Average Error: 25.8 → 24.3
Time: 16.9s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{y.re}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{y.re}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\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 r47026 = x_im;
        double r47027 = y_re;
        double r47028 = r47026 * r47027;
        double r47029 = x_re;
        double r47030 = y_im;
        double r47031 = r47029 * r47030;
        double r47032 = r47028 - r47031;
        double r47033 = r47027 * r47027;
        double r47034 = r47030 * r47030;
        double r47035 = r47033 + r47034;
        double r47036 = r47032 / r47035;
        return r47036;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r47037 = x_im;
        double r47038 = y_re;
        double r47039 = r47038 * r47038;
        double r47040 = y_im;
        double r47041 = r47040 * r47040;
        double r47042 = r47039 + r47041;
        double r47043 = sqrt(r47042);
        double r47044 = sqrt(r47043);
        double r47045 = r47037 / r47044;
        double r47046 = r47038 / r47044;
        double r47047 = r47045 * r47046;
        double r47048 = x_re;
        double r47049 = r47048 * r47040;
        double r47050 = r47049 / r47043;
        double r47051 = r47047 - r47050;
        double r47052 = r47044 * r47044;
        double r47053 = r47051 / r47052;
        return r47053;
}

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 25.8

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

  3. Applied add-sqr-sqrt25.8

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

  4. Applied associate-/r*25.7

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

  6. Applied div-sub25.7

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

  8. Applied add-sqr-sqrt25.7

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

  9. Applied sqrt-prod25.8

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

  10. Applied times-frac24.1

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

  12. Applied add-sqr-sqrt24.1

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

  13. Applied sqrt-prod24.3

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

  14. Final simplification24.3

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

Reproduce

herbie shell --seed 2019303 
(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))))