Average Error: 29.9 → 16.6
Time: 2.9s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.1114247610392124 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.284608256973942 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.785268620035206 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4702178548627831 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.1114247610392124 \cdot 10^{+153}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 5.284608256973942 \cdot 10^{-233}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 5.785268620035206 \cdot 10^{-190}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.4702178548627831 \cdot 10^{+140}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r801894 = re;
        double r801895 = r801894 * r801894;
        double r801896 = im;
        double r801897 = r801896 * r801896;
        double r801898 = r801895 + r801897;
        double r801899 = sqrt(r801898);
        return r801899;
}
↓
double f(double re, double im) {
        double r801900 = re;
        double r801901 = -6.1114247610392124e+153;
        bool r801902 = r801900 <= r801901;
        double r801903 = -r801900;
        double r801904 = 5.284608256973942e-233;
        bool r801905 = r801900 <= r801904;
        double r801906 = im;
        double r801907 = r801906 * r801906;
        double r801908 = r801900 * r801900;
        double r801909 = r801907 + r801908;
        double r801910 = sqrt(r801909);
        double r801911 = 5.785268620035206e-190;
        bool r801912 = r801900 <= r801911;
        double r801913 = 1.4702178548627831e+140;
        bool r801914 = r801900 <= r801913;
        double r801915 = r801914 ? r801910 : r801900;
        double r801916 = r801912 ? r801906 : r801915;
        double r801917 = r801905 ? r801910 : r801916;
        double r801918 = r801902 ? r801903 : r801917;
        return r801918;
}

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 < -6.1114247610392124e+153

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 6.8

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

    3. Simplified6.8

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

    if -6.1114247610392124e+153 < re < 5.284608256973942e-233 or 5.785268620035206e-190 < re < 1.4702178548627831e+140

    1. Initial program 19.1

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

    if 5.284608256973942e-233 < re < 5.785268620035206e-190

    1. Initial program 32.7

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

    2. Taylor expanded around 0 32.3

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

    if 1.4702178548627831e+140 < re

    1. Initial program 56.3

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

    2. Taylor expanded around inf 9.1

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.1114247610392124 \cdot 10^{+153}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 5.284608256973942 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 5.785268620035206 \cdot 10^{-190}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4702178548627831 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))