Average Error: 26.4 → 26.4
Time: 11.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.221597588228970858235819909623175745754 \cdot 10^{229}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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 1.221597588228970858235819909623175745754 \cdot 10^{229}:\\
\;\;\;\;\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{-1 \cdot 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 r40954 = x_im;
        double r40955 = y_re;
        double r40956 = r40954 * r40955;
        double r40957 = x_re;
        double r40958 = y_im;
        double r40959 = r40957 * r40958;
        double r40960 = r40956 - r40959;
        double r40961 = r40955 * r40955;
        double r40962 = r40958 * r40958;
        double r40963 = r40961 + r40962;
        double r40964 = r40960 / r40963;
        return r40964;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r40965 = x_im;
        double r40966 = y_re;
        double r40967 = r40965 * r40966;
        double r40968 = x_re;
        double r40969 = y_im;
        double r40970 = r40968 * r40969;
        double r40971 = r40967 - r40970;
        double r40972 = r40966 * r40966;
        double r40973 = r40969 * r40969;
        double r40974 = r40972 + r40973;
        double r40975 = r40971 / r40974;
        double r40976 = 1.2215975882289709e+229;
        bool r40977 = r40975 <= r40976;
        double r40978 = sqrt(r40974);
        double r40979 = r40971 / r40978;
        double r40980 = r40979 / r40978;
        double r40981 = -1.0;
        double r40982 = r40981 * r40968;
        double r40983 = r40982 / r40978;
        double r40984 = r40977 ? r40980 : r40983;
        return r40984;
}

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))) < 1.2215975882289709e+229

    1. Initial program 14.8

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

    3. Applied add-sqr-sqrt14.8

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

    4. Applied associate-/r*14.7

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

    if 1.2215975882289709e+229 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 59.2

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

    3. Applied add-sqr-sqrt59.2

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

    4. Applied associate-/r*59.2

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

    5. Taylor expanded around 0 59.5

      \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.221597588228970858235819909623175745754 \cdot 10^{229}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))