Average Error: 30.1 → 17.0
Time: 11.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.0334299372921848 \cdot 10^{+147}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.120853424431282 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.7106634379274166 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.4751344550795556 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.0334299372921848 \cdot 10^{+147}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -6.120853424431282 \cdot 10^{-307}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.7106634379274166 \cdot 10^{-227}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 4.4751344550795556 \cdot 10^{+147}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1444750 = re;
        double r1444751 = r1444750 * r1444750;
        double r1444752 = im;
        double r1444753 = r1444752 * r1444752;
        double r1444754 = r1444751 + r1444753;
        double r1444755 = sqrt(r1444754);
        return r1444755;
}
↓
double f(double re, double im) {
        double r1444756 = re;
        double r1444757 = -2.0334299372921848e+147;
        bool r1444758 = r1444756 <= r1444757;
        double r1444759 = -r1444756;
        double r1444760 = -6.120853424431282e-307;
        bool r1444761 = r1444756 <= r1444760;
        double r1444762 = im;
        double r1444763 = r1444762 * r1444762;
        double r1444764 = r1444756 * r1444756;
        double r1444765 = r1444763 + r1444764;
        double r1444766 = sqrt(r1444765);
        double r1444767 = 1.7106634379274166e-227;
        bool r1444768 = r1444756 <= r1444767;
        double r1444769 = 4.4751344550795556e+147;
        bool r1444770 = r1444756 <= r1444769;
        double r1444771 = r1444770 ? r1444766 : r1444756;
        double r1444772 = r1444768 ? r1444762 : r1444771;
        double r1444773 = r1444761 ? r1444766 : r1444772;
        double r1444774 = r1444758 ? r1444759 : r1444773;
        return r1444774;
}

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 < -2.0334299372921848e+147

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 7.5

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

    3. Simplified7.5

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

    if -2.0334299372921848e+147 < re < -6.120853424431282e-307 or 1.7106634379274166e-227 < re < 4.4751344550795556e+147

    1. Initial program 18.8

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

    if -6.120853424431282e-307 < re < 1.7106634379274166e-227

    1. Initial program 31.8

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt32.0

      \[\leadsto \color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}}\]

    4. Taylor expanded around 0 34.2

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

    if 4.4751344550795556e+147 < re

    1. Initial program 57.9

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt57.9

      \[\leadsto \color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}}\]

    4. Taylor expanded around inf 8.6

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.0334299372921848 \cdot 10^{+147}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -6.120853424431282 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.7106634379274166 \cdot 10^{-227}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.4751344550795556 \cdot 10^{+147}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))