_divideComplex, real part

Average Error: 25.3 → 11.9

Time: 19.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -4.434994773334765 \cdot 10^{+162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.2908775229370544 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2695665 = x_re;
        double r2695666 = y_re;
        double r2695667 = r2695665 * r2695666;
        double r2695668 = x_im;
        double r2695669 = y_im;
        double r2695670 = r2695668 * r2695669;
        double r2695671 = r2695667 + r2695670;
        double r2695672 = r2695666 * r2695666;
        double r2695673 = r2695669 * r2695669;
        double r2695674 = r2695672 + r2695673;
        double r2695675 = r2695671 / r2695674;
        return r2695675;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2695676 = y_im;
        double r2695677 = -4.434994773334765e+162;
        bool r2695678 = r2695676 <= r2695677;
        double r2695679 = x_im;
        double r2695680 = -r2695679;
        double r2695681 = y_re;
        double r2695682 = hypot(r2695676, r2695681);
        double r2695683 = r2695680 / r2695682;
        double r2695684 = 2.2908775229370544e+141;
        bool r2695685 = r2695676 <= r2695684;
        double r2695686 = x_re;
        double r2695687 = r2695681 * r2695686;
        double r2695688 = fma(r2695676, r2695679, r2695687);
        double r2695689 = r2695688 / r2695682;
        double r2695690 = r2695689 / r2695682;
        double r2695691 = r2695679 / r2695682;
        double r2695692 = r2695685 ? r2695690 : r2695691;
        double r2695693 = r2695678 ? r2695683 : r2695692;
        return r2695693;
}

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 < -4.434994773334765e+162

   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 29.1

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

   10. Applied *-commutative 29.1

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

   11. Applied associate-/r* 28.3

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

   12. Simplified 28.3

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

   13. Taylor expanded around -inf 12.3

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

   14. Simplified 12.3

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

   if -4.434994773334765e+162 < y.im < 2.2908775229370544e+141

   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

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

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied associate-/r* 18.7

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

   7. Applied clear-num 19.0

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

   8. Simplified 12.1

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

   10. Applied *-commutative 12.1

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

   11. Applied associate-/r* 11.7

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

   12. Simplified 11.6

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

   if 2.2908775229370544e+141 < y.im

   1. Initial program 43.5

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

   2. Simplified 43.5

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

   4. Applied add-sqr-sqrt 43.5

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

   5. Applied associate-/r* 43.5

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

   7. Applied clear-num 43.6

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

   8. Simplified 28.7

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

   10. Applied *-commutative 28.7

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

   11. Applied associate-/r* 28.0

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

   12. Simplified 28.0

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

   13. Taylor expanded around inf 13.5

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -4.434994773334765 \cdot 10^{+162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.2908775229370544 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))