Average Error: 31.7 → 17.5
Time: 886.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.3722084465878742 \cdot 10^{86}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.4667592323261061 \cdot 10^{131}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 1.3722084465878742 \cdot 10^{86}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r47685 = re;
        double r47686 = r47685 * r47685;
        double r47687 = im;
        double r47688 = r47687 * r47687;
        double r47689 = r47686 + r47688;
        double r47690 = sqrt(r47689);
        return r47690;
}
↓
double f(double re, double im) {
        double r47691 = re;
        double r47692 = -1.4667592323261061e+131;
        bool r47693 = r47691 <= r47692;
        double r47694 = -1.0;
        double r47695 = r47694 * r47691;
        double r47696 = 1.3722084465878742e+86;
        bool r47697 = r47691 <= r47696;
        double r47698 = r47691 * r47691;
        double r47699 = im;
        double r47700 = r47699 * r47699;
        double r47701 = r47698 + r47700;
        double r47702 = sqrt(r47701);
        double r47703 = r47697 ? r47702 : r47691;
        double r47704 = r47693 ? r47695 : r47703;
        return r47704;
}

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 < -1.4667592323261061e+131

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 8.8

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

    if -1.4667592323261061e+131 < re < 1.3722084465878742e+86

    1. Initial program 21.0

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

    if 1.3722084465878742e+86 < re

    1. Initial program 50.0

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

    2. Taylor expanded around inf 10.9

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4667592323261061 \cdot 10^{131}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 1.3722084465878742 \cdot 10^{86}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))