Average Error: 31.6 → 17.7
Time: 1.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.725651091581785482443039514213583051037 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.336206432500624299310601884853873391011 \cdot 10^{-202}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.964655061894335407151343370923293101746 \cdot 10^{123}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.725651091581785482443039514213583051037 \cdot 10^{-210}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 1.336206432500624299310601884853873391011 \cdot 10^{-202}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 4.964655061894335407151343370923293101746 \cdot 10^{123}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r87179 = re;
        double r87180 = r87179 * r87179;
        double r87181 = im;
        double r87182 = r87181 * r87181;
        double r87183 = r87180 + r87182;
        double r87184 = sqrt(r87183);
        return r87184;
}
↓
double f(double re, double im) {
        double r87185 = re;
        double r87186 = -6.754060706975556e+99;
        bool r87187 = r87185 <= r87186;
        double r87188 = -1.0;
        double r87189 = r87188 * r87185;
        double r87190 = -1.7256510915817855e-210;
        bool r87191 = r87185 <= r87190;
        double r87192 = r87185 * r87185;
        double r87193 = im;
        double r87194 = r87193 * r87193;
        double r87195 = r87192 + r87194;
        double r87196 = sqrt(r87195);
        double r87197 = 1.3362064325006243e-202;
        bool r87198 = r87185 <= r87197;
        double r87199 = 4.9646550618943354e+123;
        bool r87200 = r87185 <= r87199;
        double r87201 = r87200 ? r87196 : r87185;
        double r87202 = r87198 ? r87193 : r87201;
        double r87203 = r87191 ? r87196 : r87202;
        double r87204 = r87187 ? r87189 : r87203;
        return r87204;
}

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 < -6.754060706975556e+99

    1. Initial program 50.7

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

    2. Taylor expanded around -inf 10.4

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

    if -6.754060706975556e+99 < re < -1.7256510915817855e-210 or 1.3362064325006243e-202 < re < 4.9646550618943354e+123

    1. Initial program 17.6

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

    if -1.7256510915817855e-210 < re < 1.3362064325006243e-202

    1. Initial program 31.1

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

    2. Taylor expanded around 0 33.2

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

    if 4.9646550618943354e+123 < re

    1. Initial program 56.4

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

    2. Taylor expanded around inf 9.3

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.725651091581785482443039514213583051037 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 1.336206432500624299310601884853873391011 \cdot 10^{-202}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 4.964655061894335407151343370923293101746 \cdot 10^{123}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))