_divideComplex, real part

Average Error: 25.3 → 12.0

Time: 1.6m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -3.712028832851069 \cdot 10^{+161}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;y.re \le 9.529062653614642 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.im^2 + y.re^2}^*}{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r3329623 = x_re;
        double r3329624 = y_re;
        double r3329625 = r3329623 * r3329624;
        double r3329626 = x_im;
        double r3329627 = y_im;
        double r3329628 = r3329626 * r3329627;
        double r3329629 = r3329625 + r3329628;
        double r3329630 = r3329624 * r3329624;
        double r3329631 = r3329627 * r3329627;
        double r3329632 = r3329630 + r3329631;
        double r3329633 = r3329629 / r3329632;
        return r3329633;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r3329634 = y_re;
        double r3329635 = -3.712028832851069e+161;
        bool r3329636 = r3329634 <= r3329635;
        double r3329637 = x_re;
        double r3329638 = -r3329637;
        double r3329639 = y_im;
        double r3329640 = hypot(r3329639, r3329634);
        double r3329641 = r3329638 / r3329640;
        double r3329642 = 9.529062653614642e+142;
        bool r3329643 = r3329634 <= r3329642;
        double r3329644 = 1.0;
        double r3329645 = x_im;
        double r3329646 = r3329645 * r3329639;
        double r3329647 = fma(r3329637, r3329634, r3329646);
        double r3329648 = r3329640 / r3329647;
        double r3329649 = r3329644 / r3329648;
        double r3329650 = r3329649 / r3329640;
        double r3329651 = r3329637 / r3329640;
        double r3329652 = r3329643 ? r3329650 : r3329651;
        double r3329653 = r3329636 ? r3329641 : r3329652;
        return r3329653;
}

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 < -3.712028832851069e+161

   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

      \[\leadsto \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 44.3

      \[\leadsto \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   5. Applied associate-/r* 44.3

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

   7. Applied *-un-lft-identity 44.3

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   8. Applied *-un-lft-identity 44.3

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   9. Applied *-un-lft-identity 44.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   10. Applied times-frac 44.3

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   11. Applied times-frac 44.3

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   12. Simplified 44.3

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   13. Simplified 30.0

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]

   14. Taylor expanded around -inf 12.2

       \[\leadsto 1 \cdot \frac{\color{blue}{-1 \cdot x.re}}{\sqrt{y.im^2 + y.re^2}^*}\]

   15. Simplified 12.2

       \[\leadsto 1 \cdot \frac{\color{blue}{-x.re}}{\sqrt{y.im^2 + y.re^2}^*}\]

   if -3.712028832851069e+161 < y.re < 9.529062653614642e+142

   1. Initial program 18.8

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

   2. Simplified 18.8

      \[\leadsto \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 18.8

      \[\leadsto \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   5. Applied associate-/r* 18.7

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

   7. Applied *-un-lft-identity 18.7

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   8. Applied *-un-lft-identity 18.7

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   9. Applied *-un-lft-identity 18.7

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   10. Applied times-frac 18.7

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   11. Applied times-frac 18.7

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   12. Simplified 18.7

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   13. Simplified 11.7

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]
   14. Using strategy rm

   15. Applied clear-num 11.8

       \[\leadsto 1 \cdot \frac{\color{blue}{\frac{1}{\frac{\sqrt{y.im^2 + y.re^2}^*}{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}}}{\sqrt{y.im^2 + y.re^2}^*}\]

   if 9.529062653614642e+142 < y.re

   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

      \[\leadsto \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 43.3

      \[\leadsto \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   5. Applied associate-/r* 43.3

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

   7. Applied *-un-lft-identity 43.3

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   8. Applied *-un-lft-identity 43.3

      \[\leadsto \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\color{blue}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   9. Applied *-un-lft-identity 43.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   10. Applied times-frac 43.3

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{1 \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   11. Applied times-frac 43.3

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   12. Simplified 43.3

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   13. Simplified 26.7

       \[\leadsto 1 \cdot \color{blue}{\frac{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]
   14. Using strategy rm

   15. Applied clear-num 26.7

       \[\leadsto 1 \cdot \frac{\color{blue}{\frac{1}{\frac{\sqrt{y.im^2 + y.re^2}^*}{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}}}{\sqrt{y.im^2 + y.re^2}^*}\]

   16. Taylor expanded around 0 12.8

       \[\leadsto 1 \cdot \frac{\color{blue}{x.re}}{\sqrt{y.im^2 + y.re^2}^*}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -3.712028832851069 \cdot 10^{+161}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;y.re \le 9.529062653614642 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.im^2 + y.re^2}^*}{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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))))