Average Error: 31.8 → 18.3
Time: 950.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.5902457227401156 \cdot 10^{122}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.05078470045216924 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.45616683745643486 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.68206633937327155 \cdot 10^{123}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.5902457227401156 \cdot 10^{122}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.05078470045216924 \cdot 10^{-159}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 4.45616683745643486 \cdot 10^{-306}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.68206633937327155 \cdot 10^{123}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r36838 = re;
        double r36839 = r36838 * r36838;
        double r36840 = im;
        double r36841 = r36840 * r36840;
        double r36842 = r36839 + r36841;
        double r36843 = sqrt(r36842);
        return r36843;
}
↓
double f(double re, double im) {
        double r36844 = re;
        double r36845 = -3.5902457227401156e+122;
        bool r36846 = r36844 <= r36845;
        double r36847 = -1.0;
        double r36848 = r36847 * r36844;
        double r36849 = -1.0507847004521692e-159;
        bool r36850 = r36844 <= r36849;
        double r36851 = r36844 * r36844;
        double r36852 = im;
        double r36853 = r36852 * r36852;
        double r36854 = r36851 + r36853;
        double r36855 = sqrt(r36854);
        double r36856 = 4.456166837456435e-306;
        bool r36857 = r36844 <= r36856;
        double r36858 = 3.6820663393732715e+123;
        bool r36859 = r36844 <= r36858;
        double r36860 = r36859 ? r36855 : r36844;
        double r36861 = r36857 ? r36852 : r36860;
        double r36862 = r36850 ? r36855 : r36861;
        double r36863 = r36846 ? r36848 : r36862;
        return r36863;
}

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 < -3.5902457227401156e+122

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 10.1

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

    if -3.5902457227401156e+122 < re < -1.0507847004521692e-159 or 4.456166837456435e-306 < re < 3.6820663393732715e+123

    1. Initial program 19.3

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

    if -1.0507847004521692e-159 < re < 4.456166837456435e-306

    1. Initial program 30.6

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

    2. Taylor expanded around 0 35.1

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

    if 3.6820663393732715e+123 < re

    1. Initial program 55.9

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification18.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.5902457227401156 \cdot 10^{122}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.05078470045216924 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 4.45616683745643486 \cdot 10^{-306}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.68206633937327155 \cdot 10^{123}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))