Average Error: 31.7 → 18.2
Time: 979.0ms
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.561866723012927 \cdot 10^{-267}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.10774424975994648 \cdot 10^{54}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le -3.561866723012927 \cdot 10^{-267}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.10774424975994648 \cdot 10^{54}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r27787 = re;
        double r27788 = r27787 * r27787;
        double r27789 = im;
        double r27790 = r27789 * r27789;
        double r27791 = r27788 + r27790;
        double r27792 = sqrt(r27791);
        return r27792;
}
↓
double f(double re, double im) {
        double r27793 = re;
        double r27794 = -8.76230374871451e+143;
        bool r27795 = r27793 <= r27794;
        double r27796 = -1.0;
        double r27797 = r27796 * r27793;
        double r27798 = -1.180833083521909e-161;
        bool r27799 = r27793 <= r27798;
        double r27800 = r27793 * r27793;
        double r27801 = im;
        double r27802 = r27801 * r27801;
        double r27803 = r27800 + r27802;
        double r27804 = sqrt(r27803);
        double r27805 = -3.561866723012927e-267;
        bool r27806 = r27793 <= r27805;
        double r27807 = 1.1077442497599465e+54;
        bool r27808 = r27793 <= r27807;
        double r27809 = r27808 ? r27804 : r27793;
        double r27810 = r27806 ? r27801 : r27809;
        double r27811 = r27799 ? r27804 : r27810;
        double r27812 = r27795 ? r27797 : r27811;
        return r27812;
}

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 < -8.76230374871451e+143

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 7.9

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

    if -8.76230374871451e+143 < re < -1.180833083521909e-161 or -3.561866723012927e-267 < re < 1.1077442497599465e+54

    1. Initial program 20.3

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

    if -1.180833083521909e-161 < re < -3.561866723012927e-267

    1. Initial program 31.5

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

    2. Taylor expanded around 0 35.0

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

    if 1.1077442497599465e+54 < re

    1. Initial program 43.8

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

    2. Taylor expanded around inf 12.5

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.7623037487145096 \cdot 10^{143}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -1.1808330835219091 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le -3.561866723012927 \cdot 10^{-267}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.10774424975994648 \cdot 10^{54}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))