Average Error: 31.3 → 17.5
Time: 917.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.066866671709845106834464667975644144779 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.80613782352259500769138488086471841751 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.066866671709845106834464667975644144779 \cdot 10^{108}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.80613782352259500769138488086471841751 \cdot 10^{109}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r105386 = re;
        double r105387 = r105386 * r105386;
        double r105388 = im;
        double r105389 = r105388 * r105388;
        double r105390 = r105387 + r105389;
        double r105391 = sqrt(r105390);
        return r105391;
}
↓
double f(double re, double im) {
        double r105392 = re;
        double r105393 = -2.066866671709845e+108;
        bool r105394 = r105392 <= r105393;
        double r105395 = -1.0;
        double r105396 = r105395 * r105392;
        double r105397 = 4.806137823522595e+109;
        bool r105398 = r105392 <= r105397;
        double r105399 = r105392 * r105392;
        double r105400 = im;
        double r105401 = r105400 * r105400;
        double r105402 = r105399 + r105401;
        double r105403 = sqrt(r105402);
        double r105404 = r105398 ? r105403 : r105392;
        double r105405 = r105394 ? r105396 : r105404;
        return r105405;
}

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 < -2.066866671709845e+108

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 10.0

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

    if -2.066866671709845e+108 < re < 4.806137823522595e+109

    1. Initial program 21.0

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

    if 4.806137823522595e+109 < re

    1. Initial program 53.6

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.066866671709845106834464667975644144779 \cdot 10^{108}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.80613782352259500769138488086471841751 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))