Average Error: 31.7 → 18.2
Time: 3.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r65182 = re;
        double r65183 = r65182 * r65182;
        double r65184 = im;
        double r65185 = r65184 * r65184;
        double r65186 = r65183 + r65185;
        double r65187 = sqrt(r65186);
        return r65187;
}
↓
double f(double re, double im) {
        double r65188 = re;
        double r65189 = -2.2710920961423764e+59;
        bool r65190 = r65188 <= r65189;
        double r65191 = -1.0;
        double r65192 = r65191 * r65188;
        double r65193 = 9.634966723363643e+131;
        bool r65194 = r65188 <= r65193;
        double r65195 = r65188 * r65188;
        double r65196 = im;
        double r65197 = r65196 * r65196;
        double r65198 = r65195 + r65197;
        double r65199 = sqrt(r65198);
        double r65200 = r65194 ? r65199 : r65188;
        double r65201 = r65190 ? r65192 : r65200;
        return r65201;
}

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 < -2.2710920961423764e+59

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 12.9

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

    if -2.2710920961423764e+59 < re < 9.634966723363643e+131

    1. Initial program 22.0

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

    if 9.634966723363643e+131 < re

    1. Initial program 57.8

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))