Average Error: 26.1 → 23.9
Time: 3.0s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -1.9497673481821607 \cdot 10^{115} \lor \neg \left(y.im \le 6.67079582011057737 \cdot 10^{138}\right):\\ \;\;\;\;\frac{-1 \cdot {x.im}^{2}}{x.re \cdot y.re - x.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]
\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}\;y.im \le -1.9497673481821607 \cdot 10^{115} \lor \neg \left(y.im \le 6.67079582011057737 \cdot 10^{138}\right):\\
\;\;\;\;\frac{-1 \cdot {x.im}^{2}}{x.re \cdot y.re - x.im \cdot y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\

\end{array}
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double VAR;
	if (((y_46_im <= -1.9497673481821607e+115) || !(y_46_im <= 6.670795820110577e+138))) {
		VAR = ((-1.0 * pow(x_46_im, 2.0)) / ((x_46_re * y_46_re) - (x_46_im * y_46_im)));
	} else {
		VAR = (((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	}
	return VAR;
}

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 y.im < -1.9497673481821607e+115 or 6.670795820110577e+138 < y.im

    1. Initial program 41.7

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

    3. Applied clear-num41.8

      \[\leadsto \color{blue}{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re \cdot y.re + x.im \cdot y.im}}}\]
    4. Using strategy rm

    5. Applied flip-+48.6

      \[\leadsto \frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{\color{blue}{\frac{\left(x.re \cdot y.re\right) \cdot \left(x.re \cdot y.re\right) - \left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right)}{x.re \cdot y.re - x.im \cdot y.im}}}}\]

    6. Applied associate-/r/48.7

      \[\leadsto \frac{1}{\color{blue}{\frac{y.re \cdot y.re + y.im \cdot y.im}{\left(x.re \cdot y.re\right) \cdot \left(x.re \cdot y.re\right) - \left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right)} \cdot \left(x.re \cdot y.re - x.im \cdot y.im\right)}}\]

    7. Applied associate-/r*48.7

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{\left(x.re \cdot y.re\right) \cdot \left(x.re \cdot y.re\right) - \left(x.im \cdot y.im\right) \cdot \left(x.im \cdot y.im\right)}}}{x.re \cdot y.re - x.im \cdot y.im}}\]

    8. Simplified48.6

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \left(x.re \cdot y.re - x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}}{x.re \cdot y.re - x.im \cdot y.im}\]

    9. Taylor expanded around 0 34.7

      \[\leadsto \frac{\color{blue}{-1 \cdot {x.im}^{2}}}{x.re \cdot y.re - x.im \cdot y.im}\]

    if -1.9497673481821607e+115 < y.im < 6.670795820110577e+138

    1. Initial program 19.0

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -1.9497673481821607 \cdot 10^{115} \lor \neg \left(y.im \le 6.67079582011057737 \cdot 10^{138}\right):\\ \;\;\;\;\frac{-1 \cdot {x.im}^{2}}{x.re \cdot y.re - x.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))