Average Error: 31.8 → 17.4
Time: 878.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.17117052004380529 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.0168808923777661 \cdot 10^{94}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.17117052004380529 \cdot 10^{89}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 2.0168808923777661 \cdot 10^{94}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r50066 = re;
        double r50067 = r50066 * r50066;
        double r50068 = im;
        double r50069 = r50068 * r50068;
        double r50070 = r50067 + r50069;
        double r50071 = sqrt(r50070);
        return r50071;
}
↓
double f(double re, double im) {
        double r50072 = re;
        double r50073 = -1.1711705200438053e+89;
        bool r50074 = r50072 <= r50073;
        double r50075 = -1.0;
        double r50076 = r50075 * r50072;
        double r50077 = 2.016880892377766e+94;
        bool r50078 = r50072 <= r50077;
        double r50079 = r50072 * r50072;
        double r50080 = im;
        double r50081 = r50080 * r50080;
        double r50082 = r50079 + r50081;
        double r50083 = sqrt(r50082);
        double r50084 = r50078 ? r50083 : r50072;
        double r50085 = r50074 ? r50076 : r50084;
        return r50085;
}

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 < -1.1711705200438053e+89

    1. Initial program 50.6

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

    2. Taylor expanded around -inf 9.7

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

    if -1.1711705200438053e+89 < re < 2.016880892377766e+94

    1. Initial program 21.6

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

    if 2.016880892377766e+94 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 9.6

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.17117052004380529 \cdot 10^{89}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 2.0168808923777661 \cdot 10^{94}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))