_divideComplex, imaginary part

Average Error: 25.9 → 12.4

Time: 8.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.8184599620892983 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.im \leq 1.8848197063060136 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.354325968273711 \cdot 10^{-221}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 7.169555933218119 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.8184599620892983e+108)
   (/ x.re (hypot y.im y.re))
   (let* ((t_0
           (/
            (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.im y.re))
            (hypot y.im y.re))))
     (if (<= y.im 1.8848197063060136e-306)
       t_0
       (if (<= y.im 3.354325968273711e-221)
         (/ x.im y.re)
         (if (<= y.im 7.169555933218119e+166)
           t_0
           (/ (- (/ (* y.re x.im) y.im) x.re) (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_im * y_46_re) - (x_46_re * 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 <= -1.8184599620892983e+108) {
		tmp = x_46_re / hypot(y_46_im, y_46_re);
	} else {
		double t_0 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
		double tmp_1;
		if (y_46_im <= 1.8848197063060136e-306) {
			tmp_1 = t_0;
		} else if (y_46_im <= 3.354325968273711e-221) {
			tmp_1 = x_46_im / y_46_re;
		} else if (y_46_im <= 7.169555933218119e+166) {
			tmp_1 = t_0;
		} else {
			tmp_1 = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / hypot(y_46_im, y_46_re);
		}
		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 < -1.8184599620892983e108

   1. Initial program 40.8

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

   2. Simplified 40.8

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

   3. Applied add-sqr-sqrt_binary64 40.8

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

   4. Applied *-un-lft-identity_binary64 40.8

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

   5. Applied times-frac_binary64 40.8

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

   6. Simplified 40.8

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

   7. Simplified 27.4

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

   8. Applied associate-*l/_binary64 27.3

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

   9. Simplified 27.3

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

   10. Taylor expanded in y.im around -inf 15.7

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

   if -1.8184599620892983e108 < y.im < 1.8848197063060136e-306 or 3.35432596827371092e-221 < y.im < 7.1695559332181193e166

   1. Initial program 19.0

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

   2. Simplified 19.0

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

   3. Applied add-sqr-sqrt_binary64 19.0

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

   4. Applied *-un-lft-identity_binary64 19.0

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

   5. Applied times-frac_binary64 19.0

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

   6. Simplified 19.0

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

   7. Simplified 12.1

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

   8. Applied associate-*l/_binary64 11.9

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

   if 1.8848197063060136e-306 < y.im < 3.35432596827371092e-221

   1. Initial program 22.6

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

   2. Simplified 22.6

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

   3. Taylor expanded in y.re around inf 10.9

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

   if 7.1695559332181193e166 < y.im

   1. Initial program 44.1

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

   2. Simplified 44.1

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

   3. Applied add-sqr-sqrt_binary64 44.1

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

   4. Applied *-un-lft-identity_binary64 44.1

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

   5. Applied times-frac_binary64 44.1

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

   6. Simplified 44.1

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

   7. Simplified 29.8

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

   8. Applied associate-*l/_binary64 29.8

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

   9. Simplified 29.8

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

   10. Taylor expanded in y.re around 0 11.2

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8184599620892983 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.8848197063060136 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 3.354325968273711 \cdot 10^{-221}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 7.169555933218119 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))