Average Error: 26.2 → 25.4
Time: 4.1s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.355127360692946109502946832371947729711 \cdot 10^{283}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.355127360692946109502946832371947729711 \cdot 10^{283}:\\
\;\;\;\;\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{-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 r111714 = x_re;
        double r111715 = y_re;
        double r111716 = r111714 * r111715;
        double r111717 = x_im;
        double r111718 = y_im;
        double r111719 = r111717 * r111718;
        double r111720 = r111716 + r111719;
        double r111721 = r111715 * r111715;
        double r111722 = r111718 * r111718;
        double r111723 = r111721 + r111722;
        double r111724 = r111720 / r111723;
        return r111724;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r111725 = x_re;
        double r111726 = y_re;
        double r111727 = r111725 * r111726;
        double r111728 = x_im;
        double r111729 = y_im;
        double r111730 = r111728 * r111729;
        double r111731 = r111727 + r111730;
        double r111732 = r111726 * r111726;
        double r111733 = r111729 * r111729;
        double r111734 = r111732 + r111733;
        double r111735 = r111731 / r111734;
        double r111736 = 7.355127360692946e+283;
        bool r111737 = r111735 <= r111736;
        double r111738 = sqrt(r111734);
        double r111739 = r111731 / r111738;
        double r111740 = r111739 / r111738;
        double r111741 = -1.0;
        double r111742 = r111741 * r111725;
        double r111743 = r111742 / r111738;
        double r111744 = r111737 ? r111740 : r111743;
        return r111744;
}

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.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < 7.355127360692946e+283

    1. Initial program 14.2

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

    3. Applied add-sqr-sqrt14.2

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

    4. Applied associate-/r*14.1

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

    if 7.355127360692946e+283 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 62.9

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

    3. Applied add-sqr-sqrt62.9

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

    4. Applied associate-/r*62.9

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

    5. Taylor expanded around -inf 59.8

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

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.355127360692946109502946832371947729711 \cdot 10^{283}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))