Average Error: 26.3 → 0.5
Time: 2.8m
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{1}{\frac{1}{\sqrt{1}}} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{1}}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{1}{\frac{1}{\sqrt{1}}} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{1}}}
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 (((1.0 / (1.0 / sqrt(1.0))) * ((x_46_im / (hypot(y_46_re, y_46_im) / y_46_re)) - (x_46_re / (hypot(y_46_re, y_46_im) / y_46_im)))) / (hypot(y_46_re, y_46_im) / sqrt(1.0)));
}

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 26.3

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

  3. Applied add-sqr-sqrt26.3

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

  4. Applied *-un-lft-identity26.3

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

  5. Applied times-frac26.3

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

  6. Simplified26.3

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

  7. Simplified16.8

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{1}} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}}\]
  8. Using strategy rm

  9. Applied div-sub16.8

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

  10. Simplified9.3

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

  11. Simplified0.7

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{1}} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}} - \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}}\right)\]
  12. Using strategy rm

  13. Applied add-sqr-sqrt0.7

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

  14. Applied *-un-lft-identity0.7

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

  15. Applied times-frac0.7

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

  16. Applied associate-/r*0.7

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

  17. Applied associate-*l/0.5

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

  18. Final simplification0.5

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

Reproduce

herbie shell --seed 2020066 +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))))