Average Error: 31.4 → 18.5
Time: 24.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.5847343597756595 \cdot 10^{+152}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.35268515513709 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.6972939667011593 \cdot 10^{-158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.5847343597756595 \cdot 10^{+152}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -2.35268515513709 \cdot 10^{-257}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.6972939667011593 \cdot 10^{-158}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r2029360 = re;
        double r2029361 = r2029360 * r2029360;
        double r2029362 = im;
        double r2029363 = r2029362 * r2029362;
        double r2029364 = r2029361 + r2029363;
        double r2029365 = sqrt(r2029364);
        return r2029365;
}
↓
double f(double re, double im) {
        double r2029366 = re;
        double r2029367 = -2.5847343597756595e+152;
        bool r2029368 = r2029366 <= r2029367;
        double r2029369 = -r2029366;
        double r2029370 = -2.35268515513709e-257;
        bool r2029371 = r2029366 <= r2029370;
        double r2029372 = im;
        double r2029373 = r2029372 * r2029372;
        double r2029374 = r2029366 * r2029366;
        double r2029375 = r2029373 + r2029374;
        double r2029376 = sqrt(r2029375);
        double r2029377 = 1.6972939667011593e-158;
        bool r2029378 = r2029366 <= r2029377;
        double r2029379 = 3.161818111636812e+106;
        bool r2029380 = r2029366 <= r2029379;
        double r2029381 = r2029380 ? r2029376 : r2029366;
        double r2029382 = r2029378 ? r2029372 : r2029381;
        double r2029383 = r2029371 ? r2029376 : r2029382;
        double r2029384 = r2029368 ? r2029369 : r2029383;
        return r2029384;
}

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 < -2.5847343597756595e+152

    1. Initial program 63.4

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -2.5847343597756595e+152 < re < -2.35268515513709e-257 or 1.6972939667011593e-158 < re < 3.161818111636812e+106

    1. Initial program 18.0

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

    if -2.35268515513709e-257 < re < 1.6972939667011593e-158

    1. Initial program 30.0

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

    2. Taylor expanded around 0 35.1

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

    if 3.161818111636812e+106 < re

    1. Initial program 52.7

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

    2. Taylor expanded around inf 10.6

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

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.5847343597756595 \cdot 10^{+152}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.35268515513709 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.6972939667011593 \cdot 10^{-158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))