Average Error: 29.6 → 18.2
Time: 3.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r3593353 = re;
        double r3593354 = r3593353 * r3593353;
        double r3593355 = im;
        double r3593356 = r3593355 * r3593355;
        double r3593357 = r3593354 + r3593356;
        double r3593358 = sqrt(r3593357);
        return r3593358;
}
↓
double f(double re, double im) {
        double r3593359 = re;
        double r3593360 = -1.2918446586536957e+154;
        bool r3593361 = r3593359 <= r3593360;
        double r3593362 = -r3593359;
        double r3593363 = -2.6219396713989246e+28;
        bool r3593364 = r3593359 <= r3593363;
        double r3593365 = im;
        double r3593366 = r3593365 * r3593365;
        double r3593367 = r3593359 * r3593359;
        double r3593368 = r3593366 + r3593367;
        double r3593369 = sqrt(r3593368);
        double r3593370 = -3.209519593925633e-28;
        bool r3593371 = r3593359 <= r3593370;
        double r3593372 = -8.056228658328031e-199;
        bool r3593373 = r3593359 <= r3593372;
        double r3593374 = -3.759150523562943e-268;
        bool r3593375 = r3593359 <= r3593374;
        double r3593376 = 1.4301923552016937e+155;
        bool r3593377 = r3593359 <= r3593376;
        double r3593378 = r3593377 ? r3593369 : r3593359;
        double r3593379 = r3593375 ? r3593365 : r3593378;
        double r3593380 = r3593373 ? r3593369 : r3593379;
        double r3593381 = r3593371 ? r3593365 : r3593380;
        double r3593382 = r3593364 ? r3593369 : r3593381;
        double r3593383 = r3593361 ? r3593362 : r3593382;
        return r3593383;
}

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 < -1.2918446586536957e+154

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 7.0

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

    3. Simplified7.0

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

    if -1.2918446586536957e+154 < re < -2.6219396713989246e+28 or -3.209519593925633e-28 < re < -8.056228658328031e-199 or -3.759150523562943e-268 < re < 1.4301923552016937e+155

    1. Initial program 19.1

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

    if -2.6219396713989246e+28 < re < -3.209519593925633e-28 or -8.056228658328031e-199 < re < -3.759150523562943e-268

    1. Initial program 23.9

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

    2. Taylor expanded around 0 40.8

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

    if 1.4301923552016937e+155 < re

    1. Initial program 59.5

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))