Average Error: 31.5 → 17.4
Time: 6.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1032025771158766 \cdot 10^{142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.1032025771158766 \cdot 10^{142}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r68947 = re;
        double r68948 = r68947 * r68947;
        double r68949 = im;
        double r68950 = r68949 * r68949;
        double r68951 = r68948 + r68950;
        double r68952 = sqrt(r68951);
        return r68952;
}
↓
double f(double re, double im) {
        double r68953 = re;
        double r68954 = -4.109494051147195e+119;
        bool r68955 = r68953 <= r68954;
        double r68956 = -r68953;
        double r68957 = 1.1032025771158766e+142;
        bool r68958 = r68953 <= r68957;
        double r68959 = r68953 * r68953;
        double r68960 = im;
        double r68961 = r68960 * r68960;
        double r68962 = r68959 + r68961;
        double r68963 = sqrt(r68962);
        double r68964 = r68958 ? r68963 : r68953;
        double r68965 = r68955 ? r68956 : r68964;
        return r68965;
}

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 < -4.109494051147195e+119

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -4.109494051147195e+119 < re < 1.1032025771158766e+142

    1. Initial program 20.7

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

    if 1.1032025771158766e+142 < re

    1. Initial program 60.2

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

    2. Taylor expanded around inf 8.9

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1094940511471951 \cdot 10^{119}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.1032025771158766 \cdot 10^{142}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))