Average Error: 31.4 → 19.0
Time: 4.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r29086 = re;
        double r29087 = r29086 * r29086;
        double r29088 = im;
        double r29089 = r29088 * r29088;
        double r29090 = r29087 + r29089;
        double r29091 = sqrt(r29090);
        return r29091;
}
↓
double f(double re, double im) {
        double r29092 = re;
        double r29093 = -5.330091552844717e+114;
        bool r29094 = r29092 <= r29093;
        double r29095 = -r29092;
        double r29096 = -4.2156616274993736e-144;
        bool r29097 = r29092 <= r29096;
        double r29098 = r29092 * r29092;
        double r29099 = im;
        double r29100 = r29099 * r29099;
        double r29101 = r29098 + r29100;
        double r29102 = sqrt(r29101);
        double r29103 = 1.0597832414692678e-253;
        bool r29104 = r29092 <= r29103;
        double r29105 = 3.0122240909363507e+56;
        bool r29106 = r29092 <= r29105;
        double r29107 = r29106 ? r29102 : r29092;
        double r29108 = r29104 ? r29099 : r29107;
        double r29109 = r29097 ? r29102 : r29108;
        double r29110 = r29094 ? r29095 : r29109;
        return r29110;
}

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 < -5.330091552844717e+114

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -5.330091552844717e+114 < re < -4.2156616274993736e-144 or 1.0597832414692678e-253 < re < 3.0122240909363507e+56

    1. Initial program 18.7

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

    if -4.2156616274993736e-144 < re < 1.0597832414692678e-253

    1. Initial program 30.2

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

    2. Taylor expanded around 0 35.6

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

    if 3.0122240909363507e+56 < re

    1. Initial program 44.3

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

    2. Taylor expanded around inf 12.9

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

  4. Final simplification19.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.330091552844717472226479932066920744645 \cdot 10^{114}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -4.215661627499373563855656419004671791113 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.05978324146926776621503694441833231193 \cdot 10^{-253}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 3.012224090936350650107808168583637972217 \cdot 10^{56}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))