Average Error: 31.3 → 17.2
Time: 7.0s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r2946746 = re;
        double r2946747 = r2946746 * r2946746;
        double r2946748 = im;
        double r2946749 = r2946748 * r2946748;
        double r2946750 = r2946747 + r2946749;
        double r2946751 = sqrt(r2946750);
        return r2946751;
}
↓
double f(double re, double im) {
        double r2946752 = re;
        double r2946753 = -1.1817931832138217e+151;
        bool r2946754 = r2946752 <= r2946753;
        double r2946755 = -r2946752;
        double r2946756 = 5.948234035126459e+127;
        bool r2946757 = r2946752 <= r2946756;
        double r2946758 = r2946752 * r2946752;
        double r2946759 = im;
        double r2946760 = r2946759 * r2946759;
        double r2946761 = r2946758 + r2946760;
        double r2946762 = sqrt(r2946761);
        double r2946763 = r2946757 ? r2946762 : r2946752;
        double r2946764 = r2946754 ? r2946755 : r2946763;
        return r2946764;
}

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 < -1.1817931832138217e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -1.1817931832138217e+151 < re < 5.948234035126459e+127

    1. Initial program 20.4

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

    if 5.948234035126459e+127 < re

    1. Initial program 56.6

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

    2. Taylor expanded around inf 9.2

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))