Average Error: 26.1 → 26.1
Time: 3.7s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r56848 = x_im;
        double r56849 = y_re;
        double r56850 = r56848 * r56849;
        double r56851 = x_re;
        double r56852 = y_im;
        double r56853 = r56851 * r56852;
        double r56854 = r56850 - r56853;
        double r56855 = r56849 * r56849;
        double r56856 = r56852 * r56852;
        double r56857 = r56855 + r56856;
        double r56858 = r56854 / r56857;
        return r56858;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r56859 = x_im;
        double r56860 = y_re;
        double r56861 = r56859 * r56860;
        double r56862 = x_re;
        double r56863 = y_im;
        double r56864 = r56862 * r56863;
        double r56865 = r56861 - r56864;
        double r56866 = 1.0;
        double r56867 = r56860 * r56860;
        double r56868 = r56863 * r56863;
        double r56869 = r56867 + r56868;
        double r56870 = sqrt(r56869);
        double r56871 = r56866 / r56870;
        double r56872 = r56865 * r56871;
        double r56873 = r56872 / r56870;
        return r56873;
}

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 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.0

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

  6. Applied div-inv26.1

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

  7. Final simplification26.1

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

Reproduce

herbie shell --seed 2020100 
(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))))