Average Error: 31.5 → 18.1
Time: 1.8s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.810291952691062 \cdot 10^{115}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -6.2167173716269886 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.95065728572471767 \cdot 10^{-235}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.97128001067495674 \cdot 10^{26}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.810291952691062 \cdot 10^{115}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -6.2167173716269886 \cdot 10^{-264}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.95065728572471767 \cdot 10^{-235}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.97128001067495674 \cdot 10^{26}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r98760 = re;
        double r98761 = r98760 * r98760;
        double r98762 = im;
        double r98763 = r98762 * r98762;
        double r98764 = r98761 + r98763;
        double r98765 = sqrt(r98764);
        return r98765;
}
↓
double f(double re, double im) {
        double r98766 = re;
        double r98767 = -5.810291952691062e+115;
        bool r98768 = r98766 <= r98767;
        double r98769 = -1.0;
        double r98770 = r98769 * r98766;
        double r98771 = -6.216717371626989e-264;
        bool r98772 = r98766 <= r98771;
        double r98773 = r98766 * r98766;
        double r98774 = im;
        double r98775 = r98774 * r98774;
        double r98776 = r98773 + r98775;
        double r98777 = sqrt(r98776);
        double r98778 = 2.9506572857247177e-235;
        bool r98779 = r98766 <= r98778;
        double r98780 = 2.9712800106749567e+26;
        bool r98781 = r98766 <= r98780;
        double r98782 = r98781 ? r98777 : r98766;
        double r98783 = r98779 ? r98774 : r98782;
        double r98784 = r98772 ? r98777 : r98783;
        double r98785 = r98768 ? r98770 : r98784;
        return r98785;
}

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 < -5.810291952691062e+115

    1. Initial program 54.1

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

    2. Taylor expanded around -inf 9.4

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

    if -5.810291952691062e+115 < re < -6.216717371626989e-264 or 2.9506572857247177e-235 < re < 2.9712800106749567e+26

    1. Initial program 20.2

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

    if -6.216717371626989e-264 < re < 2.9506572857247177e-235

    1. Initial program 29.6

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

    2. Taylor expanded around 0 32.4

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

    if 2.9712800106749567e+26 < re

    1. Initial program 42.2

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

    2. Taylor expanded around inf 13.3

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

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.810291952691062 \cdot 10^{115}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -6.2167173716269886 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.95065728572471767 \cdot 10^{-235}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.97128001067495674 \cdot 10^{26}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))