Average Error: 31.9 → 18.9
Time: 1.2s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.88783731726367163 \cdot 10^{72}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 9.88783731726367163 \cdot 10^{72}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r48854 = re;
        double r48855 = r48854 * r48854;
        double r48856 = im;
        double r48857 = r48856 * r48856;
        double r48858 = r48855 + r48857;
        double r48859 = sqrt(r48858);
        return r48859;
}
↓
double f(double re, double im) {
        double r48860 = re;
        double r48861 = -6.884691467287527e+72;
        bool r48862 = r48860 <= r48861;
        double r48863 = -1.0;
        double r48864 = r48863 * r48860;
        double r48865 = -6.1011105261268675e-93;
        bool r48866 = r48860 <= r48865;
        double r48867 = r48860 * r48860;
        double r48868 = im;
        double r48869 = r48868 * r48868;
        double r48870 = r48867 + r48869;
        double r48871 = sqrt(r48870);
        double r48872 = -3.312429260003495e-125;
        bool r48873 = r48860 <= r48872;
        double r48874 = 9.887837317263672e+72;
        bool r48875 = r48860 <= r48874;
        double r48876 = r48875 ? r48871 : r48860;
        double r48877 = r48873 ? r48868 : r48876;
        double r48878 = r48866 ? r48871 : r48877;
        double r48879 = r48862 ? r48864 : r48878;
        return r48879;
}

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 4 regimes

  2. if re < -6.884691467287527e+72

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 12.7

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

    if -6.884691467287527e+72 < re < -6.1011105261268675e-93 or -3.312429260003495e-125 < re < 9.887837317263672e+72

    1. Initial program 22.1

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

    if -6.1011105261268675e-93 < re < -3.312429260003495e-125

    1. Initial program 17.7

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

    2. Taylor expanded around 0 38.9

      \[\leadsto \color{blue}{im}\]

    if 9.887837317263672e+72 < re

    1. Initial program 48.3

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

    2. Taylor expanded around inf 12.6

      \[\leadsto \color{blue}{re}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.8846914672875272 \cdot 10^{72}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -6.10111052612686754 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.3124292600034948 \cdot 10^{-125}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 9.88783731726367163 \cdot 10^{72}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))