104305729 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.07621210558784363 \cdot 10^{93}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 1.24837089590244688 \cdot 10^{-261}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 3.46573922954344998 \cdot 10^{-184}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 4.0706681104305729 \cdot 10^{132}:\\
\;\;\;\;\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 <= -2.0762121055878436e+93)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= 1.2483708959024469e-261)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= 3.46573922954345e-184)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 4.070668110430573e+132)) {
					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 < -2.07621210558784363e93

    1. Initial program 51.1

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

    2. Taylor expanded around -inf 10.7

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

    if -2.07621210558784363e93 < re < 1.24837089590244688e-261 or 3.46573922954344998e-184 < re < 4.0706681104305729e132

    1. Initial program 19.9

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

    if 1.24837089590244688e-261 < re < 3.46573922954344998e-184

    1. Initial program 30.7

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

    2. Taylor expanded around 0 32.9

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

    if 4.0706681104305729e132 < re

    1. Initial program 57.5

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.07621210558784363 \cdot 10^{93}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.24837089590244688 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.46573922954344998 \cdot 10^{-184}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.0706681104305729 \cdot 10^{132}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))