Average Error: 26.1 → 25.3
Time: 5.0s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\
\;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{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 r85475 = x_im;
        double r85476 = y_re;
        double r85477 = r85475 * r85476;
        double r85478 = x_re;
        double r85479 = y_im;
        double r85480 = r85478 * r85479;
        double r85481 = r85477 - r85480;
        double r85482 = r85476 * r85476;
        double r85483 = r85479 * r85479;
        double r85484 = r85482 + r85483;
        double r85485 = r85481 / r85484;
        return r85485;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r85486 = x_im;
        double r85487 = y_re;
        double r85488 = r85486 * r85487;
        double r85489 = x_re;
        double r85490 = y_im;
        double r85491 = r85489 * r85490;
        double r85492 = r85488 - r85491;
        double r85493 = r85487 * r85487;
        double r85494 = r85490 * r85490;
        double r85495 = r85493 + r85494;
        double r85496 = r85492 / r85495;
        double r85497 = 6.918361099356831e+305;
        bool r85498 = r85496 <= r85497;
        double r85499 = sqrt(r85495);
        double r85500 = r85486 / r85499;
        double r85501 = r85498 ? r85496 : r85500;
        return r85501;
}

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.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))) < 6.918361099356831e+305

    1. Initial program 13.9

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

    if 6.918361099356831e+305 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 64.0

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

    3. Applied add-sqr-sqrt64.0

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

    4. Applied associate-/r*64.0

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

    5. Taylor expanded around inf 60.6

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

  4. Final simplification25.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 6.918361099356831363837397929747071264075 \cdot 10^{305}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))