Average Error: 32.2 → 18.4
Time: 1.8s
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 r56317 = re;
        double r56318 = r56317 * r56317;
        double r56319 = im;
        double r56320 = r56319 * r56319;
        double r56321 = r56318 + r56320;
        double r56322 = sqrt(r56321);
        return r56322;
}
↓
double f(double re, double im) {
        double r56323 = re;
        double r56324 = -4.2696195727379345e+139;
        bool r56325 = r56323 <= r56324;
        double r56326 = -r56323;
        double r56327 = -3.5543765182763856e-161;
        bool r56328 = r56323 <= r56327;
        double r56329 = r56323 * r56323;
        double r56330 = im;
        double r56331 = r56330 * r56330;
        double r56332 = r56329 + r56331;
        double r56333 = sqrt(r56332);
        double r56334 = 2.243609177547311e-248;
        bool r56335 = r56323 <= r56334;
        double r56336 = 6.3015272029718245e+96;
        bool r56337 = r56323 <= r56336;
        double r56338 = r56337 ? r56333 : r56323;
        double r56339 = r56335 ? r56330 : r56338;
        double r56340 = r56328 ? r56333 : r56339;
        double r56341 = r56325 ? r56326 : r56340;
        return r56341;
}

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))))