Average Error: 31.6 → 17.5
Time: 3.8s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.229656712329738935737994487338299567174 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -5.08697951464219784890620928937007113919 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.196597309232864908492796515168661297752 \cdot 10^{-301}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.828273761560498175312293875676889182986 \cdot 10^{75}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.229656712329738935737994487338299567174 \cdot 10^{112}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -5.08697951464219784890620928937007113919 \cdot 10^{-205}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -2.196597309232864908492796515168661297752 \cdot 10^{-301}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.828273761560498175312293875676889182986 \cdot 10^{75}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r32241 = re;
        double r32242 = r32241 * r32241;
        double r32243 = im;
        double r32244 = r32243 * r32243;
        double r32245 = r32242 + r32244;
        double r32246 = sqrt(r32245);
        return r32246;
}
↓
double f(double re, double im) {
        double r32247 = re;
        double r32248 = -3.229656712329739e+112;
        bool r32249 = r32247 <= r32248;
        double r32250 = -1.0;
        double r32251 = r32250 * r32247;
        double r32252 = -5.086979514642198e-205;
        bool r32253 = r32247 <= r32252;
        double r32254 = r32247 * r32247;
        double r32255 = im;
        double r32256 = r32255 * r32255;
        double r32257 = r32254 + r32256;
        double r32258 = sqrt(r32257);
        double r32259 = -2.196597309232865e-301;
        bool r32260 = r32247 <= r32259;
        double r32261 = 2.828273761560498e+75;
        bool r32262 = r32247 <= r32261;
        double r32263 = r32262 ? r32258 : r32247;
        double r32264 = r32260 ? r32255 : r32263;
        double r32265 = r32253 ? r32258 : r32264;
        double r32266 = r32249 ? r32251 : r32265;
        return r32266;
}

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 < -3.229656712329739e+112

    1. Initial program 54.5

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

    2. Taylor expanded around -inf 9.7

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

    if -3.229656712329739e+112 < re < -5.086979514642198e-205 or -2.196597309232865e-301 < re < 2.828273761560498e+75

    1. Initial program 19.8

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

    if -5.086979514642198e-205 < re < -2.196597309232865e-301

    1. Initial program 30.5

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

    2. Taylor expanded around 0 34.4

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

    if 2.828273761560498e+75 < re

    1. Initial program 47.2

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

    2. Taylor expanded around inf 10.7

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.229656712329738935737994487338299567174 \cdot 10^{112}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -5.08697951464219784890620928937007113919 \cdot 10^{-205}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -2.196597309232864908492796515168661297752 \cdot 10^{-301}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.828273761560498175312293875676889182986 \cdot 10^{75}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))