Average Error: 31.5 → 18.3
Time: 2.8s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.4451944394226077 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.61825031122827605 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -4.6382787383428388 \cdot 10^{-284}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.5717864824332804 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.4451944394226077 \cdot 10^{121}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.61825031122827605 \cdot 10^{-247}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -4.6382787383428388 \cdot 10^{-284}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.5717864824332804 \cdot 10^{-8}:\\
\;\;\;\;\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 <= -4.445194439422608e+121)) {
		VAR = ((double) (-1.0 * re));
	} else {
		double VAR_1;
		if ((re <= -1.618250311228276e-247)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= -4.638278738342839e-284)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 2.5717864824332804e-08)) {
					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 < -4.4451944394226077e121

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 8.3

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

    if -4.4451944394226077e121 < re < -1.61825031122827605e-247 or -4.6382787383428388e-284 < re < 2.5717864824332804e-8

    1. Initial program 21.3

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

    if -1.61825031122827605e-247 < re < -4.6382787383428388e-284

    1. Initial program 34.5

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

    2. Taylor expanded around 0 33.2

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

    if 2.5717864824332804e-8 < re

    1. Initial program 39.3

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

    2. Taylor expanded around inf 15.8

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.4451944394226077 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.61825031122827605 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -4.6382787383428388 \cdot 10^{-284}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.5717864824332804 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))