Average Error: 29.8 → 17.8
Time: 19.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.635632655464899 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.1997069165942805 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.9066979284450217 \cdot 10^{-165}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.6018930323419904 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.635632655464899 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -2.1997069165942805 \cdot 10^{-243}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 2.9066979284450217 \cdot 10^{-165}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.6018930323419904 \cdot 10^{+98}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r2005742 = re;
        double r2005743 = r2005742 * r2005742;
        double r2005744 = im;
        double r2005745 = r2005744 * r2005744;
        double r2005746 = r2005743 + r2005745;
        double r2005747 = sqrt(r2005746);
        return r2005747;
}
↓
double f(double re, double im) {
        double r2005748 = re;
        double r2005749 = -7.635632655464899e+153;
        bool r2005750 = r2005748 <= r2005749;
        double r2005751 = -r2005748;
        double r2005752 = -2.1997069165942805e-243;
        bool r2005753 = r2005748 <= r2005752;
        double r2005754 = im;
        double r2005755 = r2005754 * r2005754;
        double r2005756 = r2005748 * r2005748;
        double r2005757 = r2005755 + r2005756;
        double r2005758 = sqrt(r2005757);
        double r2005759 = 2.9066979284450217e-165;
        bool r2005760 = r2005748 <= r2005759;
        double r2005761 = 3.6018930323419904e+98;
        bool r2005762 = r2005748 <= r2005761;
        double r2005763 = r2005762 ? r2005758 : r2005748;
        double r2005764 = r2005760 ? r2005754 : r2005763;
        double r2005765 = r2005753 ? r2005758 : r2005764;
        double r2005766 = r2005750 ? r2005751 : r2005765;
        return r2005766;
}

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 4 regimes

  2. if re < -7.635632655464899e+153

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -7.635632655464899e+153 < re < -2.1997069165942805e-243 or 2.9066979284450217e-165 < re < 3.6018930323419904e+98

    1. Initial program 17.2

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

    if -2.1997069165942805e-243 < re < 2.9066979284450217e-165

    1. Initial program 29.3

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

    2. Taylor expanded around 0 35.6

      \[\leadsto \color{blue}{im}\]

    if 3.6018930323419904e+98 < re

    1. Initial program 47.8

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

    2. Taylor expanded around inf 9.6

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification17.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.635632655464899 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.1997069165942805 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 2.9066979284450217 \cdot 10^{-165}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.6018930323419904 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))