Average Error: 31.8 → 18.0
Time: 1.6s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r52706 = re;
        double r52707 = r52706 * r52706;
        double r52708 = im;
        double r52709 = r52708 * r52708;
        double r52710 = r52707 + r52709;
        double r52711 = sqrt(r52710);
        return r52711;
}
↓
double f(double re, double im) {
        double r52712 = re;
        double r52713 = -3.0018675033187654e+143;
        bool r52714 = r52712 <= r52713;
        double r52715 = -1.0;
        double r52716 = r52715 * r52712;
        double r52717 = -2.0278572522938575e-184;
        bool r52718 = r52712 <= r52717;
        double r52719 = r52712 * r52712;
        double r52720 = im;
        double r52721 = r52720 * r52720;
        double r52722 = r52719 + r52721;
        double r52723 = sqrt(r52722);
        double r52724 = -9.306218889451781e-290;
        bool r52725 = r52712 <= r52724;
        double r52726 = 1.749481499166478e+126;
        bool r52727 = r52712 <= r52726;
        double r52728 = r52727 ? r52723 : r52712;
        double r52729 = r52725 ? r52720 : r52728;
        double r52730 = r52718 ? r52723 : r52729;
        double r52731 = r52714 ? r52716 : r52730;
        return r52731;
}

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 < -3.0018675033187654e+143

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 8.1

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

    if -3.0018675033187654e+143 < re < -2.0278572522938575e-184 or -9.306218889451781e-290 < re < 1.749481499166478e+126

    1. Initial program 19.5

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

    if -2.0278572522938575e-184 < re < -9.306218889451781e-290

    1. Initial program 30.6

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

    2. Taylor expanded around 0 36.6

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

    if 1.749481499166478e+126 < re

    1. Initial program 56.8

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.00186750331876538 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -2.02785725229385748 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -9.306218889451781 \cdot 10^{-290}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.74948149916647803 \cdot 10^{126}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))