Average Error: 31.0 → 17.5
Time: 988.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r60973 = re;
        double r60974 = r60973 * r60973;
        double r60975 = im;
        double r60976 = r60975 * r60975;
        double r60977 = r60974 + r60976;
        double r60978 = sqrt(r60977);
        return r60978;
}
↓
double f(double re, double im) {
        double r60979 = re;
        double r60980 = -1.5057522058365376e+136;
        bool r60981 = r60979 <= r60980;
        double r60982 = -1.0;
        double r60983 = r60982 * r60979;
        double r60984 = -3.2005633984364917e-257;
        bool r60985 = r60979 <= r60984;
        double r60986 = r60979 * r60979;
        double r60987 = im;
        double r60988 = r60987 * r60987;
        double r60989 = r60986 + r60988;
        double r60990 = sqrt(r60989);
        double r60991 = 3.8197786805557845e-227;
        bool r60992 = r60979 <= r60991;
        double r60993 = 8.439330033545885e+67;
        bool r60994 = r60979 <= r60993;
        double r60995 = r60994 ? r60990 : r60979;
        double r60996 = r60992 ? r60987 : r60995;
        double r60997 = r60985 ? r60990 : r60996;
        double r60998 = r60981 ? r60983 : r60997;
        return r60998;
}

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 < -1.5057522058365376e+136

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 9.2

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

    if -1.5057522058365376e+136 < re < -3.2005633984364917e-257 or 3.8197786805557845e-227 < re < 8.439330033545885e+67

    1. Initial program 18.7

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

    if -3.2005633984364917e-257 < re < 3.8197786805557845e-227

    1. Initial program 30.2

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

    2. Taylor expanded around 0 32.1

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

    if 8.439330033545885e+67 < re

    1. Initial program 46.7

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.505752205836537605611230467447200313868 \cdot 10^{136}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.200563398436491693418328268892598073539 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 3.819778680555784511216531232393990012128 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 8.439330033545885045213726212950052594665 \cdot 10^{67}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))