_divideComplex, real part

Average Error: 25.9 → 12.6

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.28633112851537097840983624634443392687 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.543067909128725187583327964894972109151 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41147 = x_re;
        double r41148 = y_re;
        double r41149 = r41147 * r41148;
        double r41150 = x_im;
        double r41151 = y_im;
        double r41152 = r41150 * r41151;
        double r41153 = r41149 + r41152;
        double r41154 = r41148 * r41148;
        double r41155 = r41151 * r41151;
        double r41156 = r41154 + r41155;
        double r41157 = r41153 / r41156;
        return r41157;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41158 = y_re;
        double r41159 = -5.286331128515371e+162;
        bool r41160 = r41158 <= r41159;
        double r41161 = x_re;
        double r41162 = -r41161;
        double r41163 = y_im;
        double r41164 = hypot(r41158, r41163);
        double r41165 = r41162 / r41164;
        double r41166 = 1.5430679091287252e+185;
        bool r41167 = r41158 <= r41166;
        double r41168 = x_im;
        double r41169 = r41163 * r41168;
        double r41170 = fma(r41158, r41161, r41169);
        double r41171 = r41170 / r41164;
        double r41172 = r41171 / r41164;
        double r41173 = r41161 / r41164;
        double r41174 = r41167 ? r41172 : r41173;
        double r41175 = r41160 ? r41165 : r41174;
        return r41175;
}

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 < -5.286331128515371e+162

   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

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

   4. Applied add-sqr-sqrt 44.6

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

   5. Applied *-un-lft-identity 44.6

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

   6. Applied times-frac 44.6

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

   7. Simplified 44.6

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

   8. Simplified 29.7

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

   10. Applied *-un-lft-identity 29.7

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

   11. Applied *-un-lft-identity 29.7

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

   12. Applied times-frac 29.7

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

   13. Applied associate-*l* 29.7

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

   14. Simplified 29.7

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

   15. Taylor expanded around -inf 12.7

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

   16. Simplified 12.7

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

   if -5.286331128515371e+162 < y.re < 1.5430679091287252e+185

   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

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

   4. Applied add-sqr-sqrt 20.4

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

   5. Applied *-un-lft-identity 20.4

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

   6. Applied times-frac 20.4

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

   7. Simplified 20.4

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

   8. Simplified 12.7

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

   10. Applied *-un-lft-identity 12.7

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

   11. Applied *-un-lft-identity 12.7

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

   12. Applied times-frac 12.7

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

   13. Applied associate-*l* 12.7

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

   14. Simplified 12.6

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

   if 1.5430679091287252e+185 < y.re

   1. Initial program 44.3

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

   2. Simplified 44.3

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

   4. Applied add-sqr-sqrt 44.3

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

   5. Applied *-un-lft-identity 44.3

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

   6. Applied times-frac 44.3

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

   7. Simplified 44.3

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

   8. Simplified 30.5

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

   10. Applied *-un-lft-identity 30.5

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

   11. Applied *-un-lft-identity 30.5

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

   12. Applied times-frac 30.5

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

   13. Applied associate-*l* 30.5

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

   14. Simplified 30.4

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

   15. Taylor expanded around inf 12.8

       \[\leadsto \frac{1}{1} \cdot \frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.28633112851537097840983624634443392687 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.543067909128725187583327964894972109151 \cdot 10^{185}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))