_divideComplex, real part

Average Error: 26.8 → 13.9

Time: 45.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.0930594930595556 \cdot 10^{+87}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 2.1006973305354477 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2328788 = x_re;
        double r2328789 = y_re;
        double r2328790 = r2328788 * r2328789;
        double r2328791 = x_im;
        double r2328792 = y_im;
        double r2328793 = r2328791 * r2328792;
        double r2328794 = r2328790 + r2328793;
        double r2328795 = r2328789 * r2328789;
        double r2328796 = r2328792 * r2328792;
        double r2328797 = r2328795 + r2328796;
        double r2328798 = r2328794 / r2328797;
        return r2328798;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2328799 = y_re;
        double r2328800 = -1.0930594930595556e+87;
        bool r2328801 = r2328799 <= r2328800;
        double r2328802 = x_re;
        double r2328803 = -r2328802;
        double r2328804 = y_im;
        double r2328805 = hypot(r2328804, r2328799);
        double r2328806 = r2328803 / r2328805;
        double r2328807 = 2.1006973305354477e+171;
        bool r2328808 = r2328799 <= r2328807;
        double r2328809 = x_im;
        double r2328810 = r2328804 * r2328809;
        double r2328811 = fma(r2328802, r2328799, r2328810);
        double r2328812 = r2328811 / r2328805;
        double r2328813 = r2328812 / r2328805;
        double r2328814 = r2328802 / r2328805;
        double r2328815 = r2328808 ? r2328813 : r2328814;
        double r2328816 = r2328801 ? r2328806 : r2328815;
        return r2328816;
}

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 3 regimes

2. if y.re < -1.0930594930595556e+87

   1. Initial program 38.6

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

   2. Simplified 38.6

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

   4. Applied add-sqr-sqrt 38.6

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

   5. Applied associate-/r* 38.6

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

   7. Applied clear-num 38.8

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

   8. Simplified 27.7

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

   10. Applied associate-/r* 27.0

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

   11. Simplified 27.0

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

   12. Taylor expanded around -inf 17.4

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

   13. Simplified 17.4

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

   if -1.0930594930595556e+87 < y.re < 2.1006973305354477e+171

   1. Initial program 20.4

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

   2. Simplified 20.4

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

   4. Applied add-sqr-sqrt 20.5

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

   5. Applied associate-/r* 20.4

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

   7. Applied clear-num 20.6

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

   8. Simplified 13.4

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

   10. Applied associate-/r* 13.0

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

   11. Simplified 12.9

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

   if 2.1006973305354477e+171 < y.re

   1. Initial program 44.5

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

   2. Simplified 44.5

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

   4. Applied add-sqr-sqrt 44.5

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

   5. Applied associate-/r* 44.5

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

   7. Applied clear-num 44.5

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

   8. Simplified 30.1

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

   10. Applied associate-/r* 29.5

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

   11. Simplified 29.5

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

   12. Taylor expanded around inf 14.1

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

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.0930594930595556 \cdot 10^{+87}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 2.1006973305354477 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))