Average Error: 25.8 → 25.8
Time: 9.7s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r115164 = x_re;
        double r115165 = y_re;
        double r115166 = r115164 * r115165;
        double r115167 = x_im;
        double r115168 = y_im;
        double r115169 = r115167 * r115168;
        double r115170 = r115166 + r115169;
        double r115171 = r115165 * r115165;
        double r115172 = r115168 * r115168;
        double r115173 = r115171 + r115172;
        double r115174 = r115170 / r115173;
        return r115174;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r115175 = x_re;
        double r115176 = y_re;
        double r115177 = r115175 * r115176;
        double r115178 = x_im;
        double r115179 = y_im;
        double r115180 = r115178 * r115179;
        double r115181 = r115177 + r115180;
        double r115182 = r115176 * r115176;
        double r115183 = r115179 * r115179;
        double r115184 = r115182 + r115183;
        double r115185 = r115181 / r115184;
        double r115186 = 3.723435793259181e+232;
        bool r115187 = r115185 <= r115186;
        double r115188 = 1.0;
        double r115189 = sqrt(r115184);
        double r115190 = r115179 * r115178;
        double r115191 = r115176 * r115175;
        double r115192 = r115190 + r115191;
        double r115193 = r115189 / r115192;
        double r115194 = r115188 / r115193;
        double r115195 = r115194 / r115189;
        double r115196 = -r115175;
        double r115197 = r115196 / r115189;
        double r115198 = r115187 ? r115195 : r115197;
        return r115198;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < 3.723435793259181e+232

    1. Initial program 13.9

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt13.9

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*13.8

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
    5. Using strategy rm

    6. Applied clear-num13.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{x.re \cdot y.re + x.im \cdot y.im}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    7. Simplified13.8

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    if 3.723435793259181e+232 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 60.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt60.0

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*60.0

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    5. Taylor expanded around -inf 59.9

      \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    6. Simplified59.9

      \[\leadsto \frac{\color{blue}{-x.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))