Average Error: 31.6 → 17.7
Time: 10.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.967077572269142663773808601587033057531 \cdot 10^{83}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.967077572269142663773808601587033057531 \cdot 10^{83}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1655733 = re;
        double r1655734 = r1655733 * r1655733;
        double r1655735 = im;
        double r1655736 = r1655735 * r1655735;
        double r1655737 = r1655734 + r1655736;
        double r1655738 = sqrt(r1655737);
        return r1655738;
}
↓
double f(double re, double im) {
        double r1655739 = re;
        double r1655740 = -2.6008532152452964e+111;
        bool r1655741 = r1655739 <= r1655740;
        double r1655742 = -r1655739;
        double r1655743 = 2.9670775722691427e+83;
        bool r1655744 = r1655739 <= r1655743;
        double r1655745 = im;
        double r1655746 = r1655745 * r1655745;
        double r1655747 = r1655739 * r1655739;
        double r1655748 = r1655746 + r1655747;
        double r1655749 = sqrt(r1655748);
        double r1655750 = r1655744 ? r1655749 : r1655739;
        double r1655751 = r1655741 ? r1655742 : r1655750;
        return r1655751;
}

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 < -2.6008532152452964e+111

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 8.9

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

    3. Simplified8.9

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

    if -2.6008532152452964e+111 < re < 2.9670775722691427e+83

    1. Initial program 21.6

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

    if 2.9670775722691427e+83 < re

    1. Initial program 47.9

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

    2. Taylor expanded around inf 11.5

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.967077572269142663773808601587033057531 \cdot 10^{83}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))