Average Error: 29.2 → 16.6
Time: 3.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.303816533194875 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.8713230114732984 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6369844480501635 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.303816533194875 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -1.8713230114732984 \cdot 10^{-249}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.6369844480501635 \cdot 10^{+116}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r2129841 = re;
        double r2129842 = r2129841 * r2129841;
        double r2129843 = im;
        double r2129844 = r2129843 * r2129843;
        double r2129845 = r2129842 + r2129844;
        double r2129846 = sqrt(r2129845);
        return r2129846;
}
↓
double f(double re, double im) {
        double r2129847 = re;
        double r2129848 = -9.303816533194875e+153;
        bool r2129849 = r2129847 <= r2129848;
        double r2129850 = -r2129847;
        double r2129851 = -1.8713230114732984e-249;
        bool r2129852 = r2129847 <= r2129851;
        double r2129853 = im;
        double r2129854 = r2129853 * r2129853;
        double r2129855 = r2129847 * r2129847;
        double r2129856 = r2129854 + r2129855;
        double r2129857 = sqrt(r2129856);
        double r2129858 = 1.0730314248253915e-288;
        bool r2129859 = r2129847 <= r2129858;
        double r2129860 = 1.6369844480501635e+116;
        bool r2129861 = r2129847 <= r2129860;
        double r2129862 = r2129861 ? r2129857 : r2129847;
        double r2129863 = r2129859 ? r2129853 : r2129862;
        double r2129864 = r2129852 ? r2129857 : r2129863;
        double r2129865 = r2129849 ? r2129850 : r2129864;
        return r2129865;
}

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 < -9.303816533194875e+153

    1. Initial program 59.3

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

    2. Taylor expanded around -inf 8.3

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

    3. Simplified8.3

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

    if -9.303816533194875e+153 < re < -1.8713230114732984e-249 or 1.0730314248253915e-288 < re < 1.6369844480501635e+116

    1. Initial program 18.7

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

    if -1.8713230114732984e-249 < re < 1.0730314248253915e-288

    1. Initial program 29.4

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

    2. Taylor expanded around 0 29.9

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

    if 1.6369844480501635e+116 < re

    1. Initial program 49.2

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.303816533194875 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.8713230114732984 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.6369844480501635 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))