_divideComplex, real part

Average Error: 26.4 → 10.2

Time: 10.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\\ \mathbf{if}\;y.re \leq -2.937760174171857 \cdot 10^{+153}:\\ \;\;\;\;\frac{-t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.050794049494874 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.8636600701744714 \cdot 10^{-111}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6628009337917042 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]

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 (/ y.im y.re) x.im x.re)))
   (if (<= y.re -2.937760174171857e+153)
     (/ (- t_0) (hypot y.im y.re))
     (if (<= y.re -1.050794049494874e-197)
       (/
        (/ (fma y.im x.im (* y.re x.re)) (hypot y.im y.re))
        (hypot y.im y.re))
       (if (<= y.re 2.8636600701744714e-111)
         (+ (/ (* y.re x.re) (pow y.im 2.0)) (/ x.im y.im))
         (if (<= y.re 1.6628009337917042e+133)
           (/
            (/ (fma x.re y.re (* y.im x.im)) (hypot y.im y.re))
            (hypot y.im y.re))
           (/ t_0 (hypot y.im y.re))))))))

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((y_46_im / y_46_re), x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -2.937760174171857e+153) {
		tmp = -t_0 / hypot(y_46_im, y_46_re);
	} else if (y_46_re <= -1.050794049494874e-197) {
		tmp = (fma(y_46_im, x_46_im, (y_46_re * x_46_re)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
	} else if (y_46_re <= 2.8636600701744714e-111) {
		tmp = ((y_46_re * x_46_re) / pow(y_46_im, 2.0)) + (x_46_im / y_46_im);
	} else if (y_46_re <= 1.6628009337917042e+133) {
		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 {
		tmp = t_0 / hypot(y_46_im, y_46_re);
	}
	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.re < -2.93776017417185701e153

   1. Initial program 44.9

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

   2. Applied add-sqr-sqrt_binary64 44.9

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

   3. Applied *-un-lft-identity_binary64 44.9

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

   4. Applied times-frac_binary64 44.9

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

   5. Simplified 44.9

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

   6. Simplified 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. Applied associate-*l/_binary64 28.1

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

   8. Simplified 28.1

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

   9. Taylor expanded in y.re around -inf 10.9

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

   10. Simplified 7.0

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

   if -2.93776017417185701e153 < y.re < -1.05079404949487398e-197

   1. Initial program 18.7

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

   2. Applied add-sqr-sqrt_binary64 18.7

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

   3. Applied *-un-lft-identity_binary64 18.7

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

   4. Applied times-frac_binary64 18.7

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

   5. Simplified 18.7

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

   6. Simplified 12.9

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

   7. Applied associate-*l/_binary64 12.8

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

   if -1.05079404949487398e-197 < y.re < 2.86366007017447139e-111

   1. Initial program 21.9

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

   2. Taylor expanded in y.re around 0 9.3

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

   if 2.86366007017447139e-111 < y.re < 1.6628009337917042e133

   1. Initial program 17.1

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

   2. Applied add-sqr-sqrt_binary64 17.1

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

   3. Applied *-un-lft-identity_binary64 17.1

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

   4. Applied times-frac_binary64 17.1

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

   5. Simplified 17.1

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

   6. Simplified 11.8

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

   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 1.6628009337917042e133 < y.re

   1. Initial program 44.1

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

   2. Applied add-sqr-sqrt_binary64 44.1

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

   3. Applied *-un-lft-identity_binary64 44.1

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

   4. Applied times-frac_binary64 44.1

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

   5. Simplified 44.1

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

   6. Simplified 29.6

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

   7. Applied associate-*l/_binary64 29.5

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

   8. Simplified 29.5

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

   9. Taylor expanded in y.re around inf 11.9

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

   10. Simplified 7.6

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.937760174171857 \cdot 10^{+153}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.050794049494874 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq 2.8636600701744714 \cdot 10^{-111}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6628009337917042 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021307 
(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))))