Average Error: 26.3 → 15.1
Time: 7.0s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.2232339396640187 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.164382113824996 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 6.598497970238571 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 3.0690585850140043 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]
\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 -1.2232339396640187 \cdot 10^{+115}:\\
\;\;\;\;\frac{x.re}{y.re}\\

\mathbf{elif}\;y.re \leq -5.164382113824996 \cdot 10^{-80}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.re \leq 6.598497970238571 \cdot 10^{-145}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\

\mathbf{elif}\;y.re \leq 3.0690585850140043 \cdot 10^{+27}:\\
\;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{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 -1.2232339396640187e+115)
   (/ x.re y.re)
   (if (<= y.re -5.164382113824996e-80)
     (/
      (/
       (+ (* y.re x.re) (* x.im y.im))
       (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.re 6.598497970238571e-145)
       (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
       (if (<= y.re 3.0690585850140043e+27)
         (/ (+ (* y.re x.re) (* x.im y.im)) (+ (* 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 <= -1.2232339396640187e+115) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.164382113824996e-80) {
		tmp = (((y_46_re * x_46_re) + (x_46_im * y_46_im)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.598497970238571e-145) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_re <= 3.0690585850140043e+27) {
		tmp = ((y_46_re * x_46_re) + (x_46_im * y_46_im)) / ((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 5 regimes

  2. if y.re < -1.2232339396640187e115

    1. Initial program 39.2

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

    2. Taylor expanded around inf 15.0

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

    if -1.2232339396640187e115 < y.re < -5.16438211382499615e-80

    1. Initial program 16.5

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

    3. Applied add-sqr-sqrt_binary64_146416.5

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

    4. Applied associate-/r*_binary64_138616.4

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

    5. Simplified16.4

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

    if -5.16438211382499615e-80 < y.re < 6.59849797023857136e-145

    1. Initial program 21.5

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

    2. Taylor expanded around 0 11.0

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

    3. Simplified11.0

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

    if 6.59849797023857136e-145 < y.re < 3.0690585850140043e27

    1. Initial program 15.6

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

    if 3.0690585850140043e27 < y.re

    1. Initial program 35.4

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

    3. Applied div-inv_binary64_143935.4

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

    4. Simplified35.4

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

    6. Applied add-sqr-sqrt_binary64_146435.4

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

    7. Applied add-sqr-sqrt_binary64_146435.4

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

    8. Applied times-frac_binary64_144835.5

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

    9. Simplified35.5

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

    10. Simplified35.5

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

    11. Taylor expanded around inf 19.1

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

    12. Simplified18.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2232339396640187 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.164382113824996 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 6.598497970238571 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 3.0690585850140043 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(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))))