Average Error: 31.2 → 17.5
Time: 16.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r47257 = re;
        double r47258 = r47257 * r47257;
        double r47259 = im;
        double r47260 = r47259 * r47259;
        double r47261 = r47258 + r47260;
        double r47262 = sqrt(r47261);
        return r47262;
}
↓
double f(double re, double im) {
        double r47263 = re;
        double r47264 = -8.953163933293596e+119;
        bool r47265 = r47263 <= r47264;
        double r47266 = -r47263;
        double r47267 = 2.931608767883357e+138;
        bool r47268 = r47263 <= r47267;
        double r47269 = r47263 * r47263;
        double r47270 = im;
        double r47271 = r47270 * r47270;
        double r47272 = r47269 + r47271;
        double r47273 = sqrt(r47272);
        double r47274 = r47268 ? r47273 : r47263;
        double r47275 = r47265 ? r47266 : r47274;
        return r47275;
}

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 3 regimes

  2. if re < -8.953163933293596e+119

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 9.5

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

    3. Simplified9.5

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

    if -8.953163933293596e+119 < re < 2.931608767883357e+138

    1. Initial program 20.9

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

    if 2.931608767883357e+138 < re

    1. Initial program 59.0

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.93160876788335701324895973715263720284 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))