Average Error: 29.5 → 17.2
Time: 8.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.6423327785059521 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.9219302233631307 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.660720867938078 \cdot 10^{-293}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.360456255819538 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.6423327785059521 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -1.9219302233631307 \cdot 10^{-249}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 4.660720867938078 \cdot 10^{-293}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 5.360456255819538 \cdot 10^{+96}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r963625 = re;
        double r963626 = r963625 * r963625;
        double r963627 = im;
        double r963628 = r963627 * r963627;
        double r963629 = r963626 + r963628;
        double r963630 = sqrt(r963629);
        return r963630;
}
↓
double f(double re, double im) {
        double r963631 = re;
        double r963632 = -1.6423327785059521e+153;
        bool r963633 = r963631 <= r963632;
        double r963634 = -r963631;
        double r963635 = -1.9219302233631307e-249;
        bool r963636 = r963631 <= r963635;
        double r963637 = im;
        double r963638 = r963637 * r963637;
        double r963639 = r963631 * r963631;
        double r963640 = r963638 + r963639;
        double r963641 = sqrt(r963640);
        double r963642 = 4.660720867938078e-293;
        bool r963643 = r963631 <= r963642;
        double r963644 = 5.360456255819538e+96;
        bool r963645 = r963631 <= r963644;
        double r963646 = r963645 ? r963641 : r963631;
        double r963647 = r963643 ? r963637 : r963646;
        double r963648 = r963636 ? r963641 : r963647;
        double r963649 = r963633 ? r963634 : r963648;
        return r963649;
}

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 < -1.6423327785059521e+153

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -1.6423327785059521e+153 < re < -1.9219302233631307e-249 or 4.660720867938078e-293 < re < 5.360456255819538e+96

    1. Initial program 19.1

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

    if -1.9219302233631307e-249 < re < 4.660720867938078e-293

    1. Initial program 30.7

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

    2. Taylor expanded around 0 34.0

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

    if 5.360456255819538e+96 < re

    1. Initial program 47.5

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

    2. Taylor expanded around inf 11.9

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6423327785059521 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.9219302233631307 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 4.660720867938078 \cdot 10^{-293}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.360456255819538 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))