Average Error: 31.7 → 18.1
Time: 1.7s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]
\sqrt{re \cdot re + im \cdot im}
↓
\begin{array}{l}
\mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\
\;\;\;\;-1 \cdot im\\

\mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\
\;\;\;\;re\\

\mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\
\;\;\;\;re\\

\mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\
\;\;\;\;-1 \cdot re\\

\mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\
\;\;\;\;\sqrt{re \cdot re + im \cdot im}\\

\mathbf{else}:\\
\;\;\;\;im\\

\end{array}
double f(double re, double im) {
        double r62877 = re;
        double r62878 = r62877 * r62877;
        double r62879 = im;
        double r62880 = r62879 * r62879;
        double r62881 = r62878 + r62880;
        double r62882 = sqrt(r62881);
        return r62882;
}
↓
double f(double re, double im) {
        double r62883 = im;
        double r62884 = -3.436284107232327e+150;
        bool r62885 = r62883 <= r62884;
        double r62886 = -1.0;
        double r62887 = r62886 * r62883;
        double r62888 = -1.7874287404230692e-225;
        bool r62889 = r62883 <= r62888;
        double r62890 = re;
        double r62891 = r62890 * r62890;
        double r62892 = r62883 * r62883;
        double r62893 = r62891 + r62892;
        double r62894 = sqrt(r62893);
        double r62895 = 5.899377614408183e-308;
        bool r62896 = r62883 <= r62895;
        double r62897 = 2.4345784437110915e-199;
        bool r62898 = r62883 <= r62897;
        double r62899 = r62886 * r62890;
        double r62900 = 1.7970794289179904e-166;
        bool r62901 = r62883 <= r62900;
        double r62902 = 1.0688990210562475e-162;
        bool r62903 = r62883 <= r62902;
        double r62904 = 2.2340209789651782e+109;
        bool r62905 = r62883 <= r62904;
        double r62906 = r62905 ? r62894 : r62883;
        double r62907 = r62903 ? r62899 : r62906;
        double r62908 = r62901 ? r62890 : r62907;
        double r62909 = r62898 ? r62899 : r62908;
        double r62910 = r62896 ? r62890 : r62909;
        double r62911 = r62889 ? r62894 : r62910;
        double r62912 = r62885 ? r62887 : r62911;
        return r62912;
}

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 5 regimes

  2. if im < -3.436284107232327e+150

    1. Initial program 62.6

      \[\sqrt{re \cdot re + im \cdot im}\]
    2. Using strategy rm

    3. Applied flip-+64.0

      \[\leadsto \sqrt{\color{blue}{\frac{\left(re \cdot re\right) \cdot \left(re \cdot re\right) - \left(im \cdot im\right) \cdot \left(im \cdot im\right)}{re \cdot re - im \cdot im}}}\]

    4. Simplified64.0

      \[\leadsto \sqrt{\frac{\color{blue}{\left(-{im}^{3}\right) \cdot im + {re}^{4}}}{re \cdot re - im \cdot im}}\]

    5. Taylor expanded around -inf 8.8

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

    if -3.436284107232327e+150 < im < -1.7874287404230692e-225 or 1.0688990210562475e-162 < im < 2.2340209789651782e+109

    1. Initial program 17.2

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

    if -1.7874287404230692e-225 < im < 5.899377614408183e-308 or 2.4345784437110915e-199 < im < 1.7970794289179904e-166

    1. Initial program 31.5

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

    2. Taylor expanded around inf 33.7

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

    if 5.899377614408183e-308 < im < 2.4345784437110915e-199 or 1.7970794289179904e-166 < im < 1.0688990210562475e-162

    1. Initial program 31.3

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

    2. Taylor expanded around -inf 35.2

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

    if 2.2340209789651782e+109 < im

    1. Initial program 53.4

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

    2. Taylor expanded around 0 9.8

      \[\leadsto \color{blue}{im}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.4362841072323272 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot im\\ \mathbf{elif}\;im \le -1.7874287404230692 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{elif}\;im \le 5.8993776144081826 \cdot 10^{-308}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 2.4345784437110915 \cdot 10^{-199}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 1.7970794289179904 \cdot 10^{-166}:\\ \;\;\;\;re\\ \mathbf{elif}\;im \le 1.0688990210562475 \cdot 10^{-162}:\\ \;\;\;\;-1 \cdot re\\ \mathbf{elif}\;im \le 2.23402097896517821 \cdot 10^{109}:\\ \;\;\;\;\sqrt{re \cdot re + im \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (re im)
  :name "math.abs on complex"
  :precision binary64
  (sqrt (+ (* re re) (* im im))))