Average Error: 26.0 → 3.4
Time: 2.0s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[x.im \cdot \frac{1}{\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)} - \frac{x.re}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
x.im \cdot \frac{1}{\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)} - \frac{x.re}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}
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 * (1.0 / fma((y_46_im / y_46_re), y_46_im, y_46_re))) - (x_46_re / fma((y_46_re / y_46_im), y_46_re, y_46_im)));
}

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 26.0

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

  3. Applied div-sub26.0

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

  4. Simplified24.8

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

  5. Simplified23.2

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

  6. Taylor expanded around 0 15.7

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

  7. Simplified14.0

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

  8. Taylor expanded around 0 5.7

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

  9. Simplified3.3

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

  11. Applied div-inv3.4

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

  12. Final simplification3.4

    \[\leadsto x.im \cdot \frac{1}{\mathsf{fma}\left(\frac{y.im}{y.re}, y.im, y.re\right)} - \frac{x.re}{\mathsf{fma}\left(\frac{y.re}{y.im}, y.re, y.im\right)}\]

Reproduce

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