Average Error: 25.7 → 12.6
Time: 16.4s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -7.729339549361261 \cdot 10^{+115}:\\ \;\;\;\;-\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.512622379546738 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \le -7.729339549361261 \cdot 10^{+115}:\\
\;\;\;\;-\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.re \le 1.512622379546738 \cdot 10^{+155}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1988803 = x_im;
        double r1988804 = y_re;
        double r1988805 = r1988803 * r1988804;
        double r1988806 = x_re;
        double r1988807 = y_im;
        double r1988808 = r1988806 * r1988807;
        double r1988809 = r1988805 - r1988808;
        double r1988810 = r1988804 * r1988804;
        double r1988811 = r1988807 * r1988807;
        double r1988812 = r1988810 + r1988811;
        double r1988813 = r1988809 / r1988812;
        return r1988813;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1988814 = y_re;
        double r1988815 = -7.729339549361261e+115;
        bool r1988816 = r1988814 <= r1988815;
        double r1988817 = x_im;
        double r1988818 = y_im;
        double r1988819 = hypot(r1988814, r1988818);
        double r1988820 = r1988817 / r1988819;
        double r1988821 = -r1988820;
        double r1988822 = 1.512622379546738e+155;
        bool r1988823 = r1988814 <= r1988822;
        double r1988824 = r1988817 * r1988814;
        double r1988825 = x_re;
        double r1988826 = r1988818 * r1988825;
        double r1988827 = r1988824 - r1988826;
        double r1988828 = r1988827 / r1988819;
        double r1988829 = r1988828 / r1988819;
        double r1988830 = r1988823 ? r1988829 : r1988820;
        double r1988831 = r1988816 ? r1988821 : r1988830;
        return r1988831;
}

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. Split input into 3 regimes

  2. if y.re < -7.729339549361261e+115

    1. Initial program 40.9

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

    3. Applied add-sqr-sqrt40.9

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

    4. Applied associate-/r*40.9

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

    6. Applied hypot-def40.9

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

    8. Applied hypot-def26.5

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

    9. Taylor expanded around -inf 15.5

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

    10. Simplified15.5

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

    if -7.729339549361261e+115 < y.re < 1.512622379546738e+155

    1. Initial program 18.5

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

    3. Applied add-sqr-sqrt18.5

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

    4. Applied associate-/r*18.4

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

    6. Applied hypot-def18.4

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

    8. Applied hypot-def11.4

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

    9. Taylor expanded around inf 11.4

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

    if 1.512622379546738e+155 < y.re

    1. Initial program 45.9

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

    3. Applied add-sqr-sqrt45.9

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

    4. Applied associate-/r*45.9

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

    6. Applied hypot-def45.9

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

    8. Applied hypot-def27.9

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

    9. Taylor expanded around inf 15.7

      \[\leadsto \frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -7.729339549361261 \cdot 10^{+115}:\\ \;\;\;\;-\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.512622379546738 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))