_divideComplex, real part

Average Error: 26.3 → 10.8

Time: 10.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \mathbf{if}\;y.im \leq -1.4352057347664762 \cdot 10^{+138}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.im \leq -3.2946202192791303 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.1327597853587381 \cdot 10^{-247}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.re}^{2}} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.032187018528407 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot t_0\\ \end{array}\\ \end{array} \]

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
 (let* ((t_0 (fma (/ x.re y.im) y.re x.im)))
   (if (<= y.im -1.4352057347664762e+138)
     (* t_0 (/ -1.0 (hypot y.im y.re)))
     (let* ((t_1
             (/
              (/ (fma x.re y.re (* y.im x.im)) (hypot y.im y.re))
              (hypot y.im y.re))))
       (if (<= y.im -3.2946202192791303e-195)
         t_1
         (if (<= y.im 1.1327597853587381e-247)
           (+ (/ (* y.im x.im) (pow y.re 2.0)) (/ x.re y.re))
           (if (<= y.im 4.032187018528407e+74)
             t_1
             (* (/ 1.0 (hypot y.im 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 t_0 = fma((x_46_re / y_46_im), y_46_re, x_46_im);
	double tmp;
	if (y_46_im <= -1.4352057347664762e+138) {
		tmp = t_0 * (-1.0 / hypot(y_46_im, y_46_re));
	} else {
		double t_1 = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
		double tmp_1;
		if (y_46_im <= -3.2946202192791303e-195) {
			tmp_1 = t_1;
		} else if (y_46_im <= 1.1327597853587381e-247) {
			tmp_1 = ((y_46_im * x_46_im) / pow(y_46_re, 2.0)) + (x_46_re / y_46_re);
		} else if (y_46_im <= 4.032187018528407e+74) {
			tmp_1 = t_1;
		} else {
			tmp_1 = (1.0 / hypot(y_46_im, 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 < -1.43520573476647616e138

   1. Initial program 42.8

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

   2. Applied add-sqr-sqrt_binary64 42.8

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

   3. Applied *-un-lft-identity_binary64 42.8

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

   4. Applied times-frac_binary64 42.8

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

   5. Simplified 42.8

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

   6. Simplified 28.1

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

   7. Taylor expanded in y.im around -inf 11.8

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

   8. Simplified 7.9

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

   if -1.43520573476647616e138 < y.im < -3.2946202192791303e-195 or 1.13275978535873808e-247 < y.im < 4.0321870185284071e74

   1. Initial program 17.9

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

   2. Applied add-sqr-sqrt_binary64 17.9

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

   3. Applied *-un-lft-identity_binary64 17.9

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

   4. Applied times-frac_binary64 17.9

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

   5. Simplified 17.9

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

   6. Simplified 11.9

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

   7. Applied associate-*l/_binary64 11.7

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

   8. Simplified 11.7

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

   if -3.2946202192791303e-195 < y.im < 1.13275978535873808e-247

   1. Initial program 25.0

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

   2. Taylor expanded in y.re around inf 9.4

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

   if 4.0321870185284071e74 < y.im

   1. Initial program 37.6

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

   2. Applied add-sqr-sqrt_binary64 37.6

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

   3. Applied *-un-lft-identity_binary64 37.6

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

   4. Applied times-frac_binary64 37.6

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

   5. Simplified 37.6

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

   6. Simplified 24.6

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

   7. Taylor expanded in y.im around inf 14.6

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

   8. Simplified 11.4

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4352057347664762 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -3.2946202192791303 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.1327597853587381 \cdot 10^{-247}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.re}^{2}} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.032187018528407 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))