Average Error: 31.9 → 18.8
Time: 2.2s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -4.3382686603473446 \cdot 10^{+135}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.3620464573579147 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 9.758403782058524 \cdot 10^{-175}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.392709190784314 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -4.3382686603473446 \cdot 10^{+135}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -2.3620464573579147 \cdot 10^{-209}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 9.758403782058524 \cdot 10^{-175}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 7.392709190784314 \cdot 10^{+91}:\\
\;\;\;\;\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 -4.3382686603473446e+135)
   (- re)
   (if (<= re -2.3620464573579147e-209)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 9.758403782058524e-175)
       im
       (if (<= re 7.392709190784314e+91) (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 <= -4.3382686603473446e+135)) {
		tmp = ((double) -(re));
	} else {
		double tmp_1;
		if ((re <= -2.3620464573579147e-209)) {
			tmp_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double tmp_2;
			if ((re <= 9.758403782058524e-175)) {
				tmp_2 = im;
			} else {
				double tmp_3;
				if ((re <= 7.392709190784314e+91)) {
					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 < -4.3382686603473446e135

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 9.4

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

    3. Simplified9.4

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

    if -4.3382686603473446e135 < re < -2.36204645735791475e-209 or 9.75840378205852385e-175 < re < 7.39270919078431371e91

    1. Initial program 18.2

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

    if -2.36204645735791475e-209 < re < 9.75840378205852385e-175

    1. Initial program 30.6

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

    2. Taylor expanded around 0 33.6

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

    if 7.39270919078431371e91 < re

    1. Initial program 50.8

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

    2. Taylor expanded around inf 11.8

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.3382686603473446 \cdot 10^{+135}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.3620464573579147 \cdot 10^{-209}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 9.758403782058524 \cdot 10^{-175}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.392709190784314 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))