Average Error: 26.2 → 12.1
Time: 7.5s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.3591271380374226 \cdot 10^{+150}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8263209635399904 \cdot 10^{-128}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{elif}\;y.re \leq 4.876955999112138 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2} + {y.re}^{2}}\\ \mathbf{elif}\;y.re \leq 5.930099859978519 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \leq -1.3591271380374226 \cdot 10^{+150}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\

\mathbf{elif}\;y.re \leq -1.8263209635399904 \cdot 10^{-128}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\

\mathbf{elif}\;y.re \leq 4.876955999112138 \cdot 10^{-70}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2} + {y.re}^{2}}\\

\mathbf{elif}\;y.re \leq 5.930099859978519 \cdot 10^{+115}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\

\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
 (if (<= y.re -1.3591271380374226e+150)
   (+
    (*
     (/ y.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
     (/ x.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0)))))
    (/ x.re y.re))
   (if (<= y.re -1.8263209635399904e-128)
     (+
      (*
       (/ y.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
       (/ x.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0)))))
      (*
       (/ y.re (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
       (/ x.re (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))))
     (if (<= y.re 4.876955999112138e-70)
       (+ (/ x.im y.im) (/ (* y.re x.re) (+ (pow y.im 2.0) (pow y.re 2.0))))
       (if (<= y.re 5.930099859978519e+115)
         (+
          (*
           (/ y.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
           (/ x.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0)))))
          (*
           (/ y.re (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
           (/ x.re (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))))
         (+
          (*
           (/ y.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
           (/ x.im (sqrt (+ (pow y.im 2.0) (pow y.re 2.0)))))
          (/ x.re 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 tmp;
	if (y_46_re <= -1.3591271380374226e+150) {
		tmp = ((y_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0)))) + (x_46_re / y_46_re);
	} else if (y_46_re <= -1.8263209635399904e-128) {
		tmp = ((y_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0)))) + ((y_46_re / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_re / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))));
	} else if (y_46_re <= 4.876955999112138e-70) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / (pow(y_46_im, 2.0) + pow(y_46_re, 2.0)));
	} else if (y_46_re <= 5.930099859978519e+115) {
		tmp = ((y_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0)))) + ((y_46_re / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_re / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))));
	} else {
		tmp = ((y_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (x_46_im / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0)))) + (x_46_re / 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if y.re < -1.35912713803742261e150 or 5.9300998599785194e115 < y.re

    1. Initial program 41.9

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

    2. Taylor expanded around 0 41.9

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

    3. Simplified41.9

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

    5. Applied add-sqr-sqrt_binary6441.9

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

    6. Applied times-frac_binary6441.7

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

    7. Simplified41.7

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

    8. Simplified41.7

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

    9. Taylor expanded around inf 13.7

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

    if -1.35912713803742261e150 < y.re < -1.82632096353999042e-128 or 4.8769559991121382e-70 < y.re < 5.9300998599785194e115

    1. Initial program 17.5

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

    2. Taylor expanded around 0 17.5

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

    3. Simplified17.5

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

    5. Applied add-sqr-sqrt_binary6417.5

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

    6. Applied times-frac_binary6416.0

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

    7. Simplified16.0

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

    8. Simplified16.0

      \[\leadsto \frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \color{blue}{\frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}} + \frac{y.re \cdot x.re}{{y.re}^{2} + {y.im}^{2}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt_binary6416.0

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

    11. Applied times-frac_binary6412.4

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

    if -1.82632096353999042e-128 < y.re < 4.8769559991121382e-70

    1. Initial program 22.0

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

    2. Taylor expanded around 0 22.0

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

    3. Simplified22.0

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

    4. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} + \frac{y.re \cdot x.re}{{y.re}^{2} + {y.im}^{2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3591271380374226 \cdot 10^{+150}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8263209635399904 \cdot 10^{-128}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{elif}\;y.re \leq 4.876955999112138 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2} + {y.re}^{2}}\\ \mathbf{elif}\;y.re \leq 5.930099859978519 \cdot 10^{+115}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{y.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}} + \frac{x.re}{y.re}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))))