Average Error: 29.6 → 18.2
Time: 3.4s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r1705181 = re;
        double r1705182 = r1705181 * r1705181;
        double r1705183 = im;
        double r1705184 = r1705183 * r1705183;
        double r1705185 = r1705182 + r1705184;
        double r1705186 = sqrt(r1705185);
        return r1705186;
}
↓
double f(double re, double im) {
        double r1705187 = re;
        double r1705188 = -1.2918446586536957e+154;
        bool r1705189 = r1705187 <= r1705188;
        double r1705190 = -r1705187;
        double r1705191 = -2.6219396713989246e+28;
        bool r1705192 = r1705187 <= r1705191;
        double r1705193 = im;
        double r1705194 = r1705193 * r1705193;
        double r1705195 = r1705187 * r1705187;
        double r1705196 = r1705194 + r1705195;
        double r1705197 = sqrt(r1705196);
        double r1705198 = -3.209519593925633e-28;
        bool r1705199 = r1705187 <= r1705198;
        double r1705200 = -8.056228658328031e-199;
        bool r1705201 = r1705187 <= r1705200;
        double r1705202 = -3.759150523562943e-268;
        bool r1705203 = r1705187 <= r1705202;
        double r1705204 = 1.4301923552016937e+155;
        bool r1705205 = r1705187 <= r1705204;
        double r1705206 = r1705205 ? r1705197 : r1705187;
        double r1705207 = r1705203 ? r1705193 : r1705206;
        double r1705208 = r1705201 ? r1705197 : r1705207;
        double r1705209 = r1705199 ? r1705193 : r1705208;
        double r1705210 = r1705192 ? r1705197 : r1705209;
        double r1705211 = r1705189 ? r1705190 : r1705210;
        return r1705211;
}

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 < -1.2918446586536957e+154

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 7.0

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

    3. Simplified7.0

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

    if -1.2918446586536957e+154 < re < -2.6219396713989246e+28 or -3.209519593925633e-28 < re < -8.056228658328031e-199 or -3.759150523562943e-268 < re < 1.4301923552016937e+155

    1. Initial program 19.1

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

    if -2.6219396713989246e+28 < re < -3.209519593925633e-28 or -8.056228658328031e-199 < re < -3.759150523562943e-268

    1. Initial program 23.9

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

    2. Taylor expanded around 0 40.8

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

    if 1.4301923552016937e+155 < re

    1. Initial program 59.5

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2918446586536957 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -2.6219396713989246 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.209519593925633 \cdot 10^{-28}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le -8.056228658328031 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le -3.759150523562943 \cdot 10^{-268}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.4301923552016937 \cdot 10^{+155}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))