Average Error: 31.9 → 17.5
Time: 3.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.534314230587066174958520593782804386096 \cdot 10^{96}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.534314230587066174958520593782804386096 \cdot 10^{96}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r3052095 = re;
        double r3052096 = r3052095 * r3052095;
        double r3052097 = im;
        double r3052098 = r3052097 * r3052097;
        double r3052099 = r3052096 + r3052098;
        double r3052100 = sqrt(r3052099);
        return r3052100;
}
↓
double f(double re, double im) {
        double r3052101 = re;
        double r3052102 = -4.8445064813097935e+101;
        bool r3052103 = r3052101 <= r3052102;
        double r3052104 = -r3052101;
        double r3052105 = 2.534314230587066e+96;
        bool r3052106 = r3052101 <= r3052105;
        double r3052107 = im;
        double r3052108 = r3052107 * r3052107;
        double r3052109 = r3052101 * r3052101;
        double r3052110 = r3052108 + r3052109;
        double r3052111 = sqrt(r3052110);
        double r3052112 = r3052106 ? r3052111 : r3052101;
        double r3052113 = r3052103 ? r3052104 : r3052112;
        return r3052113;
}

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 < -4.8445064813097935e+101

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 9.1

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

    3. Simplified9.1

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

    if -4.8445064813097935e+101 < re < 2.534314230587066e+96

    1. Initial program 21.7

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

    if 2.534314230587066e+96 < re

    1. Initial program 51.2

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

    2. Taylor expanded around inf 10.1

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.844506481309793477015811086474695118347 \cdot 10^{101}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.534314230587066174958520593782804386096 \cdot 10^{96}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))