Average Error: 29.9 → 16.6
Time: 2.7s
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 r962289 = re;
        double r962290 = r962289 * r962289;
        double r962291 = im;
        double r962292 = r962291 * r962291;
        double r962293 = r962290 + r962292;
        double r962294 = sqrt(r962293);
        return r962294;
}
↓
double f(double re, double im) {
        double r962295 = re;
        double r962296 = -6.1114247610392124e+153;
        bool r962297 = r962295 <= r962296;
        double r962298 = -r962295;
        double r962299 = 5.284608256973942e-233;
        bool r962300 = r962295 <= r962299;
        double r962301 = im;
        double r962302 = r962301 * r962301;
        double r962303 = r962295 * r962295;
        double r962304 = r962302 + r962303;
        double r962305 = sqrt(r962304);
        double r962306 = 5.785268620035206e-190;
        bool r962307 = r962295 <= r962306;
        double r962308 = 1.4702178548627831e+140;
        bool r962309 = r962295 <= r962308;
        double r962310 = r962309 ? r962305 : r962295;
        double r962311 = r962307 ? r962301 : r962310;
        double r962312 = r962300 ? r962305 : r962311;
        double r962313 = r962297 ? r962298 : r962312;
        return r962313;
}

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))))