Average Error: 26.0 → 10.9
Time: 5.6s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \mathbf{if}\;y.im \leq -7.183038243223042 \cdot 10^{+168}:\\ \;\;\;\;\frac{-t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.106771931752182 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
\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 -7.183038243223042 \cdot 10^{+168}:\\
\;\;\;\;\frac{-t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{elif}\;y.im \leq 5.106771931752182 \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{else}:\\
\;\;\;\;\frac{t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\


\end{array}

(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 -7.183038243223042e+168)
     (/ (- t_0) (hypot y.im y.re))
     (if (<= y.im 5.106771931752182e+135)
       (/
        (/ (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))))))
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 <= -7.183038243223042e+168) {
		tmp = -t_0 / hypot(y_46_im, y_46_re);
	} else if (y_46_im <= 5.106771931752182e+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 {
		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 3 regimes

  2. if y.im < -7.18303824322304229e168

    1. Initial program 44.8

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

    2. Simplified44.8

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

    3. Applied add-sqr-sqrt_binary6444.8

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

    4. Applied *-un-lft-identity_binary6444.8

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

    5. Applied times-frac_binary6444.8

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

    6. Simplified44.8

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

    7. Simplified29.5

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

    8. Applied associate-*r/_binary6429.5

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

    9. Taylor expanded in y.im around -inf 11.2

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

    10. Simplified6.1

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

    if -7.18303824322304229e168 < y.im < 5.1067719317521821e135

    1. Initial program 19.7

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

    2. Simplified19.7

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

    3. Applied add-sqr-sqrt_binary6419.7

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

    4. Applied *-un-lft-identity_binary6419.7

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

    5. Applied times-frac_binary6419.7

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

    6. Simplified19.7

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

    7. Simplified12.5

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

    8. Applied associate-*l/_binary6412.4

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

    9. Simplified12.4

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

    if 5.1067719317521821e135 < y.im

    1. Initial program 42.1

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

    2. Simplified42.1

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

    3. Applied add-sqr-sqrt_binary6442.1

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

    4. Applied *-un-lft-identity_binary6442.1

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

    5. Applied times-frac_binary6442.1

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

    6. Simplified42.1

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

    7. Simplified28.9

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

    8. Applied associate-*r/_binary6428.9

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

    9. Taylor expanded in y.im around inf 12.0

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

    10. Simplified7.6

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.183038243223042 \cdot 10^{+168}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.106771931752182 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]

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))))