Average Error: 26.3 → 26.3
Time: 16.6s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le 1.937670017341036534919481795915749319496 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\ \end{array}\]
\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.im \le 1.937670017341036534919481795915749319496 \cdot 10^{62}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x.re}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r4268111 = x_im;
        double r4268112 = y_re;
        double r4268113 = r4268111 * r4268112;
        double r4268114 = x_re;
        double r4268115 = y_im;
        double r4268116 = r4268114 * r4268115;
        double r4268117 = r4268113 - r4268116;
        double r4268118 = r4268112 * r4268112;
        double r4268119 = r4268115 * r4268115;
        double r4268120 = r4268118 + r4268119;
        double r4268121 = r4268117 / r4268120;
        return r4268121;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r4268122 = y_im;
        double r4268123 = 1.9376700173410365e+62;
        bool r4268124 = r4268122 <= r4268123;
        double r4268125 = 1.0;
        double r4268126 = y_re;
        double r4268127 = r4268126 * r4268126;
        double r4268128 = fma(r4268122, r4268122, r4268127);
        double r4268129 = sqrt(r4268128);
        double r4268130 = r4268125 / r4268129;
        double r4268131 = x_im;
        double r4268132 = r4268131 * r4268126;
        double r4268133 = x_re;
        double r4268134 = r4268122 * r4268133;
        double r4268135 = r4268132 - r4268134;
        double r4268136 = r4268130 * r4268135;
        double r4268137 = r4268136 / r4268129;
        double r4268138 = -r4268133;
        double r4268139 = r4268138 / r4268129;
        double r4268140 = r4268124 ? r4268137 : r4268139;
        return r4268140;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if y.im < 1.9376700173410365e+62

    1. Initial program 23.4

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

    2. Simplified23.4

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

    4. Applied add-sqr-sqrt23.4

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

    5. Applied associate-/r*23.3

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

    7. Applied div-inv23.4

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

    if 1.9376700173410365e+62 < y.im

    1. Initial program 37.3

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

    2. Simplified37.3

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

    4. Applied add-sqr-sqrt37.3

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

    5. Applied associate-/r*37.2

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

    6. Taylor expanded around 0 37.5

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

    7. Simplified37.5

      \[\leadsto \frac{\color{blue}{-x.re}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 1.937670017341036534919481795915749319496 \cdot 10^{62}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))))