_divideComplex, real part

Average Error: 25.6 → 12.9

Time: 18.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.086126503497258 \cdot 10^{+188}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 6.204871618360054 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2430887 = x_re;
        double r2430888 = y_re;
        double r2430889 = r2430887 * r2430888;
        double r2430890 = x_im;
        double r2430891 = y_im;
        double r2430892 = r2430890 * r2430891;
        double r2430893 = r2430889 + r2430892;
        double r2430894 = r2430888 * r2430888;
        double r2430895 = r2430891 * r2430891;
        double r2430896 = r2430894 + r2430895;
        double r2430897 = r2430893 / r2430896;
        return r2430897;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2430898 = y_re;
        double r2430899 = -5.086126503497258e+188;
        bool r2430900 = r2430898 <= r2430899;
        double r2430901 = x_re;
        double r2430902 = -r2430901;
        double r2430903 = y_im;
        double r2430904 = hypot(r2430903, r2430898);
        double r2430905 = r2430902 / r2430904;
        double r2430906 = 6.204871618360054e+176;
        bool r2430907 = r2430898 <= r2430906;
        double r2430908 = x_im;
        double r2430909 = r2430903 * r2430908;
        double r2430910 = fma(r2430901, r2430898, r2430909);
        double r2430911 = r2430910 / r2430904;
        double r2430912 = r2430911 / r2430904;
        double r2430913 = r2430901 / r2430904;
        double r2430914 = r2430907 ? r2430912 : r2430913;
        double r2430915 = r2430900 ? r2430905 : r2430914;
        return r2430915;
}

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 < -5.086126503497258e+188

   1. Initial program 41.2

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

   2. Simplified 41.2

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

   4. Applied add-sqr-sqrt 41.2

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

   5. Applied associate-/r* 41.2

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

   7. Applied clear-num 41.2

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

   8. Simplified 29.6

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

   10. Applied associate-/r* 29.0

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

   11. Simplified 29.0

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

   12. Taylor expanded around -inf 12.0

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

   13. Simplified 12.0

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

   if -5.086126503497258e+188 < y.re < 6.204871618360054e+176

   1. Initial program 20.9

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

   2. Simplified 20.9

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

   4. Applied add-sqr-sqrt 20.9

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

   5. Applied associate-/r* 20.8

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

   7. Applied clear-num 21.1

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

   8. Simplified 13.6

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

   10. Applied associate-/r* 13.1

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

   11. Simplified 13.0

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

   if 6.204871618360054e+176 < y.re

   1. Initial program 44.6

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

   2. Simplified 44.6

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

   4. Applied add-sqr-sqrt 44.6

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

   5. Applied associate-/r* 44.6

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

   7. Applied clear-num 44.6

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

   8. Simplified 31.9

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

   10. Applied associate-/r* 31.4

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

   11. Simplified 31.4

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

   12. Taylor expanded around inf 12.5

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.086126503497258 \cdot 10^{+188}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 6.204871618360054 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))))