Average Error: 31.6 → 18.8
Time: 2.6s
Precision: binary64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \leq -6.518081105388684 \cdot 10^{+114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -4.142924173025851 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 6.004365564520266 \cdot 10^{-207}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 42148.37987214342:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \leq -6.518081105388684 \cdot 10^{+114}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \leq -4.142924173025851 \cdot 10^{-210}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \leq 6.004365564520266 \cdot 10^{-207}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \leq 42148.37987214342:\\
\;\;\;\;\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 -6.518081105388684e+114)
   (- re)
   (if (<= re -4.142924173025851e-210)
     (sqrt (+ (* re re) (* im im)))
     (if (<= re 6.004365564520266e-207)
       im
       (if (<= re 42148.37987214342) (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 <= -6.518081105388684e+114) {
		tmp = -re;
	} else if (re <= -4.142924173025851e-210) {
		tmp = sqrt((re * re) + (im * im));
	} else if (re <= 6.004365564520266e-207) {
		tmp = im;
	} else if (re <= 42148.37987214342) {
		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 < -6.51805e114

    1. Initial program 53.7

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

    2. Taylor expanded around -inf 10.5

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

    3. Simplified10.5

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

    if -6.51805e114 < re < -4.14295e-210 or 6.00441e-207 < re < 42148

    1. Initial program 18.8

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

    if -4.14295e-210 < re < 6.00441e-207

    1. Initial program 30.0

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

    2. Taylor expanded around 0 32.9

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

    if 42148 < re

    1. Initial program 40.5

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

    2. Taylor expanded around inf 14.7

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.518081105388684 \cdot 10^{+114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \leq -4.142924173025851 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \leq 6.004365564520266 \cdot 10^{-207}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \leq 42148.37987214342:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))