Average Error: 31.7 → 17.9
Time: 3.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.491740461538726219683531543082814061521 \cdot 10^{111}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.342031878168628069071012321852054396987 \cdot 10^{75}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.491740461538726219683531543082814061521 \cdot 10^{111}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le 4.342031878168628069071012321852054396987 \cdot 10^{75}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r42049 = re;
        double r42050 = r42049 * r42049;
        double r42051 = im;
        double r42052 = r42051 * r42051;
        double r42053 = r42050 + r42052;
        double r42054 = sqrt(r42053);
        return r42054;
}
↓
double f(double re, double im) {
        double r42055 = re;
        double r42056 = -1.4917404615387262e+111;
        bool r42057 = r42055 <= r42056;
        double r42058 = -1.0;
        double r42059 = r42058 * r42055;
        double r42060 = 4.342031878168628e+75;
        bool r42061 = r42055 <= r42060;
        double r42062 = r42055 * r42055;
        double r42063 = im;
        double r42064 = r42063 * r42063;
        double r42065 = r42062 + r42064;
        double r42066 = sqrt(r42065);
        double r42067 = r42061 ? r42066 : r42055;
        double r42068 = r42057 ? r42059 : r42067;
        return r42068;
}

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 3 regimes

  2. if re < -1.4917404615387262e+111

    1. Initial program 53.9

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

    2. Taylor expanded around -inf 9.8

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

    if -1.4917404615387262e+111 < re < 4.342031878168628e+75

    1. Initial program 21.7

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

    if 4.342031878168628e+75 < re

    1. Initial program 48.0

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

    2. Taylor expanded around inf 11.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.491740461538726219683531543082814061521 \cdot 10^{111}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le 4.342031878168628069071012321852054396987 \cdot 10^{75}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))