Average Error: 31.6 → 18.9
Time: 2.5s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.7569446232380357 \cdot 10^{+114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -4.576170092116779 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -5.047961066574088 \cdot 10^{-113}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5.63142386051803 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.7569446232380357 \cdot 10^{+114}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -4.576170092116779 \cdot 10^{-56}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq -5.047961066574088 \cdot 10^{-113}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 5.63142386051803 \cdot 10^{+118}:\\
\;\;\;\;\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 -2.7569446232380357e+114)
   (- re)
   (if (<= re -4.576170092116779e-56)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re -5.047961066574088e-113)
       im
       (if (<= re 5.63142386051803e+118) (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 tmp;
	if ((re <= -2.7569446232380357e+114)) {
		tmp = ((double) -(re));
	} else {
		double tmp_1;
		if ((re <= -4.576170092116779e-56)) {
			tmp_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double tmp_2;
			if ((re <= -5.047961066574088e-113)) {
				tmp_2 = im;
			} else {
				double tmp_3;
				if ((re <= 5.63142386051803e+118)) {
					tmp_3 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
				} else {
					tmp_3 = re;
				}
				tmp_2 = tmp_3;
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 < -2.75694462323803569e114

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 10.0

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

    3. Simplified10.0

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

    if -2.75694462323803569e114 < re < -4.57617009211677922e-56 or -5.04796106657408782e-113 < re < 5.63142386051802992e118

    1. Initial program 21.5

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

    if -4.57617009211677922e-56 < re < -5.04796106657408782e-113

    1. Initial program 14.8

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

    2. Taylor expanded around 0 42.0

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

    if 5.63142386051802992e118 < re

    1. Initial program 55.1

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7569446232380357 \cdot 10^{+114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -4.576170092116779 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -5.047961066574088 \cdot 10^{-113}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 5.63142386051803 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))