Average Error: 31.3 → 18.0
Time: 12.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.380870002726310342811700587071868218435 \cdot 10^{59}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.566938280750767851167015199229297462562 \cdot 10^{114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.380870002726310342811700587071868218435 \cdot 10^{59}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.566938280750767851167015199229297462562 \cdot 10^{114}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r46830 = re;
        double r46831 = r46830 * r46830;
        double r46832 = im;
        double r46833 = r46832 * r46832;
        double r46834 = r46831 + r46833;
        double r46835 = sqrt(r46834);
        return r46835;
}
↓
double f(double re, double im) {
        double r46836 = re;
        double r46837 = -6.38087000272631e+59;
        bool r46838 = r46836 <= r46837;
        double r46839 = -r46836;
        double r46840 = 2.566938280750768e+114;
        bool r46841 = r46836 <= r46840;
        double r46842 = r46836 * r46836;
        double r46843 = im;
        double r46844 = r46843 * r46843;
        double r46845 = r46842 + r46844;
        double r46846 = sqrt(r46845);
        double r46847 = r46841 ? r46846 : r46836;
        double r46848 = r46838 ? r46839 : r46847;
        return r46848;
}

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 < -6.38087000272631e+59

    1. Initial program 44.7

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

    2. Taylor expanded around -inf 12.7

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

    3. Simplified12.7

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

    if -6.38087000272631e+59 < re < 2.566938280750768e+114

    1. Initial program 21.7

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

    if 2.566938280750768e+114 < re

    1. Initial program 53.4

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

    2. Taylor expanded around inf 9.6

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.380870002726310342811700587071868218435 \cdot 10^{59}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.566938280750767851167015199229297462562 \cdot 10^{114}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))