Average Error: 29.6 → 17.0
Time: 3.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2933177868002892 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.257796562138815 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2933177868002892 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.257796562138815 \cdot 10^{+139}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1104602 = re;
        double r1104603 = r1104602 * r1104602;
        double r1104604 = im;
        double r1104605 = r1104604 * r1104604;
        double r1104606 = r1104603 + r1104605;
        double r1104607 = sqrt(r1104606);
        return r1104607;
}
↓
double f(double re, double im) {
        double r1104608 = re;
        double r1104609 = -1.2933177868002892e+153;
        bool r1104610 = r1104608 <= r1104609;
        double r1104611 = -r1104608;
        double r1104612 = 1.257796562138815e+139;
        bool r1104613 = r1104608 <= r1104612;
        double r1104614 = im;
        double r1104615 = r1104614 * r1104614;
        double r1104616 = r1104608 * r1104608;
        double r1104617 = r1104615 + r1104616;
        double r1104618 = sqrt(r1104617);
        double r1104619 = r1104613 ? r1104618 : r1104608;
        double r1104620 = r1104610 ? r1104611 : r1104619;
        return r1104620;
}

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 < -1.2933177868002892e+153

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 8.8

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

    3. Simplified8.8

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

    if -1.2933177868002892e+153 < re < 1.257796562138815e+139

    1. Initial program 19.8

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

    if 1.257796562138815e+139 < re

    1. Initial program 55.5

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

    2. Taylor expanded around inf 9.3

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2933177868002892 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.257796562138815 \cdot 10^{+139}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))