Average Error: 31.9 → 17.6
Time: 894.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.23653280905907259 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 9.23653280905907259 \cdot 10^{138}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r34711 = re;
        double r34712 = r34711 * r34711;
        double r34713 = im;
        double r34714 = r34713 * r34713;
        double r34715 = r34712 + r34714;
        double r34716 = sqrt(r34715);
        return r34716;
}
↓
double f(double re, double im) {
        double r34717 = re;
        double r34718 = -9.16501881147336e+142;
        bool r34719 = r34717 <= r34718;
        double r34720 = -1.0;
        double r34721 = r34720 * r34717;
        double r34722 = 9.236532809059073e+138;
        bool r34723 = r34717 <= r34722;
        double r34724 = r34717 * r34717;
        double r34725 = im;
        double r34726 = r34725 * r34725;
        double r34727 = r34724 + r34726;
        double r34728 = sqrt(r34727);
        double r34729 = r34723 ? r34728 : r34717;
        double r34730 = r34719 ? r34721 : r34729;
        return r34730;
}

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 < -9.16501881147336e+142

    1. Initial program 61.3

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

    2. Taylor expanded around -inf 9.1

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

    if -9.16501881147336e+142 < re < 9.236532809059073e+138

    1. Initial program 20.9

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

    if 9.236532809059073e+138 < re

    1. Initial program 59.7

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.16501881147335996 \cdot 10^{142}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 9.23653280905907259 \cdot 10^{138}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))