Average Error: 31.8 → 17.2
Time: 4.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.815000942687335399540629803602323238426 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.315991764196220167239997200279477919386 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.815000942687335399540629803602323238426 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 3.315991764196220167239997200279477919386 \cdot 10^{127}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r43507 = re;
        double r43508 = r43507 * r43507;
        double r43509 = im;
        double r43510 = r43509 * r43509;
        double r43511 = r43508 + r43510;
        double r43512 = sqrt(r43511);
        return r43512;
}
↓
double f(double re, double im) {
        double r43513 = re;
        double r43514 = -7.815000942687335e+146;
        bool r43515 = r43513 <= r43514;
        double r43516 = -1.0;
        double r43517 = r43516 * r43513;
        double r43518 = 3.31599176419622e+127;
        bool r43519 = r43513 <= r43518;
        double r43520 = r43513 * r43513;
        double r43521 = im;
        double r43522 = r43521 * r43521;
        double r43523 = r43520 + r43522;
        double r43524 = sqrt(r43523);
        double r43525 = r43519 ? r43524 : r43513;
        double r43526 = r43515 ? r43517 : r43525;
        return r43526;
}

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 < -7.815000942687335e+146

    1. Initial program 62.0

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

    2. Taylor expanded around -inf 9.0

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

    if -7.815000942687335e+146 < re < 3.31599176419622e+127

    1. Initial program 20.7

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

    if 3.31599176419622e+127 < re

    1. Initial program 56.3

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.815000942687335399540629803602323238426 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.315991764196220167239997200279477919386 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))