Average Error: 31.9 → 17.5
Time: 2.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.284483376836890693595922335560233872799 \cdot 10^{136}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 9.242348626981343105629292534441998064595 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.284483376836890693595922335560233872799 \cdot 10^{136}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 9.242348626981343105629292534441998064595 \cdot 10^{127}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r41254 = re;
        double r41255 = r41254 * r41254;
        double r41256 = im;
        double r41257 = r41256 * r41256;
        double r41258 = r41255 + r41257;
        double r41259 = sqrt(r41258);
        return r41259;
}
↓
double f(double re, double im) {
        double r41260 = re;
        double r41261 = -1.2844833768368907e+136;
        bool r41262 = r41260 <= r41261;
        double r41263 = -r41260;
        double r41264 = 9.242348626981343e+127;
        bool r41265 = r41260 <= r41264;
        double r41266 = r41260 * r41260;
        double r41267 = im;
        double r41268 = r41267 * r41267;
        double r41269 = r41266 + r41268;
        double r41270 = sqrt(r41269);
        double r41271 = r41265 ? r41270 : r41260;
        double r41272 = r41262 ? r41263 : r41271;
        return r41272;
}

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 < -1.2844833768368907e+136

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 9.0

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

    3. Simplified9.0

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

    if -1.2844833768368907e+136 < re < 9.242348626981343e+127

    1. Initial program 21.1

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

    if 9.242348626981343e+127 < re

    1. Initial program 57.1

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

    2. Taylor expanded around inf 8.6

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.284483376836890693595922335560233872799 \cdot 10^{136}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 9.242348626981343105629292534441998064595 \cdot 10^{127}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))