Average Error: 31.2 → 17.4
Time: 3.6s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -9.967998367313345 \cdot 10^{+85}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.8536898573216425 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.9260948128030526 \cdot 10^{-231}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.4612703761687805 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -9.967998367313345 \cdot 10^{+85}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -2.8536898573216425 \cdot 10^{-241}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 1.9260948128030526 \cdot 10^{-231}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 1.4612703761687805 \cdot 10^{+141}:\\
\;\;\;\;\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 -9.967998367313345e+85)
   (- re)
   (if (<= re -2.8536898573216425e-241)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 1.9260948128030526e-231)
       im
       (if (<= re 1.4612703761687805e+141)
         (sqrt (+ (* re re) (* im im)))
         re)))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -9.967998367313345e+85) {
		tmp = -re;
	} else if (re <= -2.8536898573216425e-241) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= 1.9260948128030526e-231) {
		tmp = im;
	} else if (re <= 1.4612703761687805e+141) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = re;
	}
	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 < -9.96799836731334529e85

    1. Initial program 50.0

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

    2. Taylor expanded around -inf 10.4

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

    3. Simplified10.4

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

    if -9.96799836731334529e85 < re < -2.8536898573216425e-241 or 1.9260948128030526e-231 < re < 1.4612703761687805e141

    1. Initial program 18.4

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

    if -2.8536898573216425e-241 < re < 1.9260948128030526e-231

    1. Initial program 30.7

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

    2. Taylor expanded around 0 33.5

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

    if 1.4612703761687805e141 < re

    1. Initial program 60.1

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.967998367313345 \cdot 10^{+85}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -2.8536898573216425 \cdot 10^{-241}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 1.9260948128030526 \cdot 10^{-231}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 1.4612703761687805 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))