Average Error: 29.7 → 17.4
Time: 2.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.3377295553932065 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.6726164932126896 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.590629195950438 \cdot 10^{-228}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.882396363471933 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.3377295553932065 \cdot 10^{+154}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -3.6726164932126896 \cdot 10^{-218}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.590629195950438 \cdot 10^{-228}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.882396363471933 \cdot 10^{+117}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r724511 = re;
        double r724512 = r724511 * r724511;
        double r724513 = im;
        double r724514 = r724513 * r724513;
        double r724515 = r724512 + r724514;
        double r724516 = sqrt(r724515);
        return r724516;
}
↓
double f(double re, double im) {
        double r724517 = re;
        double r724518 = -1.3377295553932065e+154;
        bool r724519 = r724517 <= r724518;
        double r724520 = -r724517;
        double r724521 = -3.6726164932126896e-218;
        bool r724522 = r724517 <= r724521;
        double r724523 = im;
        double r724524 = r724523 * r724523;
        double r724525 = r724517 * r724517;
        double r724526 = r724524 + r724525;
        double r724527 = sqrt(r724526);
        double r724528 = 1.590629195950438e-228;
        bool r724529 = r724517 <= r724528;
        double r724530 = 2.882396363471933e+117;
        bool r724531 = r724517 <= r724530;
        double r724532 = r724531 ? r724527 : r724517;
        double r724533 = r724529 ? r724523 : r724532;
        double r724534 = r724522 ? r724527 : r724533;
        double r724535 = r724519 ? r724520 : r724534;
        return r724535;
}

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 4 regimes

  2. if re < -1.3377295553932065e+154

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -1.3377295553932065e+154 < re < -3.6726164932126896e-218 or 1.590629195950438e-228 < re < 2.882396363471933e+117

    1. Initial program 17.7

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

    if -3.6726164932126896e-218 < re < 1.590629195950438e-228

    1. Initial program 29.3

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

    2. Taylor expanded around 0 33.6

      \[\leadsto \color{blue}{im}\]

    if 2.882396363471933e+117 < re

    1. Initial program 50.9

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

    2. Taylor expanded around inf 9.4

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.3377295553932065 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -3.6726164932126896 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.590629195950438 \cdot 10^{-228}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.882396363471933 \cdot 10^{+117}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))