Average Error: 26.7 → 15.8
Time: 9.8s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.3266583380699222 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.im \leq -1.3781986218689126 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.6871257722081152 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.440833307462582 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{{y.im}^{2} + {y.re}^{2}}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \end{array}\]
\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.im \leq -1.3266583380699222 \cdot 10^{+83}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\

\mathbf{elif}\;y.im \leq -1.3781986218689126 \cdot 10^{-106}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 1.6871257722081152 \cdot 10^{-101}:\\
\;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 2.440833307462582 \cdot 10^{-20}:\\
\;\;\;\;\frac{1}{\frac{{y.im}^{2} + {y.re}^{2}}{y.re \cdot x.re + y.im \cdot x.im}}\\

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

\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.im -1.3266583380699222e+83)
   (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
   (if (<= y.im -1.3781986218689126e-106)
     (/
      (/
       (+ (* y.re x.re) (* y.im x.im))
       (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 1.6871257722081152e-101)
       (+ (/ x.re y.re) (/ (* y.im x.im) (pow y.re 2.0)))
       (if (<= y.im 2.440833307462582e-20)
         (/
          1.0
          (/
           (+ (pow y.im 2.0) (pow y.re 2.0))
           (+ (* y.re x.re) (* y.im x.im))))
         (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0))))))))
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_im <= -1.3266583380699222e+83) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_im <= -1.3781986218689126e-106) {
		tmp = (((y_46_re * x_46_re) + (y_46_im * x_46_im)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.6871257722081152e-101) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 2.440833307462582e-20) {
		tmp = 1.0 / ((pow(y_46_im, 2.0) + pow(y_46_re, 2.0)) / ((y_46_re * x_46_re) + (y_46_im * x_46_im)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	}
	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 4 regimes

  2. if y.im < -1.3266583380699222e83 or 2.440833307462582e-20 < y.im

    1. Initial program 35.0

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

    2. Taylor expanded around 0 19.1

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

    if -1.3266583380699222e83 < y.im < -1.3781986218689126e-106

    1. Initial program 14.7

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_112314.7

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

    4. Applied associate-/r*_binary64_104514.6

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

    5. Simplified14.6

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

    if -1.3781986218689126e-106 < y.im < 1.6871257722081152e-101

    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. Taylor expanded around inf 11.1

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

    if 1.6871257722081152e-101 < y.im < 2.440833307462582e-20

    1. Initial program 15.7

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied clear-num_binary64_110015.8

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

    4. Simplified15.8

      \[\leadsto \frac{1}{\color{blue}{\frac{{y.re}^{2} + {y.im}^{2}}{y.im \cdot x.im + y.re \cdot x.re}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3266583380699222 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.im \leq -1.3781986218689126 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.6871257722081152 \cdot 10^{-101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.440833307462582 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{{y.im}^{2} + {y.re}^{2}}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))))