Average Error: 29.4 → 17.2
Time: 3.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.575837404870754 \cdot 10^{+111}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.965422253765463 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.575837404870754 \cdot 10^{+111}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 5.965422253765463 \cdot 10^{+140}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r780744 = re;
        double r780745 = r780744 * r780744;
        double r780746 = im;
        double r780747 = r780746 * r780746;
        double r780748 = r780745 + r780747;
        double r780749 = sqrt(r780748);
        return r780749;
}
↓
double f(double re, double im) {
        double r780750 = re;
        double r780751 = -4.575837404870754e+111;
        bool r780752 = r780750 <= r780751;
        double r780753 = -r780750;
        double r780754 = -6.3441075981154e-203;
        bool r780755 = r780750 <= r780754;
        double r780756 = im;
        double r780757 = r780756 * r780756;
        double r780758 = r780750 * r780750;
        double r780759 = r780757 + r780758;
        double r780760 = sqrt(r780759);
        double r780761 = 3.4426323618367234e-252;
        bool r780762 = r780750 <= r780761;
        double r780763 = 5.965422253765463e+140;
        bool r780764 = r780750 <= r780763;
        double r780765 = r780764 ? r780760 : r780750;
        double r780766 = r780762 ? r780756 : r780765;
        double r780767 = r780755 ? r780760 : r780766;
        double r780768 = r780752 ? r780753 : r780767;
        return r780768;
}

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 < -4.575837404870754e+111

    1. Initial program 49.7

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -4.575837404870754e+111 < re < -6.3441075981154e-203 or 3.4426323618367234e-252 < re < 5.965422253765463e+140

    1. Initial program 18.1

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

    if -6.3441075981154e-203 < re < 3.4426323618367234e-252

    1. Initial program 27.2

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

    2. Taylor expanded around 0 32.1

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

    if 5.965422253765463e+140 < re

    1. Initial program 55.9

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.575837404870754 \cdot 10^{+111}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 3.4426323618367234 \cdot 10^{-252}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.965422253765463 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))