_divideComplex, real part

Percentage Accurate: 61.7% → 85.4%

Time: 12.3s

Alternatives: 10

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(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))))

C

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));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static 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));
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(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))))

C

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));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static 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));
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      1e+283)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+283) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+283)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+283], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 9.99999999999999955e282

   1. Initial program 85.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 85.0%

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

      2. add-sqr-sqrt 85.0%

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

      3. times-frac 85.0%

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

      4. hypot-define 85.0%

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

      5. fma-define 85.0%

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

      6. hypot-define 97.2%

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

   4. Applied egg-rr 97.2%

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

   if 9.99999999999999955e282 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   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. Add Preprocessing

   3. Taylor expanded in y.re around inf 46.6%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 54.0%

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

   5. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 54.0%

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

      2. pow2 54.0%

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

      3. times-frac 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. *-commutative 58.6%

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

      2. associate-*l/ 58.6%

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

      3. *-un-lft-identity 58.6%

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

      4. associate-*l/ 60.7%

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

   9. Applied egg-rr 60.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+283}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+26}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.7 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.2e+26)
   (* (+ x.re (* x.im (/ y.im y.re))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.re -4.7e-107)
     (* (fma x.re y.re (* x.im y.im)) (/ 1.0 (pow (hypot y.re y.im) 2.0)))
     (if (<= y.re -9.5e-226)
       (+ (/ x.im y.im) (* x.re (/ y.re (pow y.im 2.0))))
       (if (<= y.re 1.05e+65)
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.2e+26) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -4.7e-107) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else if (y_46_re <= -9.5e-226) {
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.05e+65) {
		tmp = ((x_46_re * y_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 / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.2e+26)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -4.7e-107)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	elseif (y_46_re <= -9.5e-226)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(y_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 1.05e+65)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.2e+26], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.7e-107], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9.5e-226], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(y$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+65], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.20000000000000007e26

   1. Initial program 51.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 51.3%

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

      2. add-sqr-sqrt 51.3%

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

      3. times-frac 51.4%

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

      4. hypot-define 51.4%

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

      5. fma-define 51.4%

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

      6. hypot-define 64.3%

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

   4. Applied egg-rr 64.3%

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

   5. Taylor expanded in y.re around -inf 80.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} \]
   6. Step-by-step derivation

      1. neg-mul-1 80.5%

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

      2. +-commutative 80.5%

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

      3. unsub-neg 80.5%

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

      4. mul-1-neg 80.5%

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

      5. associate-/l* 84.0%

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

      6. distribute-lft-neg-in 84.0%

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

   7. Simplified 84.0%

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

   if -2.20000000000000007e26 < y.re < -4.69999999999999998e-107

   1. Initial program 88.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv 89.0%

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

      2. fma-define 89.0%

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

      3. add-sqr-sqrt 89.0%

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

      4. pow2 89.0%

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

      5. hypot-define 89.0%

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

   4. Applied egg-rr 89.0%

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

   if -4.69999999999999998e-107 < y.re < -9.5000000000000007e-226

   1. Initial program 72.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 72.2%

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

      2. add-sqr-sqrt 72.2%

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

      3. times-frac 72.2%

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

      4. hypot-define 72.2%

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

      5. fma-define 72.2%

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

      6. hypot-define 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in y.re around 0 96.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if -9.5000000000000007e-226 < y.re < 1.04999999999999996e65

   1. Initial program 86.0%

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

   if 1.04999999999999996e65 < y.re

   1. Initial program 41.6%

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

   3. Taylor expanded in y.re around inf 68.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-*r* 86.0%

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

      2. clear-num 86.1%

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

      3. un-div-inv 86.2%

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

      4. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+26}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.7 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.25 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -9e+72)
     (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
     (if (<= y.re -4.25e-106)
       t_0
       (if (<= y.re -8.6e-226)
         (+ (/ x.im y.im) (* x.re (/ y.re (pow y.im 2.0))))
         (if (<= y.re 1.25e+65)
           t_0
           (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -9e+72) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -4.25e-106) {
		tmp = t_0;
	} else if (y_46_re <= -8.6e-226) {
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.25e+65) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-9d+72)) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else if (y_46re <= (-4.25d-106)) then
        tmp = t_0
    else if (y_46re <= (-8.6d-226)) then
        tmp = (x_46im / y_46im) + (x_46re * (y_46re / (y_46im ** 2.0d0)))
    else if (y_46re <= 1.25d+65) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -9e+72) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -4.25e-106) {
		tmp = t_0;
	} else if (y_46_re <= -8.6e-226) {
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.25e+65) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -9e+72:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= -4.25e-106:
		tmp = t_0
	elif y_46_re <= -8.6e-226:
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 1.25e+65:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9e+72)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= -4.25e-106)
		tmp = t_0;
	elseif (y_46_re <= -8.6e-226)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(y_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 1.25e+65)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -9e+72)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= -4.25e-106)
		tmp = t_0;
	elseif (y_46_re <= -8.6e-226)
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 1.25e+65)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+72], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.25e-106], t$95$0, If[LessEqual[y$46$re, -8.6e-226], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(y$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e+65], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.9999999999999997e72

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around inf 74.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 75.4%

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

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 75.4%

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

      2. pow2 75.4%

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

      3. times-frac 76.9%

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

   7. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-*l/ 77.0%

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

      3. *-un-lft-identity 77.0%

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

      4. associate-*l/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -8.9999999999999997e72 < y.re < -4.2499999999999999e-106 or -8.60000000000000049e-226 < y.re < 1.24999999999999993e65

   1. Initial program 86.1%

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

   if -4.2499999999999999e-106 < y.re < -8.60000000000000049e-226

   1. Initial program 72.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 72.2%

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

      2. add-sqr-sqrt 72.2%

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

      3. times-frac 72.2%

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

      4. hypot-define 72.2%

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

      5. fma-define 72.2%

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

      6. hypot-define 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in y.re around 0 96.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if 1.24999999999999993e65 < y.re

   1. Initial program 41.6%

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

   3. Taylor expanded in y.re around inf 68.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-*r* 86.0%

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

      2. clear-num 86.1%

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

      3. un-div-inv 86.2%

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

      4. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+25}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -6.5e+25)
     (* (+ x.re (* x.im (/ y.im y.re))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -1.35e-108)
       t_0
       (if (<= y.re -8.6e-226)
         (+ (/ x.im y.im) (* x.re (/ y.re (pow y.im 2.0))))
         (if (<= y.re 1.6e+65)
           t_0
           (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -6.5e+25) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.35e-108) {
		tmp = t_0;
	} else if (y_46_re <= -8.6e-226) {
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.6e+65) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((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 tmp;
	if (y_46_re <= -6.5e+25) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.35e-108) {
		tmp = t_0;
	} else if (y_46_re <= -8.6e-226) {
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.6e+65) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -6.5e+25:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -1.35e-108:
		tmp = t_0
	elif y_46_re <= -8.6e-226:
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 1.6e+65:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6.5e+25)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.35e-108)
		tmp = t_0;
	elseif (y_46_re <= -8.6e-226)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(y_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 1.6e+65)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6.5e+25)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.35e-108)
		tmp = t_0;
	elseif (y_46_re <= -8.6e-226)
		tmp = (x_46_im / y_46_im) + (x_46_re * (y_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 1.6e+65)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e+25], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.35e-108], t$95$0, If[LessEqual[y$46$re, -8.6e-226], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(y$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e+65], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.50000000000000005e25

   1. Initial program 51.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 51.3%

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

      2. add-sqr-sqrt 51.3%

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

      3. times-frac 51.4%

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

      4. hypot-define 51.4%

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

      5. fma-define 51.4%

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

      6. hypot-define 64.3%

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

   4. Applied egg-rr 64.3%

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

   5. Taylor expanded in y.re around -inf 80.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.re + -1 \cdot \frac{x.im \cdot y.im}{y.re}\right)} \]
   6. Step-by-step derivation

      1. neg-mul-1 80.5%

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

      2. +-commutative 80.5%

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

      3. unsub-neg 80.5%

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

      4. mul-1-neg 80.5%

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

      5. associate-/l* 84.0%

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

      6. distribute-lft-neg-in 84.0%

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

   7. Simplified 84.0%

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

   if -6.50000000000000005e25 < y.re < -1.35000000000000002e-108 or -8.60000000000000049e-226 < y.re < 1.60000000000000003e65

   1. Initial program 86.6%

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

   if -1.35000000000000002e-108 < y.re < -8.60000000000000049e-226

   1. Initial program 72.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 72.2%

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

      2. add-sqr-sqrt 72.2%

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

      3. times-frac 72.2%

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

      4. hypot-define 72.2%

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

      5. fma-define 72.2%

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

      6. hypot-define 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in y.re around 0 96.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if 1.60000000000000003e65 < y.re

   1. Initial program 41.6%

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

   3. Taylor expanded in y.re around inf 68.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-*r* 86.0%

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

      2. clear-num 86.1%

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

      3. un-div-inv 86.2%

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

      4. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+25}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.6e+71)
   (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
   (if (<= y.re 1.3e+65)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.6e+71) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 1.3e+65) {
		tmp = ((x_46_re * y_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 / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3.6d+71)) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else if (y_46re <= 1.3d+65) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) / (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.6e+71) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 1.3e+65) {
		tmp = ((x_46_re * y_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 / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.6e+71:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 1.3e+65:
		tmp = ((x_46_re * y_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 / y_46_re) / (y_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.6e+71)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 1.3e+65)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.6e+71)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 1.3e+65)
		tmp = ((x_46_re * y_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 / y_46_re) / (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.6e+71], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e+65], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.6e71

   1. Initial program 45.6%

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

   3. Taylor expanded in y.re around inf 74.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 75.4%

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

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 75.4%

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

      2. pow2 75.4%

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

      3. times-frac 76.9%

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

   7. Applied egg-rr 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-*l/ 77.0%

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

      3. *-un-lft-identity 77.0%

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

      4. associate-*l/ 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -3.6e71 < y.re < 1.30000000000000001e65

   1. Initial program 83.7%

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

   if 1.30000000000000001e65 < y.re

   1. Initial program 41.6%

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

   3. Taylor expanded in y.re around inf 68.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. associate-*r* 86.0%

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

      2. clear-num 86.1%

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

      3. un-div-inv 86.2%

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

      4. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+71}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -0.37 \lor \neg \left(y.im \leq 1.3 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -0.37) (not (<= y.im 1.3e-29)))
   (/ x.im y.im)
   (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -0.37) || !(y_46_im <= 1.3e-29)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-0.37d0)) .or. (.not. (y_46im <= 1.3d-29))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -0.37) || !(y_46_im <= 1.3e-29)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -0.37) or not (y_46_im <= 1.3e-29):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -0.37) || !(y_46_im <= 1.3e-29))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -0.37) || ~((y_46_im <= 1.3e-29)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -0.37], N[Not[LessEqual[y$46$im, 1.3e-29]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -0.37 or 1.3000000000000001e-29 < y.im

   1. Initial program 55.3%

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

   3. Taylor expanded in y.re around 0 67.3%

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

   if -0.37 < y.im < 1.3000000000000001e-29

   1. Initial program 82.5%

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

   3. Taylor expanded in y.re around inf 81.3%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 79.8%

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

   5. Simplified 79.8%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 79.8%

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

      2. pow2 79.8%

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

      3. times-frac 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. associate-*l/ 80.7%

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

      2. *-lft-identity 80.7%

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

   9. Simplified 80.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.37 \lor \neg \left(y.im \leq 1.3 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{-104} \lor \neg \left(y.re \leq 1.45 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.65e-104) (not (<= y.re 1.45e-62)))
   (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.65e-104) || !(y_46_re <= 1.45e-62)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.65d-104)) .or. (.not. (y_46re <= 1.45d-62))) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.65e-104) || !(y_46_re <= 1.45e-62)) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.65e-104) or not (y_46_re <= 1.45e-62):
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.65e-104) || !(y_46_re <= 1.45e-62))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.65e-104) || ~((y_46_re <= 1.45e-62)))
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.65e-104], N[Not[LessEqual[y$46$re, 1.45e-62]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.65000000000000009e-104 or 1.44999999999999993e-62 < y.re

   1. Initial program 60.8%

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

   3. Taylor expanded in y.re around inf 67.4%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 69.5%

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

   5. Simplified 69.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 69.5%

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

      2. pow2 69.5%

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

      3. times-frac 71.5%

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

   7. Applied egg-rr 71.5%

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

      1. *-commutative 71.5%

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

      2. associate-*l/ 71.6%

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

      3. *-un-lft-identity 71.6%

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

      4. associate-*l/ 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -2.65000000000000009e-104 < y.re < 1.44999999999999993e-62

   1. Initial program 82.2%

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

   3. Taylor expanded in y.re around 0 77.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{-104} \lor \neg \left(y.re \leq 1.45 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.7e-92)
   (/ x.re y.re)
   (if (<= y.re 1.25e-40) (/ x.im y.im) (/ 1.0 (/ y.re x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.7e-92) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.25e-40) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = 1.0 / (y_46_re / x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.7d-92)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.25d-40) then
        tmp = x_46im / y_46im
    else
        tmp = 1.0d0 / (y_46re / x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.7e-92) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.25e-40) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = 1.0 / (y_46_re / x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.7e-92:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.25e-40:
		tmp = x_46_im / y_46_im
	else:
		tmp = 1.0 / (y_46_re / x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.7e-92)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.25e-40)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(1.0 / Float64(y_46_re / x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.7e-92)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.25e-40)
		tmp = x_46_im / y_46_im;
	else
		tmp = 1.0 / (y_46_re / x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.7e-92], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.25e-40], N[(x$46$im / y$46$im), $MachinePrecision], N[(1.0 / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.69999999999999995e-92

   1. Initial program 62.4%

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

   3. Taylor expanded in y.re around inf 61.3%

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

   if -2.69999999999999995e-92 < y.re < 1.24999999999999991e-40

   1. Initial program 82.5%

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

   3. Taylor expanded in y.re around 0 75.0%

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

   if 1.24999999999999991e-40 < y.re

   1. Initial program 55.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 55.9%

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

      2. add-sqr-sqrt 55.9%

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

      3. times-frac 55.9%

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

      4. hypot-define 55.9%

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

      5. fma-define 55.9%

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

      6. hypot-define 68.2%

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

   4. Applied egg-rr 68.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   5. Step-by-step derivation

      1. frac-times 55.9%

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

      2. *-un-lft-identity 55.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)} \]

      3. unpow2 55.9%

         \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]

      4. clear-num 55.8%

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

   6. Applied egg-rr 55.8%

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

   7. Taylor expanded in y.re around inf 66.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{y.re}{x.re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.re}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-95} \lor \neg \left(y.re \leq 2 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5e-95) (not (<= y.re 2e-38))) (/ x.re y.re) (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5e-95) || !(y_46_re <= 2e-38)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-5d-95)) .or. (.not. (y_46re <= 2d-38))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -5e-95) || !(y_46_re <= 2e-38)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -5e-95) or not (y_46_re <= 2e-38):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -5e-95) || !(y_46_re <= 2e-38))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -5e-95) || ~((y_46_re <= 2e-38)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -5e-95], N[Not[LessEqual[y$46$re, 2e-38]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.9999999999999998e-95 or 1.9999999999999999e-38 < y.re

   1. Initial program 59.6%

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

   3. Taylor expanded in y.re around inf 63.5%

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

   if -4.9999999999999998e-95 < y.re < 1.9999999999999999e-38

   1. Initial program 82.5%

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

   3. Taylor expanded in y.re around 0 75.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-95} \lor \neg \left(y.re \leq 2 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46im
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$im), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Taylor expanded in y.re around 0 46.0%

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

4. Final simplification 46.0%

   \[\leadsto \frac{x.im}{y.im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))