Average Error: 31.7 → 18.2
Time: 3.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r62322 = re;
        double r62323 = r62322 * r62322;
        double r62324 = im;
        double r62325 = r62324 * r62324;
        double r62326 = r62323 + r62325;
        double r62327 = sqrt(r62326);
        return r62327;
}
↓
double f(double re, double im) {
        double r62328 = re;
        double r62329 = -2.2710920961423764e+59;
        bool r62330 = r62328 <= r62329;
        double r62331 = -1.0;
        double r62332 = r62331 * r62328;
        double r62333 = 9.634966723363643e+131;
        bool r62334 = r62328 <= r62333;
        double r62335 = r62328 * r62328;
        double r62336 = im;
        double r62337 = r62336 * r62336;
        double r62338 = r62335 + r62337;
        double r62339 = sqrt(r62338);
        double r62340 = r62334 ? r62339 : r62328;
        double r62341 = r62330 ? r62332 : r62340;
        return r62341;
}

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 < -2.2710920961423764e+59

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 12.9

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

    if -2.2710920961423764e+59 < re < 9.634966723363643e+131

    1. Initial program 22.0

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

    if 9.634966723363643e+131 < re

    1. Initial program 57.8

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

    2. Taylor expanded around inf 8.5

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.6349667233636427 \cdot 10^{131}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))