Average Error: 31.5 → 17.9
Time: 12.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r40840 = re;
        double r40841 = r40840 * r40840;
        double r40842 = im;
        double r40843 = r40842 * r40842;
        double r40844 = r40841 + r40843;
        double r40845 = sqrt(r40844);
        return r40845;
}
↓
double f(double re, double im) {
        double r40846 = re;
        double r40847 = -5.292452664608309e+126;
        bool r40848 = r40846 <= r40847;
        double r40849 = -r40846;
        double r40850 = -3.6872979313797757e-267;
        bool r40851 = r40846 <= r40850;
        double r40852 = r40846 * r40846;
        double r40853 = im;
        double r40854 = r40853 * r40853;
        double r40855 = r40852 + r40854;
        double r40856 = sqrt(r40855);
        double r40857 = 1.914133431995725e-220;
        bool r40858 = r40846 <= r40857;
        double r40859 = 1.921239608950814e+104;
        bool r40860 = r40846 <= r40859;
        double r40861 = r40860 ? r40856 : r40846;
        double r40862 = r40858 ? r40853 : r40861;
        double r40863 = r40851 ? r40856 : r40862;
        double r40864 = r40848 ? r40849 : r40863;
        return r40864;
}

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 4 regimes

  2. if re < -5.292452664608309e+126

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -5.292452664608309e+126 < re < -3.6872979313797757e-267 or 1.914133431995725e-220 < re < 1.921239608950814e+104

    1. Initial program 19.7

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

    if -3.6872979313797757e-267 < re < 1.914133431995725e-220

    1. Initial program 30.3

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

    2. Taylor expanded around 0 32.9

      \[\leadsto \color{blue}{im}\]

    if 1.921239608950814e+104 < re

    1. Initial program 51.4

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

    2. Taylor expanded around inf 9.8

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.292452664608308748326184357539029329404 \cdot 10^{126}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.687297931379775690416299346624397658835 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.914133431995725133760392243420299124786 \cdot 10^{-220}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.921239608950813885870067735992694561728 \cdot 10^{104}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))