Average Error: 31.9 → 17.7
Time: 12.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r72227 = re;
        double r72228 = r72227 * r72227;
        double r72229 = im;
        double r72230 = r72229 * r72229;
        double r72231 = r72228 + r72230;
        double r72232 = sqrt(r72231);
        return r72232;
}
↓
double f(double re, double im) {
        double r72233 = re;
        double r72234 = -1.2876568362185878e+137;
        bool r72235 = r72233 <= r72234;
        double r72236 = -r72233;
        double r72237 = 2.7153468834491098e+73;
        bool r72238 = r72233 <= r72237;
        double r72239 = r72233 * r72233;
        double r72240 = im;
        double r72241 = r72240 * r72240;
        double r72242 = r72239 + r72241;
        double r72243 = sqrt(r72242);
        double r72244 = r72238 ? r72243 : r72233;
        double r72245 = r72235 ? r72236 : r72244;
        return r72245;
}

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 < -1.2876568362185878e+137

    1. Initial program 59.8

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    if -1.2876568362185878e+137 < re < 2.7153468834491098e+73

    1. Initial program 21.5

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

    if 2.7153468834491098e+73 < re

    1. Initial program 47.0

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.287656836218587817843721098850935729447 \cdot 10^{137}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 2.715346883449109812449415853977495365892 \cdot 10^{73}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))