Average Error: 31.1 → 18.9
Time: 933.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double code(double re, double im) {
	return sqrt(((re * re) + (im * im)));
}
↓
double code(double re, double im) {
	double VAR;
	if ((re <= -8.15615961666859e+125)) {
		VAR = (-1.0 * re);
	} else {
		double VAR_1;
		if ((re <= -3.8099669373079583e-103)) {
			VAR_1 = sqrt(((re * re) + (im * im)));
		} else {
			double VAR_2;
			if ((re <= 2.3367351856997066e-296)) {
				VAR_2 = im;
			} else {
				double VAR_3;
				if ((re <= 1.0193327448038136e+95)) {
					VAR_3 = sqrt(((re * re) + (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 < -8.15615961666859e+125

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 9.1

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

    if -8.15615961666859e+125 < re < -3.8099669373079583e-103 or 2.3367351856997066e-296 < re < 1.0193327448038136e+95

    1. Initial program 18.9

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

    if -3.8099669373079583e-103 < re < 2.3367351856997066e-296

    1. Initial program 27.8

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

    2. Taylor expanded around 0 36.1

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

    if 1.0193327448038136e+95 < re

    1. Initial program 50.2

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

    2. Taylor expanded around inf 10.4

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))