Average Error: 31.6 → 18.2
Time: 966.0ms
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -2.7797371401730837 \cdot 10^{+124}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -9.380222908706226 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 2.314916717899127 \cdot 10^{-148}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 2.5213214626285914 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -2.7797371401730837 \cdot 10^{+124}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -9.380222908706226 \cdot 10^{-305}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 2.314916717899127 \cdot 10^{-148}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 2.5213214626285914 \cdot 10^{+106}:\\
\;\;\;\;\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 -2.7797371401730837e+124)
   (- re)
   (if (<= re -9.380222908706226e-305)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 2.314916717899127e-148)
       im
       (if (<= re 2.5213214626285914e+106)
         (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 <= -2.7797371401730837e+124) {
		tmp = -re;
	} else if (re <= -9.380222908706226e-305) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= 2.314916717899127e-148) {
		tmp = im;
	} else if (re <= 2.5213214626285914e+106) {
		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 < -2.77973714017308372e124

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 10.1

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

    3. Simplified10.1

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

    if -2.77973714017308372e124 < re < -9.3802229087062265e-305 or 2.314916717899127e-148 < re < 2.5213214626285914e106

    1. Initial program 19.1

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

    if -9.3802229087062265e-305 < re < 2.314916717899127e-148

    1. Initial program 28.1

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

    2. Taylor expanded around 0 33.8

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

    if 2.5213214626285914e106 < 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 simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.7797371401730837 \cdot 10^{+124}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -9.380222908706226 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 2.314916717899127 \cdot 10^{-148}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 2.5213214626285914 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))