Average Error: 31.7 → 17.4
Time: 3.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.314940489788706082274641758508570148355 \cdot 10^{92}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 4.314940489788706082274641758508570148355 \cdot 10^{92}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1521345 = re;
        double r1521346 = r1521345 * r1521345;
        double r1521347 = im;
        double r1521348 = r1521347 * r1521347;
        double r1521349 = r1521346 + r1521348;
        double r1521350 = sqrt(r1521349);
        return r1521350;
}
↓
double f(double re, double im) {
        double r1521351 = re;
        double r1521352 = -1.6887213599031206e+100;
        bool r1521353 = r1521351 <= r1521352;
        double r1521354 = -r1521351;
        double r1521355 = 4.314940489788706e+92;
        bool r1521356 = r1521351 <= r1521355;
        double r1521357 = im;
        double r1521358 = r1521357 * r1521357;
        double r1521359 = r1521351 * r1521351;
        double r1521360 = r1521358 + r1521359;
        double r1521361 = sqrt(r1521360);
        double r1521362 = r1521356 ? r1521361 : r1521351;
        double r1521363 = r1521353 ? r1521354 : r1521362;
        return r1521363;
}

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 < -1.6887213599031206e+100

    1. Initial program 51.8

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

    2. Taylor expanded around -inf 10.5

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

    3. Simplified10.5

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

    if -1.6887213599031206e+100 < re < 4.314940489788706e+92

    1. Initial program 21.3

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

    if 4.314940489788706e+92 < re

    1. Initial program 50.0

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

    2. Taylor expanded around inf 10.2

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.314940489788706082274641758508570148355 \cdot 10^{92}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))