Average Error: 31.1 → 18.9
Time: 1.1s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\
\;\;\;\;im\\

\mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

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

\end{array}
double f(double re, double im) {
        double r45040 = re;
        double r45041 = r45040 * r45040;
        double r45042 = im;
        double r45043 = r45042 * r45042;
        double r45044 = r45041 + r45043;
        double r45045 = sqrt(r45044);
        return r45045;
}
↓
double f(double re, double im) {
        double r45046 = re;
        double r45047 = -8.15615961666859e+125;
        bool r45048 = r45046 <= r45047;
        double r45049 = -1.0;
        double r45050 = r45049 * r45046;
        double r45051 = -3.8099669373079583e-103;
        bool r45052 = r45046 <= r45051;
        double r45053 = r45046 * r45046;
        double r45054 = im;
        double r45055 = r45054 * r45054;
        double r45056 = r45053 + r45055;
        double r45057 = sqrt(r45056);
        double r45058 = 2.3367351856997066e-296;
        bool r45059 = r45046 <= r45058;
        double r45060 = 1.0193327448038136e+95;
        bool r45061 = r45046 <= r45060;
        double r45062 = r45061 ? r45057 : r45046;
        double r45063 = r45059 ? r45054 : r45062;
        double r45064 = r45052 ? r45057 : r45063;
        double r45065 = r45048 ? r45050 : r45064;
        return r45065;
}

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 < -8.15615961666859e+125

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 9.1

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

    if -8.15615961666859e+125 < re < -3.8099669373079583e-103 or 2.3367351856997066e-296 < re < 1.0193327448038136e+95

    1. Initial program 18.9

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

    if -3.8099669373079583e-103 < re < 2.3367351856997066e-296

    1. Initial program 27.8

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

    2. Taylor expanded around 0 36.1

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

    if 1.0193327448038136e+95 < re

    1. Initial program 50.2

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

    2. Taylor expanded around inf 10.4

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

  4. Final simplification18.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.1561596166685901 \cdot 10^{125}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;re \le -3.80996693730795831 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;re \le 2.33673518569970664 \cdot 10^{-296}:\\ \;\;\;\;im\\ \mathbf{elif}\;re \le 1.0193327448038136 \cdot 10^{95}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;re\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))