Average Error: 31.3 → 17.2
Time: 6.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.94823403512645934829207680164770844431 \cdot 10^{127}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\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{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1694356 = re;
        double r1694357 = r1694356 * r1694356;
        double r1694358 = im;
        double r1694359 = r1694358 * r1694358;
        double r1694360 = r1694357 + r1694359;
        double r1694361 = sqrt(r1694360);
        return r1694361;
}
↓
double f(double re, double im) {
        double r1694362 = re;
        double r1694363 = -1.1817931832138217e+151;
        bool r1694364 = r1694362 <= r1694363;
        double r1694365 = -r1694362;
        double r1694366 = 5.948234035126459e+127;
        bool r1694367 = r1694362 <= r1694366;
        double r1694368 = im;
        double r1694369 = r1694368 * r1694368;
        double r1694370 = r1694362 * r1694362;
        double r1694371 = r1694369 + r1694370;
        double r1694372 = sqrt(r1694371);
        double r1694373 = r1694367 ? r1694372 : r1694362;
        double r1694374 = r1694364 ? r1694365 : r1694373;
        return r1694374;
}

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

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

Reproduce

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