Average Error: 31.9 → 17.7
Time: 4.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r49579 = re;
        double r49580 = r49579 * r49579;
        double r49581 = im;
        double r49582 = r49581 * r49581;
        double r49583 = r49580 + r49582;
        double r49584 = sqrt(r49583);
        return r49584;
}
↓
double f(double re, double im) {
        double r49585 = re;
        double r49586 = -1.2876568362185878e+137;
        bool r49587 = r49585 <= r49586;
        double r49588 = -1.0;
        double r49589 = r49588 * r49585;
        double r49590 = 2.7153468834491098e+73;
        bool r49591 = r49585 <= r49590;
        double r49592 = r49585 * r49585;
        double r49593 = im;
        double r49594 = r49593 * r49593;
        double r49595 = r49592 + r49594;
        double r49596 = sqrt(r49595);
        double r49597 = r49591 ? r49596 : r49585;
        double r49598 = r49587 ? r49589 : r49597;
        return r49598;
}

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 < -1.2876568362185878e+137

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 9.3

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

    if -1.2876568362185878e+137 < re < 2.7153468834491098e+73

    1. Initial program 21.5

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

    if 2.7153468834491098e+73 < re

    1. Initial program 47.0

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))