Average Error: 30.9 → 17.2
Time: 2.4s
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 r43853 = re;
        double r43854 = r43853 * r43853;
        double r43855 = im;
        double r43856 = r43855 * r43855;
        double r43857 = r43854 + r43856;
        double r43858 = sqrt(r43857);
        return r43858;
}
↓
double f(double re, double im) {
        double r43859 = re;
        double r43860 = -9.850726757232305e+116;
        bool r43861 = r43859 <= r43860;
        double r43862 = -r43859;
        double r43863 = 5.63797909970182e+53;
        bool r43864 = r43859 <= r43863;
        double r43865 = im;
        double r43866 = r43865 * r43865;
        double r43867 = r43859 * r43859;
        double r43868 = r43866 + r43867;
        double r43869 = sqrt(r43868);
        double r43870 = r43864 ? r43869 : r43859;
        double r43871 = r43861 ? r43862 : r43870;
        return r43871;
}

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))))