Average Error: 31.3 → 18.0
Time: 12.8s
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 r60838 = re;
        double r60839 = r60838 * r60838;
        double r60840 = im;
        double r60841 = r60840 * r60840;
        double r60842 = r60839 + r60841;
        double r60843 = sqrt(r60842);
        return r60843;
}
↓
double f(double re, double im) {
        double r60844 = re;
        double r60845 = -6.38087000272631e+59;
        bool r60846 = r60844 <= r60845;
        double r60847 = -r60844;
        double r60848 = 2.566938280750768e+114;
        bool r60849 = r60844 <= r60848;
        double r60850 = r60844 * r60844;
        double r60851 = im;
        double r60852 = r60851 * r60851;
        double r60853 = r60850 + r60852;
        double r60854 = sqrt(r60853);
        double r60855 = r60849 ? r60854 : r60844;
        double r60856 = r60846 ? r60847 : r60855;
        return r60856;
}

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))))