Average Error: 31.6 → 17.8
Time: 13.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r33656 = re;
        double r33657 = r33656 * r33656;
        double r33658 = im;
        double r33659 = r33658 * r33658;
        double r33660 = r33657 + r33659;
        double r33661 = sqrt(r33660);
        return r33661;
}
↓
double f(double re, double im) {
        double r33662 = re;
        double r33663 = -1.1564076018637175e+112;
        bool r33664 = r33662 <= r33663;
        double r33665 = -r33662;
        double r33666 = 1.3595155319523303e+138;
        bool r33667 = r33662 <= r33666;
        double r33668 = r33662 * r33662;
        double r33669 = im;
        double r33670 = r33669 * r33669;
        double r33671 = r33668 + r33670;
        double r33672 = sqrt(r33671);
        double r33673 = r33667 ? r33672 : r33662;
        double r33674 = r33664 ? r33665 : r33673;
        return r33674;
}

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 < -1.1564076018637175e+112

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.6

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

    3. Simplified9.6

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

    if -1.1564076018637175e+112 < re < 1.3595155319523303e+138

    1. Initial program 21.4

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

    if 1.3595155319523303e+138 < re

    1. Initial program 58.8

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.156407601863717509012505141513837828653 \cdot 10^{112}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.359515531952330295686549505956711156315 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))