Average Error: 32.3 → 8.1
Time: 1.9s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \leq 4.2886947769615204 \cdot 10^{-221}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 6.278318910904598 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \leq 4.2886947769615204 \cdot 10^{-221}:\\
\;\;\;\;-re\\

\mathbf{elif}\;im \leq 6.278318910904598 \cdot 10^{+103}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;im\\

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= im 4.2886947769615204e-221)
   (- re)
   (if (<= im 6.278318910904598e+103) (sqrt (+ (* re re) (* im im))) im)))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (im <= 4.2886947769615204e-221) {
		tmp = -re;
	} else if (im <= 6.278318910904598e+103) {
		tmp = sqrt((re * re) + (im * im));
	} else {
		tmp = im;
	}
	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 im < 4.28869477696152041e-221

    1. Initial program 31.9

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

    2. Taylor expanded around -inf 2.4

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

    if 4.28869477696152041e-221 < im < 6.27831891090459817e103

    1. Initial program 15.4

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

    if 6.27831891090459817e103 < im

    1. Initial program 52.1

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

    2. Taylor expanded around 0 5.6

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 4.2886947769615204 \cdot 10^{-221}:\\ \;\;\;\;-re\\ \mathbf{elif}\;im \leq 6.278318910904598 \cdot 10^{+103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))