Average Error: 26.0 → 16.5
Time: 9.2s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.3339225097803556 \cdot 10^{-17}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.1035576422910915 \cdot 10^{-08}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.0502000497354221 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]
\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 -2.3339225097803556 \cdot 10^{-17}:\\
\;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\

\mathbf{elif}\;y.re \leq 1.1035576422910915 \cdot 10^{-08}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\

\mathbf{elif}\;y.re \leq 1.0502000497354221 \cdot 10^{+124}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.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}}\\

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

\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 -2.3339225097803556e-17)
   (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) y.im)))
   (if (<= y.re 1.1035576422910915e-08)
     (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
     (if (<= y.re 1.0502000497354221e+124)
       (/
        (/
         (+ (* y.re x.re) (* x.im y.im))
         (sqrt (+ (* y.re y.re) (* y.im y.im))))
        (sqrt (+ (* y.re y.re) (* y.im y.im))))
       (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) y.im)))))))
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 <= -2.3339225097803556e-17) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_re <= 1.1035576422910915e-08) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_re <= 1.0502000497354221e+124) {
		tmp = (((y_46_re * x_46_re) + (x_46_im * y_46_im)) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	}
	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 < -2.33392250978035558e-17 or 1.0502000497354221e124 < y.re

    1. Initial program 35.1

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

    3. Applied clear-num_binary6435.2

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

    4. Simplified35.2

      \[\leadsto \frac{1}{\color{blue}{\frac{{y.re}^{2} + {y.im}^{2}}{y.im \cdot x.im + y.re \cdot x.re}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity_binary6435.2

      \[\leadsto \frac{1}{\frac{{y.re}^{2} + {y.im}^{2}}{\color{blue}{1 \cdot \left(y.im \cdot x.im + y.re \cdot x.re\right)}}}\]

    7. Applied add-sqr-sqrt_binary6435.2

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt{{y.re}^{2} + {y.im}^{2}}}}{1 \cdot \left(y.im \cdot x.im + y.re \cdot x.re\right)}}\]

    8. Applied times-frac_binary6435.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{y.re}^{2} + {y.im}^{2}}}{1} \cdot \frac{\sqrt{{y.re}^{2} + {y.im}^{2}}}{y.im \cdot x.im + y.re \cdot x.re}}}\]

    9. Simplified35.2

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt{{y.re}^{2} + {y.im}^{2}}}{y.im \cdot x.im + y.re \cdot x.re}}\]

    10. Simplified35.2

      \[\leadsto \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}} \cdot \color{blue}{\frac{\sqrt{{y.re}^{2} + {y.im}^{2}}}{y.re \cdot x.re + y.im \cdot x.im}}}\]

    11. Taylor expanded around inf 18.1

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

    12. Simplified17.6

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

    if -2.33392250978035558e-17 < y.re < 1.103557642291092e-8

    1. Initial program 19.1

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

    2. Taylor expanded around 0 15.2

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

    if 1.103557642291092e-8 < y.re < 1.0502000497354221e124

    1. Initial program 17.9

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

    3. Applied add-sqr-sqrt_binary6417.9

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

    4. Applied associate-/r*_binary6417.8

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3339225097803556 \cdot 10^{-17}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.1035576422910915 \cdot 10^{-08}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.0502000497354221 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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))))