Average Error: 26.0 → 25.4
Time: 13.8s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.4994875845408422 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.4994875845408422 \cdot 10^{+304}:\\
\;\;\;\;\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 r2170035 = x_im;
        double r2170036 = y_re;
        double r2170037 = r2170035 * r2170036;
        double r2170038 = x_re;
        double r2170039 = y_im;
        double r2170040 = r2170038 * r2170039;
        double r2170041 = r2170037 - r2170040;
        double r2170042 = r2170036 * r2170036;
        double r2170043 = r2170039 * r2170039;
        double r2170044 = r2170042 + r2170043;
        double r2170045 = r2170041 / r2170044;
        return r2170045;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2170046 = x_im;
        double r2170047 = y_re;
        double r2170048 = r2170046 * r2170047;
        double r2170049 = x_re;
        double r2170050 = y_im;
        double r2170051 = r2170049 * r2170050;
        double r2170052 = r2170048 - r2170051;
        double r2170053 = r2170047 * r2170047;
        double r2170054 = r2170050 * r2170050;
        double r2170055 = r2170053 + r2170054;
        double r2170056 = r2170052 / r2170055;
        double r2170057 = 1.4994875845408422e+304;
        bool r2170058 = r2170056 <= r2170057;
        double r2170059 = sqrt(r2170055);
        double r2170060 = r2170052 / r2170059;
        double r2170061 = r2170060 / r2170059;
        double r2170062 = r2170046 / r2170059;
        double r2170063 = r2170058 ? r2170061 : r2170062;
        return r2170063;
}

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))) < 1.4994875845408422e+304

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied associate-/r*13.8

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

    if 1.4994875845408422e+304 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 62.2

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

    3. Applied add-sqr-sqrt62.2

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

    4. Applied associate-/r*62.2

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

    6. Applied add-sqr-sqrt62.2

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

    7. Applied sqrt-prod62.2

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

    8. Taylor expanded around inf 59.9

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

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.4994875845408422 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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))))