Average Error: 31.2 → 17.2
Time: 893.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.5701395123872976 \cdot 10^{96}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.5292574126268538 \cdot 10^{125}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.5701395123872976 \cdot 10^{96}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 1.5292574126268538 \cdot 10^{125}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r33000 = re;
        double r33001 = r33000 * r33000;
        double r33002 = im;
        double r33003 = r33002 * r33002;
        double r33004 = r33001 + r33003;
        double r33005 = sqrt(r33004);
        return r33005;
}
↓
double f(double re, double im) {
        double r33006 = re;
        double r33007 = -1.5701395123872976e+96;
        bool r33008 = r33006 <= r33007;
        double r33009 = -1.0;
        double r33010 = r33009 * r33006;
        double r33011 = 1.5292574126268538e+125;
        bool r33012 = r33006 <= r33011;
        double r33013 = r33006 * r33006;
        double r33014 = im;
        double r33015 = r33014 * r33014;
        double r33016 = r33013 + r33015;
        double r33017 = sqrt(r33016);
        double r33018 = r33012 ? r33017 : r33006;
        double r33019 = r33008 ? r33010 : r33018;
        return r33019;
}

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 3 regimes

  2. if re < -1.5701395123872976e+96

    1. Initial program 50.6

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

    2. Taylor expanded around -inf 10.3

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

    if -1.5701395123872976e+96 < re < 1.5292574126268538e+125

    1. Initial program 20.7

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

    if 1.5292574126268538e+125 < re

    1. Initial program 57.7

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.5701395123872976 \cdot 10^{96}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.5292574126268538 \cdot 10^{125}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))