Average Error: 26.2 → 24.7
Time: 13.6s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{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 r54186 = x_im;
        double r54187 = y_re;
        double r54188 = r54186 * r54187;
        double r54189 = x_re;
        double r54190 = y_im;
        double r54191 = r54189 * r54190;
        double r54192 = r54188 - r54191;
        double r54193 = r54187 * r54187;
        double r54194 = r54190 * r54190;
        double r54195 = r54193 + r54194;
        double r54196 = r54192 / r54195;
        return r54196;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r54197 = x_im;
        double r54198 = y_re;
        double r54199 = r54197 * r54198;
        double r54200 = r54198 * r54198;
        double r54201 = y_im;
        double r54202 = r54201 * r54201;
        double r54203 = r54200 + r54202;
        double r54204 = r54199 / r54203;
        double r54205 = x_re;
        double r54206 = sqrt(r54203);
        double r54207 = r54205 / r54206;
        double r54208 = r54201 / r54206;
        double r54209 = r54207 * r54208;
        double r54210 = r54204 - r54209;
        return r54210;
}

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. Using strategy rm

  3. Applied div-sub26.2

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

  5. Applied add-sqr-sqrt26.2

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

  6. Applied times-frac24.7

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

  7. Final simplification24.7

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

Reproduce

herbie shell --seed 2019325 
(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))))