Average Error: 29.5 → 17.0
Time: 3.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2808585694765927 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.07576562131624 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2944248503587878 \cdot 10^{-158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.940868728015608 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2808585694765927 \cdot 10^{+154}:\\
\;\;\;\;-re\\

\mathbf{elif}\;re \le 7.07576562131624 \cdot 10^{-253}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

\mathbf{elif}\;re \le 1.2944248503587878 \cdot 10^{-158}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 6.940868728015608 \cdot 10^{+153}:\\
\;\;\;\;\sqrt{im \cdot im + re \cdot re}\\

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

\end{array}
double f(double re, double im) {
        double r3504657 = re;
        double r3504658 = r3504657 * r3504657;
        double r3504659 = im;
        double r3504660 = r3504659 * r3504659;
        double r3504661 = r3504658 + r3504660;
        double r3504662 = sqrt(r3504661);
        return r3504662;
}
↓
double f(double re, double im) {
        double r3504663 = re;
        double r3504664 = -1.2808585694765927e+154;
        bool r3504665 = r3504663 <= r3504664;
        double r3504666 = -r3504663;
        double r3504667 = 7.07576562131624e-253;
        bool r3504668 = r3504663 <= r3504667;
        double r3504669 = im;
        double r3504670 = r3504669 * r3504669;
        double r3504671 = r3504663 * r3504663;
        double r3504672 = r3504670 + r3504671;
        double r3504673 = sqrt(r3504672);
        double r3504674 = 1.2944248503587878e-158;
        bool r3504675 = r3504663 <= r3504674;
        double r3504676 = 6.940868728015608e+153;
        bool r3504677 = r3504663 <= r3504676;
        double r3504678 = r3504677 ? r3504673 : r3504663;
        double r3504679 = r3504675 ? r3504669 : r3504678;
        double r3504680 = r3504668 ? r3504673 : r3504679;
        double r3504681 = r3504665 ? r3504666 : r3504680;
        return r3504681;
}

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 < -1.2808585694765927e+154

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 7.8

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

    3. Simplified7.8

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

    if -1.2808585694765927e+154 < re < 7.07576562131624e-253 or 1.2944248503587878e-158 < re < 6.940868728015608e+153

    1. Initial program 18.1

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

    if 7.07576562131624e-253 < re < 1.2944248503587878e-158

    1. Initial program 29.2

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

    2. Taylor expanded around 0 36.2

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

    if 6.940868728015608e+153 < re

    1. Initial program 59.3

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2808585694765927 \cdot 10^{+154}:\\ \;\;\;\;-re\\ \mathbf{elif}\;re \le 7.07576562131624 \cdot 10^{-253}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{elif}\;re \le 1.2944248503587878 \cdot 10^{-158}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 6.940868728015608 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{im \cdot im + re \cdot re}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))