Average Error: 26.5 → 26.5
Time: 36.1s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -7.846283352613039 \cdot 10^{+90}:\\ \;\;\;\;-\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \le 5.2422713568197025 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \le -7.846283352613039 \cdot 10^{+90}:\\
\;\;\;\;-\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.re \le 5.2422713568197025 \cdot 10^{+36}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \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 r2818428 = x_im;
        double r2818429 = y_re;
        double r2818430 = r2818428 * r2818429;
        double r2818431 = x_re;
        double r2818432 = y_im;
        double r2818433 = r2818431 * r2818432;
        double r2818434 = r2818430 - r2818433;
        double r2818435 = r2818429 * r2818429;
        double r2818436 = r2818432 * r2818432;
        double r2818437 = r2818435 + r2818436;
        double r2818438 = r2818434 / r2818437;
        return r2818438;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2818439 = y_re;
        double r2818440 = -7.846283352613039e+90;
        bool r2818441 = r2818439 <= r2818440;
        double r2818442 = x_im;
        double r2818443 = r2818439 * r2818439;
        double r2818444 = y_im;
        double r2818445 = r2818444 * r2818444;
        double r2818446 = r2818443 + r2818445;
        double r2818447 = sqrt(r2818446);
        double r2818448 = r2818442 / r2818447;
        double r2818449 = -r2818448;
        double r2818450 = 5.2422713568197025e+36;
        bool r2818451 = r2818439 <= r2818450;
        double r2818452 = r2818442 * r2818439;
        double r2818453 = x_re;
        double r2818454 = r2818453 * r2818444;
        double r2818455 = r2818452 - r2818454;
        double r2818456 = r2818455 / r2818447;
        double r2818457 = r2818456 / r2818447;
        double r2818458 = r2818451 ? r2818457 : r2818448;
        double r2818459 = r2818441 ? r2818449 : r2818458;
        return r2818459;
}

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 3 regimes

  2. if y.re < -7.846283352613039e+90

    1. Initial program 39.2

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

    3. Applied add-sqr-sqrt39.2

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

    4. Applied associate-/r*39.2

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

    5. Taylor expanded around -inf 38.1

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

    6. Simplified38.1

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

    if -7.846283352613039e+90 < y.re < 5.2422713568197025e+36

    1. Initial program 18.9

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

    3. Applied add-sqr-sqrt18.9

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

    4. Applied associate-/r*18.8

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

    if 5.2422713568197025e+36 < y.re

    1. Initial program 35.0

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

    3. Applied add-sqr-sqrt35.0

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

    4. Applied associate-/r*35.0

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

    5. Taylor expanded around inf 36.1

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

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -7.846283352613039 \cdot 10^{+90}:\\ \;\;\;\;-\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \le 5.2422713568197025 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))