Average Error: 31.0 → 17.3
Time: 5.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r60650 = re;
        double r60651 = r60650 * r60650;
        double r60652 = im;
        double r60653 = r60652 * r60652;
        double r60654 = r60651 + r60653;
        double r60655 = sqrt(r60654);
        return r60655;
}
↓
double f(double re, double im) {
        double r60656 = re;
        double r60657 = -6.523076260268755e+150;
        bool r60658 = r60656 <= r60657;
        double r60659 = -r60656;
        double r60660 = 1.158407537644574e+105;
        bool r60661 = r60656 <= r60660;
        double r60662 = r60656 * r60656;
        double r60663 = im;
        double r60664 = r60663 * r60663;
        double r60665 = r60662 + r60664;
        double r60666 = sqrt(r60665);
        double r60667 = r60661 ? r60666 : r60656;
        double r60668 = r60658 ? r60659 : r60667;
        return r60668;
}

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 < -6.523076260268755e+150

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -6.523076260268755e+150 < re < 1.158407537644574e+105

    1. Initial program 20.8

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

    if 1.158407537644574e+105 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))