Average Error: 31.0 → 17.4
Time: 3.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.856770854610730383064334389991702440912 \cdot 10^{140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 4.856770854610730383064334389991702440912 \cdot 10^{140}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r2142824 = re;
        double r2142825 = r2142824 * r2142824;
        double r2142826 = im;
        double r2142827 = r2142826 * r2142826;
        double r2142828 = r2142825 + r2142827;
        double r2142829 = sqrt(r2142828);
        return r2142829;
}
↓
double f(double re, double im) {
        double r2142830 = re;
        double r2142831 = -2.222006465724332e+103;
        bool r2142832 = r2142830 <= r2142831;
        double r2142833 = -r2142830;
        double r2142834 = 4.85677085461073e+140;
        bool r2142835 = r2142830 <= r2142834;
        double r2142836 = im;
        double r2142837 = r2142836 * r2142836;
        double r2142838 = r2142830 * r2142830;
        double r2142839 = r2142837 + r2142838;
        double r2142840 = sqrt(r2142839);
        double r2142841 = r2142835 ? r2142840 : r2142830;
        double r2142842 = r2142832 ? r2142833 : r2142841;
        return r2142842;
}

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 < -2.222006465724332e+103

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 10.0

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

    3. Simplified10.0

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

    if -2.222006465724332e+103 < re < 4.85677085461073e+140

    1. Initial program 20.7

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

    if 4.85677085461073e+140 < re

    1. Initial program 59.6

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

    2. Taylor expanded around inf 8.9

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.222006465724332039862348815623896484961 \cdot 10^{103}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.856770854610730383064334389991702440912 \cdot 10^{140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))