Average Error: 31.9 → 17.8
Time: 3.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r58303 = re;
        double r58304 = r58303 * r58303;
        double r58305 = im;
        double r58306 = r58305 * r58305;
        double r58307 = r58304 + r58306;
        double r58308 = sqrt(r58307);
        return r58308;
}
↓
double f(double re, double im) {
        double r58309 = re;
        double r58310 = -6.485144449769119e+83;
        bool r58311 = r58309 <= r58310;
        double r58312 = -1.0;
        double r58313 = r58312 * r58309;
        double r58314 = 9.194803090293717e-296;
        bool r58315 = r58309 <= r58314;
        double r58316 = r58309 * r58309;
        double r58317 = im;
        double r58318 = r58317 * r58317;
        double r58319 = r58316 + r58318;
        double r58320 = sqrt(r58319);
        double r58321 = 1.1600661433813666e-208;
        bool r58322 = r58309 <= r58321;
        double r58323 = 1.2142912145306137e+146;
        bool r58324 = r58309 <= r58323;
        double r58325 = r58324 ? r58320 : r58309;
        double r58326 = r58322 ? r58317 : r58325;
        double r58327 = r58315 ? r58320 : r58326;
        double r58328 = r58311 ? r58313 : r58327;
        return r58328;
}

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 < -6.485144449769119e+83

    1. Initial program 49.8

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

    2. Taylor expanded around -inf 11.4

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

    if -6.485144449769119e+83 < re < 9.194803090293717e-296 or 1.1600661433813666e-208 < re < 1.2142912145306137e+146

    1. Initial program 19.9

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

    if 9.194803090293717e-296 < re < 1.1600661433813666e-208

    1. Initial program 31.5

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

    2. Taylor expanded around 0 34.0

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

    if 1.2142912145306137e+146 < re

    1. Initial program 62.0

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

    2. Taylor expanded around inf 7.9

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.4851444497691187 \cdot 10^{83}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.19480309029371711 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.1600661433813666 \cdot 10^{-208}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.21429121453061369 \cdot 10^{146}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))