Average Error: 31.9 → 17.7
Time: 3.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.4397722734901768 \cdot 10^{138}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.95063775952267208 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.4397722734901768 \cdot 10^{138}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.95063775952267208 \cdot 10^{126}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r37073 = re;
        double r37074 = r37073 * r37073;
        double r37075 = im;
        double r37076 = r37075 * r37075;
        double r37077 = r37074 + r37076;
        double r37078 = sqrt(r37077);
        return r37078;
}
↓
double f(double re, double im) {
        double r37079 = re;
        double r37080 = -6.439772273490177e+138;
        bool r37081 = r37079 <= r37080;
        double r37082 = -r37079;
        double r37083 = 1.950637759522672e+126;
        bool r37084 = r37079 <= r37083;
        double r37085 = r37079 * r37079;
        double r37086 = im;
        double r37087 = r37086 * r37086;
        double r37088 = r37085 + r37087;
        double r37089 = sqrt(r37088);
        double r37090 = r37084 ? r37089 : r37079;
        double r37091 = r37081 ? r37082 : r37090;
        return r37091;
}

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 < -6.439772273490177e+138

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 8.2

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

    3. Simplified8.2

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

    if -6.439772273490177e+138 < re < 1.950637759522672e+126

    1. Initial program 21.4

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

    if 1.950637759522672e+126 < re

    1. Initial program 56.2

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.4397722734901768 \cdot 10^{138}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.95063775952267208 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))