_divideComplex, real part

Average Error: 26.4 → 10.0

Time: 9.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \mathbf{if}\;y.im \leq -2.558733259505715 \cdot 10^{+128}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.0098211784491114 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.544901258368836 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.im \leq 5.209498248710266 \cdot 10^{+98}:\\ \;\;\;\;\frac{t_1 \cdot \mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_0\\ \end{array}\\ \end{array} \]

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 -2.558733259505715e+128)
     (* t_0 (/ -1.0 (hypot y.im y.re)))
     (if (<= y.im -1.0098211784491114e-135)
       (/
        (/ (fma x.re y.re (* y.im x.im)) (hypot y.im y.re))
        (hypot y.im y.re))
       (if (<= y.im 1.544901258368836e-100)
         (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
         (let* ((t_1 (/ 1.0 (hypot y.im y.re))))
           (if (<= y.im 5.209498248710266e+98)
             (/ (* t_1 (fma y.im x.im (* y.re x.re))) (hypot y.im y.re))
             (* t_1 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 <= -2.558733259505715e+128) {
		tmp = t_0 * (-1.0 / hypot(y_46_im, y_46_re));
	} else if (y_46_im <= -1.0098211784491114e-135) {
		tmp = (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);
	} else if (y_46_im <= 1.544901258368836e-100) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else {
		double t_1 = 1.0 / hypot(y_46_im, y_46_re);
		double tmp_1;
		if (y_46_im <= 5.209498248710266e+98) {
			tmp_1 = (t_1 * fma(y_46_im, x_46_im, (y_46_re * x_46_re))) / hypot(y_46_im, y_46_re);
		} else {
			tmp_1 = t_1 * 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 5 regimes

2. if y.im < -2.558733259505715e128

   1. Initial program 42.0

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

   2. Simplified 42.0

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

   3. Applied add-sqr-sqrt_binary64 41.9

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

   4. Applied *-un-lft-identity_binary64 41.9

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

   5. Applied times-frac_binary64 42.0

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

   6. Simplified 42.0

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

   7. Simplified 27.1

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

   8. Taylor expanded in y.im around -inf 10.9

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

   9. Simplified 7.1

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

   if -2.558733259505715e128 < y.im < -1.0098211784491114e-135

   1. Initial program 16.4

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

   2. Simplified 16.4

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

   3. Applied add-sqr-sqrt_binary64 16.4

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

   4. Applied *-un-lft-identity_binary64 16.4

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

   5. Applied times-frac_binary64 16.4

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

   6. Simplified 16.4

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

   7. Simplified 12.1

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

   8. Applied associate-*l/_binary64 12.0

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

   9. Simplified 12.0

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

   if -1.0098211784491114e-135 < y.im < 1.5449012583688359e-100

   1. Initial program 22.3

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

   2. Simplified 22.3

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

   3. Applied add-sqr-sqrt_binary64 22.3

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

   4. Applied *-un-lft-identity_binary64 22.3

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

   5. Applied times-frac_binary64 22.3

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

   6. Simplified 22.3

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

   7. Simplified 12.4

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

   8. Taylor expanded in y.im around 0 10.9

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

   9. Simplified 9.4

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

   if 1.5449012583688359e-100 < y.im < 5.20949824871026587e98

   1. Initial program 16.3

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

   2. Simplified 16.3

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

   3. Applied add-sqr-sqrt_binary64 16.3

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

   4. Applied *-un-lft-identity_binary64 16.3

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

   5. Applied times-frac_binary64 16.3

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

   6. Simplified 16.3

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

   7. Simplified 11.2

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

   8. Applied associate-*r/_binary64 11.2

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

   if 5.20949824871026587e98 < y.im

   1. Initial program 40.9

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

   2. Simplified 40.9

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

   3. Applied add-sqr-sqrt_binary64 40.9

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

   4. Applied *-un-lft-identity_binary64 40.9

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

   5. Applied times-frac_binary64 40.9

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

   6. Simplified 40.9

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

   7. Simplified 27.8

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

   8. Taylor expanded in y.im around inf 13.8

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

   9. Simplified 9.8

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.558733259505715 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.0098211784491114 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.544901258368836 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.209498248710266 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))