Average Error: 31.3 → 17.2
Time: 1.4s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -4.8534135319298755 \cdot 10^{+120}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.7561694933647637 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.6733231477544338 \cdot 10^{-305}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.6023651145839193 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -4.8534135319298755 \cdot 10^{+120}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -1.7561694933647637 \cdot 10^{-223}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq -1.6733231477544338 \cdot 10^{-305}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 1.6023651145839193 \cdot 10^{+102}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -4.8534135319298755e+120)
   (- re)
   (if (<= re -1.7561694933647637e-223)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re -1.6733231477544338e-305)
       im
       (if (<= re 1.6023651145839193e+102)
         (sqrt (+ (* re re) (* im im)))
         re)))))
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 <= -4.8534135319298755e+120)) {
		VAR = ((double) -(re));
	} else {
		double VAR_1;
		if ((re <= -1.7561694933647637e-223)) {
			VAR_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double VAR_2;
			if ((re <= -1.6733231477544338e-305)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.6023651145839193e+102)) {
					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 < -4.8534135319298755e120

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    if -4.8534135319298755e120 < re < -1.75616949336476367e-223 or -1.6733231477544338e-305 < re < 1.602365114583919e102

    1. Initial program 19.3

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

    if -1.75616949336476367e-223 < re < -1.6733231477544338e-305

    1. Initial program 30.8

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

    2. Taylor expanded around 0 32.6

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

    if 1.602365114583919e102 < re

    1. Initial program 53.0

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

    2. Taylor expanded around inf 10.8

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8534135319298755 \cdot 10^{+120}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.7561694933647637 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.6733231477544338 \cdot 10^{-305}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.6023651145839193 \cdot 10^{+102}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))