Average Error: 31.6 → 17.7
Time: 1.0s
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 r56312 = re;
        double r56313 = r56312 * r56312;
        double r56314 = im;
        double r56315 = r56314 * r56314;
        double r56316 = r56313 + r56315;
        double r56317 = sqrt(r56316);
        return r56317;
}
↓
double f(double re, double im) {
        double r56318 = re;
        double r56319 = -6.754060706975556e+99;
        bool r56320 = r56318 <= r56319;
        double r56321 = -1.0;
        double r56322 = r56321 * r56318;
        double r56323 = -1.7256510915817855e-210;
        bool r56324 = r56318 <= r56323;
        double r56325 = r56318 * r56318;
        double r56326 = im;
        double r56327 = r56326 * r56326;
        double r56328 = r56325 + r56327;
        double r56329 = sqrt(r56328);
        double r56330 = 1.3362064325006243e-202;
        bool r56331 = r56318 <= r56330;
        double r56332 = 4.9646550618943354e+123;
        bool r56333 = r56318 <= r56332;
        double r56334 = r56333 ? r56329 : r56318;
        double r56335 = r56331 ? r56326 : r56334;
        double r56336 = r56324 ? r56329 : r56335;
        double r56337 = r56320 ? r56322 : r56336;
        return r56337;
}

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))))