Average Error: 31.5 → 18.0
Time: 1.3s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.8473355353767562 \cdot 10^{+116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq 2.3232582542635886 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.8473355353767562 \cdot 10^{+116}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq 2.3232582542635886 \cdot 10^{+20}:\\
\;\;\;\;\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 -2.8473355353767562e+116)
   (- re)
   (if (<= re 2.3232582542635886e+20) (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 <= -2.8473355353767562e+116)) {
		tmp = ((double) -(re));
	} else {
		double tmp_1;
		if ((re <= 2.3232582542635886e+20)) {
			tmp_1 = ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))));
		} else {
			tmp_1 = re;
		}
		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 3 regimes

  2. if re < -2.8473355353767562e116

    1. Initial program Error: 54.7 bits

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

    2. Taylor expanded around -inf Error: 8.6 bits

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

    3. SimplifiedError: 8.6 bits

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

    if -2.8473355353767562e116 < re < 232325825426358860000

    1. Initial program Error: 22.0 bits

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

    if 232325825426358860000 < re

    1. Initial program Error: 40.4 bits

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

    2. Taylor expanded around inf Error: 13.8 bits

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

  4. Final simplificationError: 18.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.8473355353767562 \cdot 10^{+116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq 2.3232582542635886 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))