Average Error: 32.0 → 17.9
Time: 1.8s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -5.134590569129671 \cdot 10^{+121}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.614162643035948 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 3.623187642129605 \cdot 10^{-234}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.011260704811922 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -5.134590569129671 \cdot 10^{+121}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -1.614162643035948 \cdot 10^{-179}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 3.623187642129605 \cdot 10^{-234}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 7.011260704811922 \cdot 10^{+109}:\\
\;\;\;\;\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 -5.134590569129671e+121)
   (- re)
   (if (<= re -1.614162643035948e-179)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 3.623187642129605e-234)
       im
       (if (<= re 7.011260704811922e+109)
         (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 <= -5.134590569129671e+121) {
		tmp = -re;
	} else if (re <= -1.614162643035948e-179) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= 3.623187642129605e-234) {
		tmp = im;
	} else if (re <= 7.011260704811922e+109) {
		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 < -5.1345905691296712e121

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 9.2

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

    3. Simplified9.2

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

    if -5.1345905691296712e121 < re < -1.6141626430359481e-179 or 3.623187642129605e-234 < re < 7.0112607048119218e109

    1. Initial program 17.9

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

    if -1.6141626430359481e-179 < re < 3.623187642129605e-234

    1. Initial program 32.0

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

    2. Taylor expanded around 0 33.9

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

    if 7.0112607048119218e109 < re

    1. Initial program 53.2

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

    2. Taylor expanded around inf 9.9

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.134590569129671 \cdot 10^{+121}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -1.614162643035948 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 3.623187642129605 \cdot 10^{-234}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 7.011260704811922 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))