_divideComplex, imaginary part

Average Error: 25.6 → 12.8

Time: 15.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -2.349525322457489826589608031591607051529 \cdot 10^{135}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 9.70450990751971859322124590202865896249 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41136 = x_im;
        double r41137 = y_re;
        double r41138 = r41136 * r41137;
        double r41139 = x_re;
        double r41140 = y_im;
        double r41141 = r41139 * r41140;
        double r41142 = r41138 - r41141;
        double r41143 = r41137 * r41137;
        double r41144 = r41140 * r41140;
        double r41145 = r41143 + r41144;
        double r41146 = r41142 / r41145;
        return r41146;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41147 = y_im;
        double r41148 = -2.3495253224574898e+135;
        bool r41149 = r41147 <= r41148;
        double r41150 = x_re;
        double r41151 = y_re;
        double r41152 = hypot(r41151, r41147);
        double r41153 = r41150 / r41152;
        double r41154 = 9.704509907519719e+95;
        bool r41155 = r41147 <= r41154;
        double r41156 = 1.0;
        double r41157 = -r41147;
        double r41158 = x_im;
        double r41159 = r41158 * r41151;
        double r41160 = fma(r41157, r41150, r41159);
        double r41161 = r41152 / r41160;
        double r41162 = r41156 / r41161;
        double r41163 = r41162 / r41152;
        double r41164 = -r41153;
        double r41165 = r41155 ? r41163 : r41164;
        double r41166 = r41149 ? r41153 : r41165;
        return r41166;
}

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.im < -2.3495253224574898e+135

   1. Initial program 41.2

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

   2. Simplified 41.2

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

   4. Applied add-sqr-sqrt 41.2

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

   5. Applied *-un-lft-identity 41.2

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

   6. Applied times-frac 41.2

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

   7. Simplified 41.2

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

   8. Simplified 27.1

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

   10. Applied associate-*r/ 27.0

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

   11. Simplified 27.0

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

   12. Taylor expanded around -inf 13.8

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

   if -2.3495253224574898e+135 < y.im < 9.704509907519719e+95

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   4. Applied add-sqr-sqrt 18.7

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

   5. Applied *-un-lft-identity 18.7

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

   6. Applied times-frac 18.7

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

   7. Simplified 18.7

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

   8. Simplified 11.6

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

   10. Applied associate-*r/ 11.6

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

   11. Simplified 11.5

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

   13. Applied clear-num 11.6

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

   14. Simplified 11.6

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

   if 9.704509907519719e+95 < y.im

   1. Initial program 39.2

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

   2. Simplified 39.2

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

   4. Applied add-sqr-sqrt 39.2

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

   5. Applied *-un-lft-identity 39.2

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

   6. Applied times-frac 39.2

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

   7. Simplified 39.2

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

   8. Simplified 26.6

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

   10. Applied associate-*r/ 26.6

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

   11. Simplified 26.6

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

   12. Taylor expanded around inf 16.3

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

   13. Simplified 16.3

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -2.349525322457489826589608031591607051529 \cdot 10^{135}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 9.70450990751971859322124590202865896249 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))