Average Error: 31.8 → 18.0
Time: 1.5s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -4.864167289730826 \cdot 10^{+136}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.4704911317770355 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.6020683207984553 \cdot 10^{-95}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.1635286815108139 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -4.864167289730826 \cdot 10^{+136}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -2.4704911317770355 \cdot 10^{-70}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq -1.6020683207984553 \cdot 10^{-95}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 1.1635286815108139 \cdot 10^{+92}:\\
\;\;\;\;\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 -4.864167289730826e+136)
   (- re)
   (if (<= re -2.4704911317770355e-70)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re -1.6020683207984553e-95)
       im
       (if (<= re 1.1635286815108139e+92)
         (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 <= -4.864167289730826e+136)) {
		tmp = ((double) -(re));
	} else {
		double tmp_1;
		if ((re <= -2.4704911317770355e-70)) {
			tmp_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			double tmp_2;
			if ((re <= -1.6020683207984553e-95)) {
				tmp_2 = im;
			} else {
				double tmp_3;
				if ((re <= 1.1635286815108139e+92)) {
					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 < -4.8641672897308265e136

    1. Initial program Error: 59.4 bits

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

    2. Taylor expanded around -inf Error: 9.6 bits

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

    3. SimplifiedError: 9.6 bits

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

    if -4.8641672897308265e136 < re < -2.4704911317770355e-70 or -1.6020683207984553e-95 < re < 1.1635286815108139e92

    1. Initial program Error: 21.0 bits

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

    if -2.4704911317770355e-70 < re < -1.6020683207984553e-95

    1. Initial program Error: 20.5 bits

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

    2. Taylor expanded around 0 Error: 39.2 bits

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

    if 1.1635286815108139e92 < re

    1. Initial program Error: 49.8 bits

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

    2. Taylor expanded around inf Error: 11.6 bits

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

  4. Final simplificationError: 18.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.864167289730826 \cdot 10^{+136}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.4704911317770355 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.6020683207984553 \cdot 10^{-95}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.1635286815108139 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))