_divideComplex, imaginary part

Average Error: 26.0 → 0.7

Time: 1.6m

Precision: 64

Math

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

↓

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) (((double) (((double) (x_46_im * y_46_re)) - ((double) (x_46_re * y_46_im)))) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (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 ((double) (((double) (1.0 / ((double) (((double) hypot(y_46_re, y_46_im)) / 1.0)))) * ((double) (((double) fma(((double) (x_46_im / 1.0)), ((double) (y_46_re / ((double) hypot(y_46_re, y_46_im)))), ((double) -(((double) (((double) (y_46_im / ((double) hypot(y_46_re, y_46_im)))) * ((double) (x_46_re / 1.0)))))))) + ((double) fma(((double) -(((double) (y_46_im / ((double) hypot(y_46_re, y_46_im)))))), ((double) (x_46_re / 1.0)), ((double) (((double) (y_46_im / ((double) hypot(y_46_re, y_46_im)))) * ((double) (x_46_re / 1.0))))))))));
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

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 add-sqr-sqrt 26.0

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

4. Applied *-un-lft-identity 26.0

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

5. Applied times-frac 26.0

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

6. Simplified 26.0

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

7. Simplified 16.9

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

9. Applied div-sub 16.9

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

10. Simplified 9.2

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

11. Simplified 0.8

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

13. Applied clear-num 0.8

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

14. Applied associate-/r/ 0.7

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

15. Applied clear-num 0.8

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

16. Applied associate-/r/ 0.7

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

17. Applied prod-diff 0.7

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

18. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020114 +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))))