Average Error: 31.9 → 18.0
Time: 1.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r86783 = re;
        double r86784 = r86783 * r86783;
        double r86785 = im;
        double r86786 = r86785 * r86785;
        double r86787 = r86784 + r86786;
        double r86788 = sqrt(r86787);
        return r86788;
}
↓
double f(double re, double im) {
        double r86789 = re;
        double r86790 = -4.0084779007164915e+147;
        bool r86791 = r86789 <= r86790;
        double r86792 = -1.0;
        double r86793 = r86792 * r86789;
        double r86794 = 4.734467921936515e+65;
        bool r86795 = r86789 <= r86794;
        double r86796 = r86789 * r86789;
        double r86797 = im;
        double r86798 = r86797 * r86797;
        double r86799 = r86796 + r86798;
        double r86800 = sqrt(r86799);
        double r86801 = r86795 ? r86800 : r86789;
        double r86802 = r86791 ? r86793 : r86801;
        return r86802;
}

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 < -4.0084779007164915e+147

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 8.5

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

    if -4.0084779007164915e+147 < re < 4.734467921936515e+65

    1. Initial program 21.7

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

    if 4.734467921936515e+65 < re

    1. Initial program 47.4

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

    2. Taylor expanded around inf 11.9

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.00847790071649149 \cdot 10^{147}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.7344679219365152 \cdot 10^{65}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))