Average Error: 31.2 → 17.7
Time: 3.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r40764 = re;
        double r40765 = r40764 * r40764;
        double r40766 = im;
        double r40767 = r40766 * r40766;
        double r40768 = r40765 + r40767;
        double r40769 = sqrt(r40768);
        return r40769;
}
↓
double f(double re, double im) {
        double r40770 = re;
        double r40771 = -2.4740957117892876e+117;
        bool r40772 = r40770 <= r40771;
        double r40773 = -r40770;
        double r40774 = 5.435137585363575e+84;
        bool r40775 = r40770 <= r40774;
        double r40776 = r40770 * r40770;
        double r40777 = im;
        double r40778 = r40777 * r40777;
        double r40779 = r40776 + r40778;
        double r40780 = sqrt(r40779);
        double r40781 = r40775 ? r40780 : r40770;
        double r40782 = r40772 ? r40773 : r40781;
        return r40782;
}

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 < -2.4740957117892876e+117

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 9.4

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

    3. Simplified9.4

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

    if -2.4740957117892876e+117 < re < 5.435137585363575e+84

    1. Initial program 21.3

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

    if 5.435137585363575e+84 < re

    1. Initial program 47.8

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.47409571178928762 \cdot 10^{117}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.43513758536357538 \cdot 10^{84}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))