Average Error: 31.0 → 17.3
Time: 10.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r98541 = re;
        double r98542 = r98541 * r98541;
        double r98543 = im;
        double r98544 = r98543 * r98543;
        double r98545 = r98542 + r98544;
        double r98546 = sqrt(r98545);
        return r98546;
}
↓
double f(double re, double im) {
        double r98547 = re;
        double r98548 = -6.523076260268755e+150;
        bool r98549 = r98547 <= r98548;
        double r98550 = -r98547;
        double r98551 = 1.158407537644574e+105;
        bool r98552 = r98547 <= r98551;
        double r98553 = r98547 * r98547;
        double r98554 = im;
        double r98555 = r98554 * r98554;
        double r98556 = r98553 + r98555;
        double r98557 = sqrt(r98556);
        double r98558 = r98552 ? r98557 : r98547;
        double r98559 = r98549 ? r98550 : r98558;
        return r98559;
}

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 < -6.523076260268755e+150

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 7.6

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

    3. Simplified7.6

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

    if -6.523076260268755e+150 < re < 1.158407537644574e+105

    1. Initial program 20.8

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

    if 1.158407537644574e+105 < re

    1. Initial program 51.1

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

    2. Taylor expanded around inf 9.5

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.52307626026875473 \cdot 10^{150}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 1.15840753764457407 \cdot 10^{105}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))