Average Error: 31.1 → 17.2
Time: 2.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.4968122243654856 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.7350730255718 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.2918788242355971 \cdot 10^{-219}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.23023028256962109 \cdot 10^{151}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.4968122243654856 \cdot 10^{112}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -8.7350730255718 \cdot 10^{-293}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.2918788242355971 \cdot 10^{-219}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.23023028256962109 \cdot 10^{151}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r50965 = re;
        double r50966 = r50965 * r50965;
        double r50967 = im;
        double r50968 = r50967 * r50967;
        double r50969 = r50966 + r50968;
        double r50970 = sqrt(r50969);
        return r50970;
}
↓
double f(double re, double im) {
        double r50971 = re;
        double r50972 = -3.4968122243654856e+112;
        bool r50973 = r50971 <= r50972;
        double r50974 = -1.0;
        double r50975 = r50974 * r50971;
        double r50976 = -8.7350730255718e-293;
        bool r50977 = r50971 <= r50976;
        double r50978 = r50971 * r50971;
        double r50979 = im;
        double r50980 = r50979 * r50979;
        double r50981 = r50978 + r50980;
        double r50982 = sqrt(r50981);
        double r50983 = 1.2918788242355971e-219;
        bool r50984 = r50971 <= r50983;
        double r50985 = 1.2302302825696211e+151;
        bool r50986 = r50971 <= r50985;
        double r50987 = r50986 ? r50982 : r50971;
        double r50988 = r50984 ? r50979 : r50987;
        double r50989 = r50977 ? r50982 : r50988;
        double r50990 = r50973 ? r50975 : r50989;
        return r50990;
}

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 < -3.4968122243654856e+112

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.3

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

    if -3.4968122243654856e+112 < re < -8.7350730255718e-293 or 1.2918788242355971e-219 < re < 1.2302302825696211e+151

    1. Initial program 19.1

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

    if -8.7350730255718e-293 < re < 1.2918788242355971e-219

    1. Initial program 31.1

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

    2. Taylor expanded around 0 31.9

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

    if 1.2302302825696211e+151 < re

    1. Initial program 63.0

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

    2. Taylor expanded around inf 7.2

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.4968122243654856 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -8.7350730255718 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.2918788242355971 \cdot 10^{-219}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.23023028256962109 \cdot 10^{151}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))