Average Error: 25.8 → 12.3
Time: 10.9s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.553879560626287 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \left(\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)\right)\\ \mathbf{elif}\;y.im \leq -7.875830746956647 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 2.5365321914959593 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 7.162264487896466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \left(\left(x.re - \frac{x.im}{y.im} \cdot y.re\right) \cdot \frac{\sqrt[3]{y.re}}{y.im}\right)\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \leq -7.553879560626287 \cdot 10^{+93}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \left(\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)\right)\\

\mathbf{elif}\;y.im \leq -7.875830746956647 \cdot 10^{-107}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 2.5365321914959593 \cdot 10^{-134}:\\
\;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 7.162264487896466 \cdot 10^{+81}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.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.im}{y.im} + \frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \left(\left(x.re - \frac{x.im}{y.im} \cdot y.re\right) \cdot \frac{\sqrt[3]{y.re}}{y.im}\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
 (if (<= y.im -7.553879560626287e+93)
   (+
    (/ x.im y.im)
    (* (/ 1.0 y.im) (* (/ y.re y.im) (- x.re (* (/ x.im y.im) y.re)))))
   (if (<= y.im -7.875830746956647e-107)
     (/
      (/
       (+ (* y.re x.re) (* y.im x.im))
       (sqrt (+ (* y.re y.re) (* y.im y.im))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 2.5365321914959593e-134)
       (+ (/ x.re y.re) (/ (* y.im x.im) (pow y.re 2.0)))
       (if (<= y.im 7.162264487896466e+81)
         (/
          (/
           (+ (* y.re x.re) (* y.im x.im))
           (sqrt (+ (* y.re y.re) (* y.im y.im))))
          (sqrt (+ (* y.re y.re) (* y.im y.im))))
         (+
          (/ x.im y.im)
          (*
           (/ (* (cbrt y.re) (cbrt y.re)) y.im)
           (* (- x.re (* (/ x.im y.im) y.re)) (/ (cbrt y.re) 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_im <= -7.553879560626287e+93) {
		tmp = (x_46_im / y_46_im) + ((1.0 / y_46_im) * ((y_46_re / y_46_im) * (x_46_re - ((x_46_im / y_46_im) * y_46_re))));
	} else if (y_46_im <= -7.875830746956647e-107) {
		tmp = (((y_46_re * x_46_re) + (y_46_im * x_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 if (y_46_im <= 2.5365321914959593e-134) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 7.162264487896466e+81) {
		tmp = (((y_46_re * x_46_re) + (y_46_im * x_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_im / y_46_im) + (((cbrt(y_46_re) * cbrt(y_46_re)) / y_46_im) * ((x_46_re - ((x_46_im / y_46_im) * y_46_re)) * (cbrt(y_46_re) / 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 4 regimes

  2. if y.im < -7.5538795606262866e93

    1. Initial program 40.4

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

    3. Applied div-inv_binary64_143940.4

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

    4. Simplified40.4

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

    5. Taylor expanded around 0 23.2

      \[\leadsto \color{blue}{\left(\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\right) - \frac{{y.re}^{2} \cdot x.im}{{y.im}^{3}}}\]

    6. Simplified18.8

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

    8. Applied unpow2_binary64_150718.8

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

    9. Applied *-un-lft-identity_binary64_144218.8

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

    10. Applied times-frac_binary64_144816.6

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

    11. Applied associate-*l*_binary64_138313.6

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

    12. Simplified10.4

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

    if -7.5538795606262866e93 < y.im < -7.87583074695664708e-107 or 2.5365321914959593e-134 < y.im < 7.16226448789646589e81

    1. Initial program 15.1

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

    3. Applied add-sqr-sqrt_binary64_146415.1

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

    4. Applied associate-/r*_binary64_138615.0

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

    if -7.87583074695664708e-107 < y.im < 2.5365321914959593e-134

    1. Initial program 21.9

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

    2. Taylor expanded around inf 10.6

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

    if 7.16226448789646589e81 < y.im

    1. Initial program 37.4

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

    3. Applied div-inv_binary64_143937.4

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

    4. Simplified37.4

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

    5. Taylor expanded around 0 24.2

      \[\leadsto \color{blue}{\left(\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\right) - \frac{{y.re}^{2} \cdot x.im}{{y.im}^{3}}}\]

    6. Simplified19.5

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

    8. Applied unpow2_binary64_150719.5

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

    9. Applied add-cube-cbrt_binary64_147719.5

      \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{\left(\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}\right) \cdot \sqrt[3]{y.re}}}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)\]

    10. Applied times-frac_binary64_144818.0

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(\frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \frac{\sqrt[3]{y.re}}{y.im}\right)} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)\]

    11. Applied associate-*l*_binary64_138315.1

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \left(\frac{\sqrt[3]{y.re}}{y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)\right)}\]

    12. Simplified11.7

      \[\leadsto \frac{x.im}{y.im} + \frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \color{blue}{\left(\frac{\sqrt[3]{y.re}}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.553879560626287 \cdot 10^{+93}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{y.im} \cdot \left(\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)\right)\\ \mathbf{elif}\;y.im \leq -7.875830746956647 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 2.5365321914959593 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 7.162264487896466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + y.im \cdot x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\sqrt[3]{y.re} \cdot \sqrt[3]{y.re}}{y.im} \cdot \left(\left(x.re - \frac{x.im}{y.im} \cdot y.re\right) \cdot \frac{\sqrt[3]{y.re}}{y.im}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))