Average Error: 31.1 → 17.7
Time: 1.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.133313689967870834445494013025035211846 \cdot 10^{114}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.285382656044220860536640397267855783932 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.133313689967870834445494013025035211846 \cdot 10^{114}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.285382656044220860536640397267855783932 \cdot 10^{67}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r52296 = re;
        double r52297 = r52296 * r52296;
        double r52298 = im;
        double r52299 = r52298 * r52298;
        double r52300 = r52297 + r52299;
        double r52301 = sqrt(r52300);
        return r52301;
}
↓
double f(double re, double im) {
        double r52302 = re;
        double r52303 = -3.133313689967871e+114;
        bool r52304 = r52302 <= r52303;
        double r52305 = -1.0;
        double r52306 = r52305 * r52302;
        double r52307 = -8.357854099413859e-296;
        bool r52308 = r52302 <= r52307;
        double r52309 = r52302 * r52302;
        double r52310 = im;
        double r52311 = r52310 * r52310;
        double r52312 = r52309 + r52311;
        double r52313 = sqrt(r52312);
        double r52314 = 7.758435713562232e-281;
        bool r52315 = r52302 <= r52314;
        double r52316 = 1.2853826560442209e+67;
        bool r52317 = r52302 <= r52316;
        double r52318 = r52317 ? r52313 : r52302;
        double r52319 = r52315 ? r52310 : r52318;
        double r52320 = r52308 ? r52313 : r52319;
        double r52321 = r52304 ? r52306 : r52320;
        return r52321;
}

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 < -3.133313689967871e+114

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 9.7

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

    if -3.133313689967871e+114 < re < -8.357854099413859e-296 or 7.758435713562232e-281 < re < 1.2853826560442209e+67

    1. Initial program 21.0

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

    if -8.357854099413859e-296 < re < 7.758435713562232e-281

    1. Initial program 28.8

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

    2. Taylor expanded around 0 33.9

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

    if 1.2853826560442209e+67 < re

    1. Initial program 45.0

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

    2. Taylor expanded around inf 11.3

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.133313689967870834445494013025035211846 \cdot 10^{114}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.357854099413858949922341791993400005698 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 7.758435713562231820695193588189493548713 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.285382656044220860536640397267855783932 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))