_divideComplex, real part

Average Error: 26.0 → 13.5

Time: 11.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.889595916751356 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{{y.im}^{2}}, y.re, \frac{x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)\\ t_1 := \frac{y.im}{t_0}\\ \mathbf{if}\;y.im \leq -1.138134009878751 \cdot 10^{-83}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt[3]{t_0}\\ \mathsf{fma}\left(t_1, x.im, \frac{y.re}{t_2 \cdot t_2} \cdot \frac{x.re}{t_2}\right) \end{array}\\ \mathbf{elif}\;y.im \leq 4.7526766511788395 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(t_1, x.im, \frac{x.re}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathsf{fma}\left(t_3 \cdot t_3, x.im, \frac{x.re \cdot y.re}{t_0}\right) \end{array}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.889595916751356e+153)
   (fma (/ x.re (pow y.im 2.0)) y.re (/ x.im y.im))
   (let* ((t_0 (fma y.im y.im (pow y.re 2.0))) (t_1 (/ y.im t_0)))
     (if (<= y.im -1.138134009878751e-83)
       (let* ((t_2 (cbrt t_0)))
         (fma t_1 x.im (* (/ y.re (* t_2 t_2)) (/ x.re t_2))))
       (if (<= y.im 4.7526766511788395e+30)
         (fma t_1 x.im (/ x.re y.re))
         (let* ((t_3 (/ (sqrt y.im) (hypot y.im y.re))))
           (fma (* t_3 t_3) x.im (/ (* x.re y.re) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.889595916751356e+153) {
		tmp = fma((x_46_re / pow(y_46_im, 2.0)), y_46_re, (x_46_im / y_46_im));
	} else {
		double t_0 = fma(y_46_im, y_46_im, pow(y_46_re, 2.0));
		double t_1 = y_46_im / t_0;
		double tmp_1;
		if (y_46_im <= -1.138134009878751e-83) {
			double t_2_2 = cbrt(t_0);
			tmp_1 = fma(t_1, x_46_im, ((y_46_re / (t_2_2 * t_2_2)) * (x_46_re / t_2_2)));
		} else if (y_46_im <= 4.7526766511788395e+30) {
			tmp_1 = fma(t_1, x_46_im, (x_46_re / y_46_re));
		} else {
			double t_3 = sqrt(y_46_im) / hypot(y_46_im, y_46_re);
			tmp_1 = fma((t_3 * t_3), x_46_im, ((x_46_re * y_46_re) / t_0));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 4 regimes

2. if y.im < -6.88959591675135632e153

   1. Initial program 45.4

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

   2. Simplified 45.4

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

   3. Taylor expanded in y.re around 0 15.4

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

   4. Simplified 13.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.re}{{y.im}^{2}}, y.re, \frac{x.im}{y.im}\right)} \]

   if -6.88959591675135632e153 < y.im < -1.1381340098787509e-83

   1. Initial program 18.6

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

   2. Simplified 18.6

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

   3. Taylor expanded in x.re around 0 18.6

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

   4. Simplified 14.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right)} \]

   5. Applied add-cube-cbrt_binary64 14.4

      \[\leadsto \mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{y.re \cdot x.re}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}}\right) \]

   6. Applied times-frac_binary64 12.7

      \[\leadsto \mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \color{blue}{\frac{y.re}{\sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot \frac{x.re}{\sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}}\right) \]

   if -1.1381340098787509e-83 < y.im < 4.7526766511788395e30

   1. Initial program 19.4

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

   2. Simplified 19.4

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

   3. Taylor expanded in x.re around 0 19.4

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

   4. Simplified 19.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right)} \]

   5. Taylor expanded in y.re around inf 14.1

      \[\leadsto \mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \color{blue}{\frac{x.re}{y.re}}\right) \]

   if 4.7526766511788395e30 < y.im

   1. Initial program 33.3

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

   2. Simplified 33.3

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

   3. Taylor expanded in x.re around 0 33.3

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

   4. Simplified 30.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right)} \]

   5. Applied div-inv_binary64 30.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{y.im \cdot \frac{1}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right) \]

   6. Applied add-sqr-sqrt_binary64 30.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y.im \cdot \frac{1}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot \sqrt{y.im \cdot \frac{1}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right) \]

   7. Simplified 30.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left|\frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\right|} \cdot \sqrt{y.im \cdot \frac{1}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right) \]

   8. Simplified 13.1

      \[\leadsto \mathsf{fma}\left(\left|\frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\right| \cdot \color{blue}{\left|\frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\right|}, x.im, \frac{y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.889595916751356 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{{y.im}^{2}}, y.re, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -1.138134009878751 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{y.re}{\sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}} \cdot \frac{x.re}{\sqrt[3]{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}}\right)\\ \mathbf{elif}\;y.im \leq 4.7526766511788395 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, x.im, \frac{x.re}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\sqrt{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}, x.im, \frac{x.re \cdot y.re}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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))))