Average Error: 31.4 → 19.0
Time: 4.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;re\\

\end{array}
double f(double re, double im) {
        double r43367 = re;
        double r43368 = r43367 * r43367;
        double r43369 = im;
        double r43370 = r43369 * r43369;
        double r43371 = r43368 + r43370;
        double r43372 = sqrt(r43371);
        return r43372;
}
↓
double f(double re, double im) {
        double r43373 = re;
        double r43374 = -5.330091552844717e+114;
        bool r43375 = r43373 <= r43374;
        double r43376 = -r43373;
        double r43377 = -4.2156616274993736e-144;
        bool r43378 = r43373 <= r43377;
        double r43379 = r43373 * r43373;
        double r43380 = im;
        double r43381 = r43380 * r43380;
        double r43382 = r43379 + r43381;
        double r43383 = sqrt(r43382);
        double r43384 = 1.0597832414692678e-253;
        bool r43385 = r43373 <= r43384;
        double r43386 = 3.0122240909363507e+56;
        bool r43387 = r43373 <= r43386;
        double r43388 = r43387 ? r43383 : r43373;
        double r43389 = r43385 ? r43380 : r43388;
        double r43390 = r43378 ? r43383 : r43389;
        double r43391 = r43375 ? r43376 : r43390;
        return r43391;
}

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 < -5.330091552844717e+114

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -5.330091552844717e+114 < re < -4.2156616274993736e-144 or 1.0597832414692678e-253 < re < 3.0122240909363507e+56

    1. Initial program 18.7

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

    if -4.2156616274993736e-144 < re < 1.0597832414692678e-253

    1. Initial program 30.2

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

    2. Taylor expanded around 0 35.6

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

    if 3.0122240909363507e+56 < re

    1. Initial program 44.3

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))