Average Error: 31.6 → 17.8
Time: 14.9s
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 r98115 = re;
        double r98116 = r98115 * r98115;
        double r98117 = im;
        double r98118 = r98117 * r98117;
        double r98119 = r98116 + r98118;
        double r98120 = sqrt(r98119);
        return r98120;
}
↓
double f(double re, double im) {
        double r98121 = re;
        double r98122 = -1.1564076018637175e+112;
        bool r98123 = r98121 <= r98122;
        double r98124 = -r98121;
        double r98125 = 1.3595155319523303e+138;
        bool r98126 = r98121 <= r98125;
        double r98127 = r98121 * r98121;
        double r98128 = im;
        double r98129 = r98128 * r98128;
        double r98130 = r98127 + r98129;
        double r98131 = sqrt(r98130);
        double r98132 = r98126 ? r98131 : r98121;
        double r98133 = r98123 ? r98124 : r98132;
        return r98133;
}

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))))