Average Error: 30.9 → 17.2
Time: 2.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.637979099701820103283669167726628154584 \cdot 10^{53}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 5.637979099701820103283669167726628154584 \cdot 10^{53}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r43666 = re;
        double r43667 = r43666 * r43666;
        double r43668 = im;
        double r43669 = r43668 * r43668;
        double r43670 = r43667 + r43669;
        double r43671 = sqrt(r43670);
        return r43671;
}
↓
double f(double re, double im) {
        double r43672 = re;
        double r43673 = -9.850726757232305e+116;
        bool r43674 = r43672 <= r43673;
        double r43675 = -r43672;
        double r43676 = 5.63797909970182e+53;
        bool r43677 = r43672 <= r43676;
        double r43678 = im;
        double r43679 = r43678 * r43678;
        double r43680 = r43672 * r43672;
        double r43681 = r43679 + r43680;
        double r43682 = sqrt(r43681);
        double r43683 = r43677 ? r43682 : r43672;
        double r43684 = r43674 ? r43675 : r43683;
        return r43684;
}

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 < -9.850726757232305e+116

    1. Initial program 55.5

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

    2. Taylor expanded around -inf 9.5

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

    3. Simplified9.5

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

    if -9.850726757232305e+116 < re < 5.63797909970182e+53

    1. Initial program 20.7

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

    if 5.63797909970182e+53 < re

    1. Initial program 43.9

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

    2. Taylor expanded around inf 12.5

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.850726757232304656097215039461175225007 \cdot 10^{116}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.637979099701820103283669167726628154584 \cdot 10^{53}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))