_divideComplex, real part

Average Error: 26.2 → 12.7

Time: 33.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -1.1964723101605788 \cdot 10^{+201}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \le 3.85213929759163 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1679269 = x_re;
        double r1679270 = y_re;
        double r1679271 = r1679269 * r1679270;
        double r1679272 = x_im;
        double r1679273 = y_im;
        double r1679274 = r1679272 * r1679273;
        double r1679275 = r1679271 + r1679274;
        double r1679276 = r1679270 * r1679270;
        double r1679277 = r1679273 * r1679273;
        double r1679278 = r1679276 + r1679277;
        double r1679279 = r1679275 / r1679278;
        return r1679279;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1679280 = y_im;
        double r1679281 = -1.1964723101605788e+201;
        bool r1679282 = r1679280 <= r1679281;
        double r1679283 = -1.0;
        double r1679284 = y_re;
        double r1679285 = hypot(r1679280, r1679284);
        double r1679286 = r1679283 / r1679285;
        double r1679287 = x_im;
        double r1679288 = r1679286 * r1679287;
        double r1679289 = 3.85213929759163e+150;
        bool r1679290 = r1679280 <= r1679289;
        double r1679291 = x_re;
        double r1679292 = r1679284 * r1679291;
        double r1679293 = fma(r1679287, r1679280, r1679292);
        double r1679294 = r1679293 / r1679285;
        double r1679295 = 1.0;
        double r1679296 = r1679295 / r1679285;
        double r1679297 = r1679294 * r1679296;
        double r1679298 = r1679287 * r1679296;
        double r1679299 = r1679290 ? r1679297 : r1679298;
        double r1679300 = r1679282 ? r1679288 : r1679299;
        return r1679300;
}

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 < -1.1964723101605788e+201

   1. Initial program 43.3

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

   2. Simplified 43.3

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

   4. Applied add-sqr-sqrt 43.3

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

   5. Applied associate-/r* 43.3

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

   7. Applied clear-num 43.3

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

   8. Simplified 32.0

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

   10. Applied div-inv 32.1

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

   11. Applied add-cube-cbrt 32.1

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   12. Applied times-frac 31.3

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}}\]

   13. Simplified 31.3

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   14. Simplified 31.3

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

   15. Taylor expanded around -inf 10.3

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

   16. Simplified 10.3

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

   if -1.1964723101605788e+201 < y.im < 3.85213929759163e+150

   1. Initial program 20.5

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

   2. Simplified 20.5

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

   4. Applied add-sqr-sqrt 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 20.7

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

   8. Simplified 13.1

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

   10. Applied div-inv 13.2

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

   11. Applied add-cube-cbrt 13.2

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   12. Applied times-frac 12.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}}\]

   13. Simplified 12.8

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   14. Simplified 12.8

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

   if 3.85213929759163e+150 < y.im

   1. Initial program 45.3

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

   2. Simplified 45.3

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

   4. Applied add-sqr-sqrt 45.3

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

   5. Applied associate-/r* 45.3

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

   7. Applied clear-num 45.3

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

   8. Simplified 29.6

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

   10. Applied div-inv 29.6

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

   11. Applied add-cube-cbrt 29.6

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   12. Applied times-frac 29.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}}\]

   13. Simplified 29.1

       \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}\]

   14. Simplified 29.1

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

   15. Taylor expanded around inf 14.0

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -1.1964723101605788 \cdot 10^{+201}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.im\\ \mathbf{elif}\;y.im \le 3.85213929759163 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +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))))