Average Error: 31.6 → 17.8
Time: 23.7s
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 r37254 = re;
        double r37255 = r37254 * r37254;
        double r37256 = im;
        double r37257 = r37256 * r37256;
        double r37258 = r37255 + r37257;
        double r37259 = sqrt(r37258);
        return r37259;
}
↓
double f(double re, double im) {
        double r37260 = re;
        double r37261 = -1.1564076018637175e+112;
        bool r37262 = r37260 <= r37261;
        double r37263 = -r37260;
        double r37264 = 1.3595155319523303e+138;
        bool r37265 = r37260 <= r37264;
        double r37266 = r37260 * r37260;
        double r37267 = im;
        double r37268 = r37267 * r37267;
        double r37269 = r37266 + r37268;
        double r37270 = sqrt(r37269);
        double r37271 = r37265 ? r37270 : r37260;
        double r37272 = r37262 ? r37263 : r37271;
        return r37272;
}

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))))