Average Error: 31.5 → 17.5
Time: 2.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.019634613598924703951801238213161238145 \cdot 10^{134}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.407306448838225519879757223457047572583 \cdot 10^{108}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.019634613598924703951801238213161238145 \cdot 10^{134}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.407306448838225519879757223457047572583 \cdot 10^{108}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r45887 = re;
        double r45888 = r45887 * r45887;
        double r45889 = im;
        double r45890 = r45889 * r45889;
        double r45891 = r45888 + r45890;
        double r45892 = sqrt(r45891);
        return r45892;
}
↓
double f(double re, double im) {
        double r45893 = re;
        double r45894 = -4.019634613598925e+134;
        bool r45895 = r45893 <= r45894;
        double r45896 = -r45893;
        double r45897 = 1.4073064488382255e+108;
        bool r45898 = r45893 <= r45897;
        double r45899 = r45893 * r45893;
        double r45900 = im;
        double r45901 = r45900 * r45900;
        double r45902 = r45899 + r45901;
        double r45903 = sqrt(r45902);
        double r45904 = r45898 ? r45903 : r45893;
        double r45905 = r45895 ? r45896 : r45904;
        return r45905;
}

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 < -4.019634613598925e+134

    1. Initial program 58.4

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    if -4.019634613598925e+134 < re < 1.4073064488382255e+108

    1. Initial program 21.2

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

    if 1.4073064488382255e+108 < re

    1. Initial program 52.1

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.019634613598924703951801238213161238145 \cdot 10^{134}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.407306448838225519879757223457047572583 \cdot 10^{108}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))