Average Error: 31.1 → 17.6
Time: 2.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.154655280967186248169160366051090225892 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.94459162722497846963223303723510052557 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 7.357011145278623595340421361957804170614 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.540206777357535753716497569649753059253 \cdot 10^{120}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.154655280967186248169160366051090225892 \cdot 10^{151}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 3.94459162722497846963223303723510052557 \cdot 10^{-301}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 7.357011145278623595340421361957804170614 \cdot 10^{-217}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 5.540206777357535753716497569649753059253 \cdot 10^{120}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r49927 = re;
        double r49928 = r49927 * r49927;
        double r49929 = im;
        double r49930 = r49929 * r49929;
        double r49931 = r49928 + r49930;
        double r49932 = sqrt(r49931);
        return r49932;
}
↓
double f(double re, double im) {
        double r49933 = re;
        double r49934 = -4.154655280967186e+151;
        bool r49935 = r49933 <= r49934;
        double r49936 = -1.0;
        double r49937 = r49936 * r49933;
        double r49938 = 3.9445916272249785e-301;
        bool r49939 = r49933 <= r49938;
        double r49940 = r49933 * r49933;
        double r49941 = im;
        double r49942 = r49941 * r49941;
        double r49943 = r49940 + r49942;
        double r49944 = sqrt(r49943);
        double r49945 = 7.357011145278624e-217;
        bool r49946 = r49933 <= r49945;
        double r49947 = 5.540206777357536e+120;
        bool r49948 = r49933 <= r49947;
        double r49949 = r49948 ? r49944 : r49933;
        double r49950 = r49946 ? r49941 : r49949;
        double r49951 = r49939 ? r49944 : r49950;
        double r49952 = r49935 ? r49937 : r49951;
        return r49952;
}

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 < -4.154655280967186e+151

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 7.8

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

    if -4.154655280967186e+151 < re < 3.9445916272249785e-301 or 7.357011145278624e-217 < re < 5.540206777357536e+120

    1. Initial program 19.5

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

    if 3.9445916272249785e-301 < re < 7.357011145278624e-217

    1. Initial program 29.9

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

    2. Taylor expanded around 0 33.6

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

    if 5.540206777357536e+120 < re

    1. Initial program 56.5

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.154655280967186248169160366051090225892 \cdot 10^{151}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 3.94459162722497846963223303723510052557 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 7.357011145278623595340421361957804170614 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 5.540206777357535753716497569649753059253 \cdot 10^{120}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))