Average Error: 31.3 → 17.2
Time: 5.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r93861 = re;
        double r93862 = r93861 * r93861;
        double r93863 = im;
        double r93864 = r93863 * r93863;
        double r93865 = r93862 + r93864;
        double r93866 = sqrt(r93865);
        return r93866;
}
↓
double f(double re, double im) {
        double r93867 = re;
        double r93868 = -1.1817931832138217e+151;
        bool r93869 = r93867 <= r93868;
        double r93870 = -r93867;
        double r93871 = 5.948234035126459e+127;
        bool r93872 = r93867 <= r93871;
        double r93873 = im;
        double r93874 = r93873 * r93873;
        double r93875 = r93867 * r93867;
        double r93876 = r93874 + r93875;
        double r93877 = sqrt(r93876);
        double r93878 = r93872 ? r93877 : r93867;
        double r93879 = r93869 ? r93870 : r93878;
        return r93879;
}

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 < -1.1817931832138217e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -1.1817931832138217e+151 < re < 5.948234035126459e+127

    1. Initial program 20.4

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

    if 5.948234035126459e+127 < re

    1. Initial program 56.6

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))