Average Error: 26.8 → 26.6
Time: 15.3s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le 1.022096285157088796449452448047526613473 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;y.im \le 1.022096285157088796449452448047526613473 \cdot 10^{92}:\\
\;\;\;\;\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}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\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 r66585 = x_re;
        double r66586 = y_re;
        double r66587 = r66585 * r66586;
        double r66588 = x_im;
        double r66589 = y_im;
        double r66590 = r66588 * r66589;
        double r66591 = r66587 + r66590;
        double r66592 = r66586 * r66586;
        double r66593 = r66589 * r66589;
        double r66594 = r66592 + r66593;
        double r66595 = r66591 / r66594;
        return r66595;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r66596 = y_im;
        double r66597 = 1.0220962851570888e+92;
        bool r66598 = r66596 <= r66597;
        double r66599 = x_re;
        double r66600 = y_re;
        double r66601 = r66599 * r66600;
        double r66602 = x_im;
        double r66603 = r66602 * r66596;
        double r66604 = r66601 + r66603;
        double r66605 = r66600 * r66600;
        double r66606 = r66596 * r66596;
        double r66607 = r66605 + r66606;
        double r66608 = sqrt(r66607);
        double r66609 = r66604 / r66608;
        double r66610 = r66609 / r66608;
        double r66611 = r66602 / r66608;
        double r66612 = r66598 ? r66610 : r66611;
        return r66612;
}

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 y.im < 1.0220962851570888e+92

    1. Initial program 23.9

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

    3. Applied add-sqr-sqrt23.9

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

    4. Applied associate-/r*23.8

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

    if 1.0220962851570888e+92 < y.im

    1. Initial program 39.6

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

    3. Applied add-sqr-sqrt39.6

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

    4. Applied associate-/r*39.6

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

    5. Taylor expanded around 0 39.1

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

  4. Final simplification26.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 1.022096285157088796449452448047526613473 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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))))