Average Error: 31.2 → 7.8
Time: 1.5s
Precision: binary64
\[[re, im]=\mathsf{sort}([re, im])\]
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.8823818060423687 \cdot 10^{+93}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -8.295868111085673 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.7085030248267635 \cdot 10^{-189}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.8823818060423687 \cdot 10^{+93}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -8.295868111085673 \cdot 10^{-165}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq -1.7085030248267635 \cdot 10^{-189}:\\
\;\;\;\;-re\\

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

\end{array}
(FPCore (re im) :precision binary64 (sqrt (+ (* re re) (* im im))))
↓
(FPCore (re im)
 :precision binary64
 (if (<= re -1.8823818060423687e+93)
   (- re)
   (if (<= re -8.295868111085673e-165)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re -1.7085030248267635e-189) (- re) im))))
double code(double re, double im) {
	return sqrt((re * re) + (im * im));
}
↓
double code(double re, double im) {
	double tmp;
	if (re <= -1.8823818060423687e+93) {
		tmp = -re;
	} else if (re <= -8.295868111085673e-165) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= -1.7085030248267635e-189) {
		tmp = -re;
	} 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 re < -1.88238180604236865e93 or -8.29586811108567327e-165 < re < -1.70850302482676351e-189

    1. Initial program 48.3

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

    2. Taylor expanded around -inf 8.6

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

    if -1.88238180604236865e93 < re < -8.29586811108567327e-165

    1. Initial program 11.7

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

    if -1.70850302482676351e-189 < re

    1. Initial program 31.5

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

    2. Taylor expanded around 0 3.9

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8823818060423687 \cdot 10^{+93}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -8.295868111085673 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq -1.7085030248267635 \cdot 10^{-189}:\\ \;\;\;\;-re\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))