Average Error: 31.8 → 18.0
Time: 1.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r66841 = re;
        double r66842 = r66841 * r66841;
        double r66843 = im;
        double r66844 = r66843 * r66843;
        double r66845 = r66842 + r66844;
        double r66846 = sqrt(r66845);
        return r66846;
}
↓
double f(double re, double im) {
        double r66847 = re;
        double r66848 = -3.0018675033187654e+143;
        bool r66849 = r66847 <= r66848;
        double r66850 = -1.0;
        double r66851 = r66850 * r66847;
        double r66852 = -2.0278572522938575e-184;
        bool r66853 = r66847 <= r66852;
        double r66854 = r66847 * r66847;
        double r66855 = im;
        double r66856 = r66855 * r66855;
        double r66857 = r66854 + r66856;
        double r66858 = sqrt(r66857);
        double r66859 = -9.306218889451781e-290;
        bool r66860 = r66847 <= r66859;
        double r66861 = 1.749481499166478e+126;
        bool r66862 = r66847 <= r66861;
        double r66863 = r66862 ? r66858 : r66847;
        double r66864 = r66860 ? r66855 : r66863;
        double r66865 = r66853 ? r66858 : r66864;
        double r66866 = r66849 ? r66851 : r66865;
        return r66866;
}

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 < -3.0018675033187654e+143

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 8.1

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

    if -3.0018675033187654e+143 < re < -2.0278572522938575e-184 or -9.306218889451781e-290 < re < 1.749481499166478e+126

    1. Initial program 19.5

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

    if -2.0278572522938575e-184 < re < -9.306218889451781e-290

    1. Initial program 30.6

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

    2. Taylor expanded around 0 36.6

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

    if 1.749481499166478e+126 < re

    1. Initial program 56.8

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))