Average Error: 31.7 → 17.7
Time: 1.2s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -1.6585338134613207 \cdot 10^{+142}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq 1.0618359535513641 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -1.6585338134613207 \cdot 10^{+142}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq 1.0618359535513641 \cdot 10^{+80}:\\
\;\;\;\;\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 -1.6585338134613207e+142)
   (- re)
   (if (<= re 1.0618359535513641e+80) (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 <= -1.6585338134613207e+142) {
		tmp = -re;
	} else if (re <= 1.0618359535513641e+80) {
		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 3 regimes

  2. if re < -1.6585338134613207e142

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 7.9

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

    3. Simplified7.9

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

    if -1.6585338134613207e142 < re < 1.0618359535513641e80

    1. Initial program 21.6

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

    if 1.0618359535513641e80 < re

    1. Initial program 47.0

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

    2. Taylor expanded around inf 11.1

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.6585338134613207 \cdot 10^{+142}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq 1.0618359535513641 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))