Average Error: 31.9 → 17.6
Time: 1.7s
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 r286 = re;
        double r287 = r286 * r286;
        double r288 = im;
        double r289 = r288 * r288;
        double r290 = r287 + r289;
        double r291 = sqrt(r290);
        return r291;
}
↓
double f(double re, double im) {
        double r292 = re;
        double r293 = -9.16501881147336e+142;
        bool r294 = r292 <= r293;
        double r295 = -1.0;
        double r296 = r295 * r292;
        double r297 = 9.236532809059073e+138;
        bool r298 = r292 <= r297;
        double r299 = r292 * r292;
        double r300 = im;
        double r301 = r300 * r300;
        double r302 = r299 + r301;
        double r303 = sqrt(r302);
        double r304 = r298 ? r303 : r292;
        double r305 = r294 ? r296 : r304;
        return r305;
}

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))))