Average Error: 31.6 → 18.7
Time: 971.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.8460535119133569 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.19099635470288769 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.08574889376971239 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.98422560465703889 \cdot 10^{39}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.8460535119133569 \cdot 10^{74}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.19099635470288769 \cdot 10^{-228}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.08574889376971239 \cdot 10^{-190}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.98422560465703889 \cdot 10^{39}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r52849 = re;
        double r52850 = r52849 * r52849;
        double r52851 = im;
        double r52852 = r52851 * r52851;
        double r52853 = r52850 + r52852;
        double r52854 = sqrt(r52853);
        return r52854;
}
↓
double f(double re, double im) {
        double r52855 = re;
        double r52856 = -3.846053511913357e+74;
        bool r52857 = r52855 <= r52856;
        double r52858 = -1.0;
        double r52859 = r52858 * r52855;
        double r52860 = -1.3504253849915568e-194;
        bool r52861 = r52855 <= r52860;
        double r52862 = r52855 * r52855;
        double r52863 = im;
        double r52864 = r52863 * r52863;
        double r52865 = r52862 + r52864;
        double r52866 = sqrt(r52865);
        double r52867 = -2.968956980813959e-266;
        bool r52868 = r52855 <= r52867;
        double r52869 = 1.1909963547028877e-228;
        bool r52870 = r52855 <= r52869;
        double r52871 = 1.0857488937697124e-190;
        bool r52872 = r52855 <= r52871;
        double r52873 = 3.984225604657039e+39;
        bool r52874 = r52855 <= r52873;
        double r52875 = r52874 ? r52866 : r52855;
        double r52876 = r52872 ? r52863 : r52875;
        double r52877 = r52870 ? r52866 : r52876;
        double r52878 = r52868 ? r52863 : r52877;
        double r52879 = r52861 ? r52866 : r52878;
        double r52880 = r52857 ? r52859 : r52879;
        return r52880;
}

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 < -3.846053511913357e+74

    1. Initial program 47.2

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

    2. Taylor expanded around -inf 10.7

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

    if -3.846053511913357e+74 < re < -1.3504253849915568e-194 or -2.968956980813959e-266 < re < 1.1909963547028877e-228 or 1.0857488937697124e-190 < re < 3.984225604657039e+39

    1. Initial program 20.9

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

    if -1.3504253849915568e-194 < re < -2.968956980813959e-266 or 1.1909963547028877e-228 < re < 1.0857488937697124e-190

    1. Initial program 31.4

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

    2. Taylor expanded around 0 34.8

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

    if 3.984225604657039e+39 < re

    1. Initial program 43.8

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

    2. Taylor expanded around inf 13.6

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

  4. Final simplification18.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.8460535119133569 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.3504253849915568 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.968956980813959 \cdot 10^{-266}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.19099635470288769 \cdot 10^{-228}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.08574889376971239 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.98422560465703889 \cdot 10^{39}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))