Average Error: 31.5 → 17.4
Time: 5.9s
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 r58030 = re;
        double r58031 = r58030 * r58030;
        double r58032 = im;
        double r58033 = r58032 * r58032;
        double r58034 = r58031 + r58033;
        double r58035 = sqrt(r58034);
        return r58035;
}
↓
double f(double re, double im) {
        double r58036 = re;
        double r58037 = -4.109494051147195e+119;
        bool r58038 = r58036 <= r58037;
        double r58039 = -r58036;
        double r58040 = 1.1032025771158766e+142;
        bool r58041 = r58036 <= r58040;
        double r58042 = r58036 * r58036;
        double r58043 = im;
        double r58044 = r58043 * r58043;
        double r58045 = r58042 + r58044;
        double r58046 = sqrt(r58045);
        double r58047 = r58041 ? r58046 : r58036;
        double r58048 = r58038 ? r58039 : r58047;
        return r58048;
}

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))))