Average Error: 31.8 → 17.9
Time: 9.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.391819308233852128501761539035118741121 \cdot 10^{130}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.082781542809915013551950598635310407664 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.350068654779021630227434578668149263935 \cdot 10^{-288}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.391819308233852128501761539035118741121 \cdot 10^{130}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le -1.082781542809915013551950598635310407664 \cdot 10^{-198}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -1.350068654779021630227434578668149263935 \cdot 10^{-288}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r43790 = re;
        double r43791 = r43790 * r43790;
        double r43792 = im;
        double r43793 = r43792 * r43792;
        double r43794 = r43791 + r43793;
        double r43795 = sqrt(r43794);
        return r43795;
}
↓
double f(double re, double im) {
        double r43796 = re;
        double r43797 = -4.391819308233852e+130;
        bool r43798 = r43796 <= r43797;
        double r43799 = -r43796;
        double r43800 = -1.082781542809915e-198;
        bool r43801 = r43796 <= r43800;
        double r43802 = r43796 * r43796;
        double r43803 = im;
        double r43804 = r43803 * r43803;
        double r43805 = r43802 + r43804;
        double r43806 = sqrt(r43805);
        double r43807 = -1.3500686547790216e-288;
        bool r43808 = r43796 <= r43807;
        double r43809 = 2.076008095509726e+76;
        bool r43810 = r43796 <= r43809;
        double r43811 = r43810 ? r43806 : r43796;
        double r43812 = r43808 ? r43803 : r43811;
        double r43813 = r43801 ? r43806 : r43812;
        double r43814 = r43798 ? r43799 : r43813;
        return r43814;
}

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 < -4.391819308233852e+130

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 8.7

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

    3. Simplified8.7

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

    if -4.391819308233852e+130 < re < -1.082781542809915e-198 or -1.3500686547790216e-288 < re < 2.076008095509726e+76

    1. Initial program 20.3

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

    if -1.082781542809915e-198 < re < -1.3500686547790216e-288

    1. Initial program 29.0

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

    2. Taylor expanded around 0 32.4

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

    if 2.076008095509726e+76 < re

    1. Initial program 48.1

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

    2. Taylor expanded around inf 12.0

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.391819308233852128501761539035118741121 \cdot 10^{130}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le -1.082781542809915013551950598635310407664 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -1.350068654779021630227434578668149263935 \cdot 10^{-288}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))