Average Error: 26.2 → 1.8
Time: 4.7s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\frac{x.im}{y.re + \frac{y.im}{\frac{y.re}{y.im}}} - \frac{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (-
  (/ x.im (+ y.re (/ y.im (/ y.re y.im))))
  (/ (* (/ y.im (hypot y.re y.im)) x.re) (hypot y.re y.im))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * 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) {
	return (x_46_im / (y_46_re + (y_46_im / (y_46_re / y_46_im)))) - (((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im));
}

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}
↓
public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (x_46_im / (y_46_re + (y_46_im / (y_46_re / y_46_im)))) - (((y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re) / Math.hypot(y_46_re, y_46_im));
}

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
↓
def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (x_46_im / (y_46_re + (y_46_im / (y_46_re / y_46_im)))) - (((y_46_im / math.hypot(y_46_re, y_46_im)) * x_46_re) / math.hypot(y_46_re, y_46_im))

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end
↓
function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(x_46_im / Float64(y_46_re + Float64(y_46_im / Float64(y_46_re / y_46_im)))) - Float64(Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im)))
end

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end
↓
function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (x_46_im / (y_46_re + (y_46_im / (y_46_re / y_46_im)))) - (((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re) / hypot(y_46_re, y_46_im));
end

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(x$46$im / N[(y$46$re + N[(y$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{x.im}{y.re + \frac{y.im}{\frac{y.re}{y.im}}} - \frac{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}

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. Initial program 26.2

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

  2. Simplified26.2

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

  3. Taylor expanded in x.re around 0 26.2

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

  4. Simplified25.2

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

  5. Taylor expanded in y.re around 0 17.8

    \[\leadsto \frac{x.im}{\color{blue}{y.re + \frac{{y.im}^{2}}{y.re}}} - y.im \cdot \frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \]

  6. Simplified16.1

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

  7. Applied egg-rr10.5

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

  8. Applied egg-rr1.8

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

  9. Final simplification1.8

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

Reproduce

herbie shell --seed 2022155 
(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))))