Average Error: 26.2 → 26.3
Time: 16.2s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 8.391246277230149 \cdot 10^{+244}:\\ \;\;\;\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 8.391246277230149 \cdot 10^{+244}:\\
\;\;\;\;\frac{x.im \cdot y.im + x.re \cdot y.re}{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 r2145144 = x_re;
        double r2145145 = y_re;
        double r2145146 = r2145144 * r2145145;
        double r2145147 = x_im;
        double r2145148 = y_im;
        double r2145149 = r2145147 * r2145148;
        double r2145150 = r2145146 + r2145149;
        double r2145151 = r2145145 * r2145145;
        double r2145152 = r2145148 * r2145148;
        double r2145153 = r2145151 + r2145152;
        double r2145154 = r2145150 / r2145153;
        return r2145154;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2145155 = x_im;
        double r2145156 = y_im;
        double r2145157 = r2145155 * r2145156;
        double r2145158 = x_re;
        double r2145159 = y_re;
        double r2145160 = r2145158 * r2145159;
        double r2145161 = r2145157 + r2145160;
        double r2145162 = r2145159 * r2145159;
        double r2145163 = r2145156 * r2145156;
        double r2145164 = r2145162 + r2145163;
        double r2145165 = r2145161 / r2145164;
        double r2145166 = 8.391246277230149e+244;
        bool r2145167 = r2145165 <= r2145166;
        double r2145168 = -r2145158;
        double r2145169 = sqrt(r2145164);
        double r2145170 = r2145168 / r2145169;
        double r2145171 = r2145167 ? r2145165 : r2145170;
        return r2145171;
}

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))) < 8.391246277230149e+244

    1. Initial program 14.2

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    if 8.391246277230149e+244 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 59.2

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

    3. Applied add-sqr-sqrt59.2

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

    4. Applied associate-/r*59.2

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

    5. Taylor expanded around -inf 59.7

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

    6. Simplified59.7

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 8.391246277230149 \cdot 10^{+244}:\\ \;\;\;\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))