Average Error: 25.9 → 13.0
Time: 17.1s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -1.47788539242579584930401717666761308705 \cdot 10^{104}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.336053818650175824019993309646752305462 \cdot 10^{147}:\\ \;\;\;\;\frac{\left(x.im \cdot y.re - y.im \cdot x.re\right) \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]
\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 -1.47788539242579584930401717666761308705 \cdot 10^{104}:\\
\;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{elif}\;y.re \le 1.336053818650175824019993309646752305462 \cdot 10^{147}:\\
\;\;\;\;\frac{\left(x.im \cdot y.re - y.im \cdot x.re\right) \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3087636 = x_im;
        double r3087637 = y_re;
        double r3087638 = r3087636 * r3087637;
        double r3087639 = x_re;
        double r3087640 = y_im;
        double r3087641 = r3087639 * r3087640;
        double r3087642 = r3087638 - r3087641;
        double r3087643 = r3087637 * r3087637;
        double r3087644 = r3087640 * r3087640;
        double r3087645 = r3087643 + r3087644;
        double r3087646 = r3087642 / r3087645;
        return r3087646;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3087647 = y_re;
        double r3087648 = -1.477885392425796e+104;
        bool r3087649 = r3087647 <= r3087648;
        double r3087650 = x_im;
        double r3087651 = -r3087650;
        double r3087652 = y_im;
        double r3087653 = hypot(r3087652, r3087647);
        double r3087654 = r3087651 / r3087653;
        double r3087655 = 1.3360538186501758e+147;
        bool r3087656 = r3087647 <= r3087655;
        double r3087657 = r3087650 * r3087647;
        double r3087658 = x_re;
        double r3087659 = r3087652 * r3087658;
        double r3087660 = r3087657 - r3087659;
        double r3087661 = 1.0;
        double r3087662 = r3087661 / r3087653;
        double r3087663 = r3087660 * r3087662;
        double r3087664 = r3087663 / r3087653;
        double r3087665 = r3087650 / r3087653;
        double r3087666 = r3087656 ? r3087664 : r3087665;
        double r3087667 = r3087649 ? r3087654 : r3087666;
        return r3087667;
}

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 < -1.477885392425796e+104

    1. Initial program 39.8

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

    2. Simplified39.8

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

    4. Applied add-sqr-sqrt39.8

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

    5. Applied associate-/r*39.7

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

    7. Applied fma-udef39.7

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

    8. Applied hypot-def39.7

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

    9. Taylor expanded around -inf 16.6

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

    10. Simplified16.6

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

    if -1.477885392425796e+104 < y.re < 1.3360538186501758e+147

    1. Initial program 19.0

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

    2. Simplified19.0

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

    4. Applied add-sqr-sqrt19.0

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

    5. Applied associate-/r*18.9

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

    7. Applied fma-udef18.9

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

    8. Applied hypot-def18.9

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

    10. Applied div-inv19.0

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

    11. Simplified11.8

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

    if 1.3360538186501758e+147 < y.re

    1. Initial program 42.8

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

    2. Simplified42.8

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

    4. Applied add-sqr-sqrt42.8

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

    5. Applied associate-/r*42.8

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

    7. Applied fma-udef42.8

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

    8. Applied hypot-def42.8

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

    9. Taylor expanded around inf 14.0

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.47788539242579584930401717666761308705 \cdot 10^{104}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.336053818650175824019993309646752305462 \cdot 10^{147}:\\ \;\;\;\;\frac{\left(x.im \cdot y.re - y.im \cdot x.re\right) \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))))