Average Error: 30.2 → 17.2
Time: 11.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.562500805767915 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1829372131501764 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 8.782161380506134 \cdot 10^{-190}:\\ \;\;\;\;re\\ \mathbf{elif}\;re \le 7.628842183490642 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.562500805767915 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.1829372131501764 \cdot 10^{-211}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 8.782161380506134 \cdot 10^{-190}:\\
\;\;\;\;re\\

\mathbf{elif}\;re \le 7.628842183490642 \cdot 10^{+130}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1962970 = re;
        double r1962971 = r1962970 * r1962970;
        double r1962972 = im;
        double r1962973 = r1962972 * r1962972;
        double r1962974 = r1962971 + r1962973;
        double r1962975 = sqrt(r1962974);
        return r1962975;
}
↓
double f(double re, double im) {
        double r1962976 = re;
        double r1962977 = -6.562500805767915e+153;
        bool r1962978 = r1962976 <= r1962977;
        double r1962979 = -r1962976;
        double r1962980 = 1.1829372131501764e-211;
        bool r1962981 = r1962976 <= r1962980;
        double r1962982 = im;
        double r1962983 = r1962982 * r1962982;
        double r1962984 = r1962976 * r1962976;
        double r1962985 = r1962983 + r1962984;
        double r1962986 = sqrt(r1962985);
        double r1962987 = 8.782161380506134e-190;
        bool r1962988 = r1962976 <= r1962987;
        double r1962989 = 7.628842183490642e+130;
        bool r1962990 = r1962976 <= r1962989;
        double r1962991 = r1962990 ? r1962986 : r1962976;
        double r1962992 = r1962988 ? r1962976 : r1962991;
        double r1962993 = r1962981 ? r1962986 : r1962992;
        double r1962994 = r1962978 ? r1962979 : r1962993;
        return r1962994;
}

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 < -6.562500805767915e+153

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -6.562500805767915e+153 < re < 1.1829372131501764e-211 or 8.782161380506134e-190 < re < 7.628842183490642e+130

    1. Initial program 20.0

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

    if 1.1829372131501764e-211 < re < 8.782161380506134e-190 or 7.628842183490642e+130 < re

    1. Initial program 52.0

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.562500805767915 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1829372131501764 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 8.782161380506134 \cdot 10^{-190}:\\ \;\;\;\;re\\ \mathbf{elif}\;re \le 7.628842183490642 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))