Average Error: 31.3 → 18.1
Time: 978.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.9350549922861674 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.230526966536829 \cdot 10^{-195}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.70673714189414653 \cdot 10^{49}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.9350549922861674 \cdot 10^{-252}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.230526966536829 \cdot 10^{-195}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.70673714189414653 \cdot 10^{49}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r52769 = re;
        double r52770 = r52769 * r52769;
        double r52771 = im;
        double r52772 = r52771 * r52771;
        double r52773 = r52770 + r52772;
        double r52774 = sqrt(r52773);
        return r52774;
}
↓
double f(double re, double im) {
        double r52775 = re;
        double r52776 = -6.5586381711852656e+44;
        bool r52777 = r52775 <= r52776;
        double r52778 = -1.0;
        double r52779 = r52778 * r52775;
        double r52780 = 4.935054992286167e-252;
        bool r52781 = r52775 <= r52780;
        double r52782 = r52775 * r52775;
        double r52783 = im;
        double r52784 = r52783 * r52783;
        double r52785 = r52782 + r52784;
        double r52786 = sqrt(r52785);
        double r52787 = 1.230526966536829e-195;
        bool r52788 = r52775 <= r52787;
        double r52789 = 1.7067371418941465e+49;
        bool r52790 = r52775 <= r52789;
        double r52791 = r52790 ? r52786 : r52775;
        double r52792 = r52788 ? r52783 : r52791;
        double r52793 = r52781 ? r52786 : r52792;
        double r52794 = r52777 ? r52779 : r52793;
        return r52794;
}

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 < -6.5586381711852656e+44

    1. Initial program 45.2

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

    2. Taylor expanded around -inf 12.8

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

    if -6.5586381711852656e+44 < re < 4.935054992286167e-252 or 1.230526966536829e-195 < re < 1.7067371418941465e+49

    1. Initial program 20.7

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

    if 4.935054992286167e-252 < re < 1.230526966536829e-195

    1. Initial program 31.4

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

    2. Taylor expanded around 0 36.0

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

    if 1.7067371418941465e+49 < re

    1. Initial program 44.4

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.9350549922861674 \cdot 10^{-252}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.230526966536829 \cdot 10^{-195}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.70673714189414653 \cdot 10^{49}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))