Average Error: 32.3 → 18.0
Time: 2.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r55881 = re;
        double r55882 = r55881 * r55881;
        double r55883 = im;
        double r55884 = r55883 * r55883;
        double r55885 = r55882 + r55884;
        double r55886 = sqrt(r55885);
        return r55886;
}
↓
double f(double re, double im) {
        double r55887 = re;
        double r55888 = -8.15024475259888e+153;
        bool r55889 = r55887 <= r55888;
        double r55890 = -r55887;
        double r55891 = -9.528172448826491e-265;
        bool r55892 = r55887 <= r55891;
        double r55893 = r55887 * r55887;
        double r55894 = im;
        double r55895 = r55894 * r55894;
        double r55896 = r55893 + r55895;
        double r55897 = sqrt(r55896);
        double r55898 = 1.047455535241276e-281;
        bool r55899 = r55887 <= r55898;
        double r55900 = 2.70835173311075e+105;
        bool r55901 = r55887 <= r55900;
        double r55902 = r55901 ? r55897 : r55887;
        double r55903 = r55899 ? r55894 : r55902;
        double r55904 = r55892 ? r55897 : r55903;
        double r55905 = r55889 ? r55890 : r55904;
        return r55905;
}

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 < -8.15024475259888e+153

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 7.8

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

    3. Simplified7.8

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

    if -8.15024475259888e+153 < re < -9.528172448826491e-265 or 1.047455535241276e-281 < re < 2.70835173311075e+105

    1. Initial program 21.0

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

    if -9.528172448826491e-265 < re < 1.047455535241276e-281

    1. Initial program 30.8

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

    2. Taylor expanded around 0 32.9

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

    if 2.70835173311075e+105 < re

    1. Initial program 52.5

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.15024475259887937 \cdot 10^{153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -9.52817244882649108 \cdot 10^{-265}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.04745553524127593 \cdot 10^{-281}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.70835173311075 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))