Average Error: 30.1 → 16.9
Time: 3.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0031590879196048 \cdot 10^{+160}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.428571781876908 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0031590879196048 \cdot 10^{+160}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 4.428571781876908 \cdot 10^{+138}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1525197 = re;
        double r1525198 = r1525197 * r1525197;
        double r1525199 = im;
        double r1525200 = r1525199 * r1525199;
        double r1525201 = r1525198 + r1525200;
        double r1525202 = sqrt(r1525201);
        return r1525202;
}
↓
double f(double re, double im) {
        double r1525203 = re;
        double r1525204 = -1.0031590879196048e+160;
        bool r1525205 = r1525203 <= r1525204;
        double r1525206 = -r1525203;
        double r1525207 = 4.428571781876908e+138;
        bool r1525208 = r1525203 <= r1525207;
        double r1525209 = im;
        double r1525210 = r1525209 * r1525209;
        double r1525211 = r1525203 * r1525203;
        double r1525212 = r1525210 + r1525211;
        double r1525213 = sqrt(r1525212);
        double r1525214 = r1525208 ? r1525213 : r1525203;
        double r1525215 = r1525205 ? r1525206 : r1525214;
        return r1525215;
}

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 < -1.0031590879196048e+160

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 7.1

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

    3. Simplified7.1

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

    if -1.0031590879196048e+160 < re < 4.428571781876908e+138

    1. Initial program 20.3

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

    if 4.428571781876908e+138 < re

    1. Initial program 55.3

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0031590879196048 \cdot 10^{+160}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 4.428571781876908 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))