Average Error: 32.2 → 18.4
Time: 2.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.2436091775473112 \cdot 10^{-248}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 6.3015272029718245 \cdot 10^{96}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r86370 = re;
        double r86371 = r86370 * r86370;
        double r86372 = im;
        double r86373 = r86372 * r86372;
        double r86374 = r86371 + r86373;
        double r86375 = sqrt(r86374);
        return r86375;
}
↓
double f(double re, double im) {
        double r86376 = re;
        double r86377 = -4.2696195727379345e+139;
        bool r86378 = r86376 <= r86377;
        double r86379 = -r86376;
        double r86380 = -3.5543765182763856e-161;
        bool r86381 = r86376 <= r86380;
        double r86382 = r86376 * r86376;
        double r86383 = im;
        double r86384 = r86383 * r86383;
        double r86385 = r86382 + r86384;
        double r86386 = sqrt(r86385);
        double r86387 = 2.243609177547311e-248;
        bool r86388 = r86376 <= r86387;
        double r86389 = 6.3015272029718245e+96;
        bool r86390 = r86376 <= r86389;
        double r86391 = r86390 ? r86386 : r86376;
        double r86392 = r86388 ? r86383 : r86391;
        double r86393 = r86381 ? r86386 : r86392;
        double r86394 = r86378 ? r86379 : r86393;
        return r86394;
}

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 < -4.2696195727379345e+139

    1. Initial program 59.5

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -4.2696195727379345e+139 < re < -3.5543765182763856e-161 or 2.243609177547311e-248 < re < 6.3015272029718245e+96

    1. Initial program 18.8

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

    if -3.5543765182763856e-161 < re < 2.243609177547311e-248

    1. Initial program 32.3

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

    2. Taylor expanded around 0 33.8

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

    if 6.3015272029718245e+96 < re

    1. Initial program 51.2

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))