Average Error: 32.2 → 18.4
Time: 975.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\
\;\;\;\;-1 \cdot 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 r56601 = re;
        double r56602 = r56601 * r56601;
        double r56603 = im;
        double r56604 = r56603 * r56603;
        double r56605 = r56602 + r56604;
        double r56606 = sqrt(r56605);
        return r56606;
}
↓
double f(double re, double im) {
        double r56607 = re;
        double r56608 = -4.2696195727379345e+139;
        bool r56609 = r56607 <= r56608;
        double r56610 = -1.0;
        double r56611 = r56610 * r56607;
        double r56612 = -3.5543765182763856e-161;
        bool r56613 = r56607 <= r56612;
        double r56614 = r56607 * r56607;
        double r56615 = im;
        double r56616 = r56615 * r56615;
        double r56617 = r56614 + r56616;
        double r56618 = sqrt(r56617);
        double r56619 = 2.243609177547311e-248;
        bool r56620 = r56607 <= r56619;
        double r56621 = 6.3015272029718245e+96;
        bool r56622 = r56607 <= r56621;
        double r56623 = r56622 ? r56618 : r56607;
        double r56624 = r56620 ? r56615 : r56623;
        double r56625 = r56613 ? r56618 : r56624;
        double r56626 = r56609 ? r56611 : r56625;
        return r56626;
}

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

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.5543765182763856 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.2436091775473112 \cdot 10^{-248}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.3015272029718245 \cdot 10^{96}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

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