Average Error: 31.5 → 18.0
Time: 6.8s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.149380376403675710903737768625835202966 \cdot 10^{95}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.149380376403675710903737768625835202966 \cdot 10^{95}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r48164 = re;
        double r48165 = r48164 * r48164;
        double r48166 = im;
        double r48167 = r48166 * r48166;
        double r48168 = r48165 + r48167;
        double r48169 = sqrt(r48168);
        return r48169;
}
↓
double f(double re, double im) {
        double r48170 = re;
        double r48171 = -1.9401195934627838e+70;
        bool r48172 = r48170 <= r48171;
        double r48173 = -r48170;
        double r48174 = 1.1493803764036757e+95;
        bool r48175 = r48170 <= r48174;
        double r48176 = im;
        double r48177 = r48176 * r48176;
        double r48178 = r48170 * r48170;
        double r48179 = r48177 + r48178;
        double r48180 = sqrt(r48179);
        double r48181 = r48175 ? r48180 : r48170;
        double r48182 = r48172 ? r48173 : r48181;
        return r48182;
}

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 < -1.9401195934627838e+70

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 12.1

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

    3. Simplified12.1

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

    if -1.9401195934627838e+70 < re < 1.1493803764036757e+95

    1. Initial program 21.8

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

    if 1.1493803764036757e+95 < re

    1. Initial program 49.0

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.149380376403675710903737768625835202966 \cdot 10^{95}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))