Average Error: 29.4 → 16.6
Time: 6.5s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2456535811590808 \cdot 10^{-300}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.8099256402278783 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.721648594770613 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.4073759586339516 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.2456535811590808 \cdot 10^{-300}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.8099256402278783 \cdot 10^{-246}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 5.721648594770613 \cdot 10^{-217}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.4073759586339516 \cdot 10^{+149}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1925141 = re;
        double r1925142 = r1925141 * r1925141;
        double r1925143 = im;
        double r1925144 = r1925143 * r1925143;
        double r1925145 = r1925142 + r1925144;
        double r1925146 = sqrt(r1925145);
        return r1925146;
}
↓
double f(double re, double im) {
        double r1925147 = re;
        double r1925148 = -1.1292868428778451e+139;
        bool r1925149 = r1925147 <= r1925148;
        double r1925150 = -r1925147;
        double r1925151 = -1.0853955874561044e-276;
        bool r1925152 = r1925147 <= r1925151;
        double r1925153 = im;
        double r1925154 = r1925153 * r1925153;
        double r1925155 = r1925147 * r1925147;
        double r1925156 = r1925154 + r1925155;
        double r1925157 = sqrt(r1925156);
        double r1925158 = 1.2456535811590808e-300;
        bool r1925159 = r1925147 <= r1925158;
        double r1925160 = 1.8099256402278783e-246;
        bool r1925161 = r1925147 <= r1925160;
        double r1925162 = 5.721648594770613e-217;
        bool r1925163 = r1925147 <= r1925162;
        double r1925164 = 2.4073759586339516e+149;
        bool r1925165 = r1925147 <= r1925164;
        double r1925166 = r1925165 ? r1925157 : r1925147;
        double r1925167 = r1925163 ? r1925153 : r1925166;
        double r1925168 = r1925161 ? r1925157 : r1925167;
        double r1925169 = r1925159 ? r1925153 : r1925168;
        double r1925170 = r1925152 ? r1925157 : r1925169;
        double r1925171 = r1925149 ? r1925150 : r1925170;
        return r1925171;
}

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 < -1.1292868428778451e+139

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified8.5

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

    if -1.1292868428778451e+139 < re < -1.0853955874561044e-276 or 1.2456535811590808e-300 < re < 1.8099256402278783e-246 or 5.721648594770613e-217 < re < 2.4073759586339516e+149

    1. Initial program 18.8

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

    if -1.0853955874561044e-276 < re < 1.2456535811590808e-300 or 1.8099256402278783e-246 < re < 5.721648594770613e-217

    1. Initial program 32.1

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

    2. Taylor expanded around 0 32.9

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

    if 2.4073759586339516e+149 < re

    1. Initial program 57.9

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

    2. Taylor expanded around inf 6.8

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2456535811590808 \cdot 10^{-300}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.8099256402278783 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.721648594770613 \cdot 10^{-217}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.4073759586339516 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))