_divideComplex, real part

Percentage Accurate: 61.0% → 84.9%

Time: 9.8s

Alternatives: 10

Speedup: 2.1×

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: 61.0% 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: 84.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+287}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}\right)\\ \end{array} \end{array} \]

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+287)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (* (/ -1.0 y.re) (- (- x.re) (/ x.im (/ 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 ((((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+287) {
		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 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im / (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 (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+287)
		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(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(x_46_im / 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[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+287], 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[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $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))) < 1.0000000000000001e287

   1. Initial program 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. add-sqr-sqrt 79.7%

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

      3. times-frac 79.7%

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

      4. hypot-def 79.7%

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

      5. fma-def 79.7%

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

      6. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 1.0000000000000001e287 < (/.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 15.2%

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

      1. *-un-lft-identity 15.2%

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

      2. add-sqr-sqrt 15.2%

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

      3. times-frac 15.2%

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

      4. hypot-def 15.2%

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

      5. fma-def 15.2%

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

      6. hypot-def 22.1%

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

   3. Applied egg-rr 22.1%

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

   4. Taylor expanded in y.re around -inf 33.3%

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

      1. neg-mul-1 33.3%

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

      2. +-commutative 33.3%

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

      3. unsub-neg 33.3%

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

      4. mul-1-neg 33.3%

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

      5. associate-/l* 35.1%

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

      6. distribute-neg-frac 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in y.re around -inf 71.5%

      \[\leadsto \color{blue}{\frac{-1}{y.re}} \cdot \left(\frac{-x.im}{\frac{y.re}{y.im}} - x.re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+287}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}\right)\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))))
        (t_1 (* (/ -1.0 y.re) (- (* y.im (/ (- x.im) y.re)) x.re))))
   (if (<= y.re -7.5e+82)
     t_1
     (if (<= y.re -1.15e-146)
       t_0
       (if (<= y.re 3.35e-87)
         (+ (/ x.im y.im) (* y.re (/ x.re (pow y.im 2.0))))
         (if (<= y.re 7.2e+47) t_0 t_1))))))

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 t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -7.5e+82) {
		tmp = t_1;
	} else if (y_46_re <= -1.15e-146) {
		tmp = t_0;
	} else if (y_46_re <= 3.35e-87) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((-1.0d0) / y_46re) * ((y_46im * (-x_46im / y_46re)) - x_46re)
    if (y_46re <= (-7.5d+82)) then
        tmp = t_1
    else if (y_46re <= (-1.15d-146)) then
        tmp = t_0
    else if (y_46re <= 3.35d-87) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im ** 2.0d0)))
    else if (y_46re <= 7.2d+47) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -7.5e+82) {
		tmp = t_1;
	} else if (y_46_re <= -1.15e-146) {
		tmp = t_0;
	} else if (y_46_re <= 3.35e-87) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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))
	t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re)
	tmp = 0
	if y_46_re <= -7.5e+82:
		tmp = t_1
	elif y_46_re <= -1.15e-146:
		tmp = t_0
	elif y_46_re <= 3.35e-87:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 7.2e+47:
		tmp = t_0
	else:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(y_46_im * Float64(Float64(-x_46_im) / y_46_re)) - x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.5e+82)
		tmp = t_1;
	elseif (y_46_re <= -1.15e-146)
		tmp = t_0;
	elseif (y_46_re <= 3.35e-87)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_1;
	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));
	t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7.5e+82)
		tmp = t_1;
	elseif (y_46_re <= -1.15e-146)
		tmp = t_0;
	elseif (y_46_re <= 3.35e-87)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(y$46$im * N[((-x$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+82], t$95$1, If[LessEqual[y$46$re, -1.15e-146], t$95$0, If[LessEqual[y$46$re, 3.35e-87], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+47], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.4999999999999999e82 or 7.20000000000000015e47 < y.re

   1. Initial program 45.0%

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

      1. *-un-lft-identity 45.0%

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

      2. add-sqr-sqrt 45.0%

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

      3. times-frac 44.9%

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

      4. hypot-def 44.9%

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

      5. fma-def 44.9%

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

      6. hypot-def 69.3%

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

   3. Applied egg-rr 69.3%

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

   4. Taylor expanded in y.re around -inf 48.8%

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

      1. neg-mul-1 48.8%

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

      2. +-commutative 48.8%

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

      3. unsub-neg 48.8%

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

      4. mul-1-neg 48.8%

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

      5. associate-/l* 48.9%

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

      6. distribute-neg-frac 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in y.re around -inf 90.5%

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

   8. Taylor expanded in x.im around 0 88.9%

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

      1. mul-1-neg 88.9%

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

      2. associate-*l/ 92.2%

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

      3. distribute-rgt-neg-out 92.2%

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

   10. Simplified 92.2%

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

   if -7.4999999999999999e82 < y.re < -1.15e-146 or 3.35e-87 < y.re < 7.20000000000000015e47

   1. Initial program 86.7%

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

   if -1.15e-146 < y.re < 3.35e-87

   1. Initial program 73.5%

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

   2. Taylor expanded in y.re around 0 88.7%

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

      1. associate-/l* 86.4%

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

      2. associate-/r/ 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.35 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+66}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -8.8 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \end{array} \]

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 -1.25e+66)
     (* (+ x.re (* y.im (/ x.im y.re))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -8.8e-147)
       t_0
       (if (<= y.re 3.8e-86)
         (+ (/ x.im y.im) (* y.re (/ x.re (pow y.im 2.0))))
         (if (<= y.re 7.2e+47)
           t_0
           (* (/ -1.0 y.re) (- (* y.im (/ (- x.im) y.re)) x.re))))))))

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 <= -1.25e+66) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -8.8e-147) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e-86) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	}
	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 <= -1.25e+66) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -8.8e-147) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e-86) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	}
	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 <= -1.25e+66:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -8.8e-147:
		tmp = t_0
	elif y_46_re <= 3.8e-86:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 7.2e+47:
		tmp = t_0
	else:
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re)
	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 <= -1.25e+66)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -8.8e-147)
		tmp = t_0;
	elseif (y_46_re <= 3.8e-86)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(y_46_im * Float64(Float64(-x_46_im) / 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)
	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 <= -1.25e+66)
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -8.8e-147)
		tmp = t_0;
	elseif (y_46_re <= 3.8e-86)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / 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_] := 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, -1.25e+66], N[(N[(x$46$re + N[(y$46$im * N[(x$46$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, -8.8e-147], t$95$0, If[LessEqual[y$46$re, 3.8e-86], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+47], t$95$0, N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(y$46$im * N[((-x$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.24999999999999998e66

   1. Initial program 43.4%

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

      1. *-un-lft-identity 43.4%

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

      2. add-sqr-sqrt 43.4%

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

      3. times-frac 43.4%

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

      4. hypot-def 43.4%

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

      5. fma-def 43.4%

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

      6. hypot-def 65.5%

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

   3. Applied egg-rr 65.5%

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

   4. Taylor expanded in y.re around -inf 86.3%

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

      1. neg-mul-1 86.3%

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

      2. +-commutative 86.3%

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

      3. unsub-neg 86.3%

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

      4. mul-1-neg 86.3%

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

      5. associate-/l* 86.4%

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

      6. distribute-neg-frac 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x.im around 0 86.3%

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

      1. mul-1-neg 86.1%

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

      2. associate-*l/ 88.1%

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

      3. distribute-rgt-neg-out 88.1%

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

   9. Simplified 88.3%

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

   if -1.24999999999999998e66 < y.re < -8.8000000000000004e-147 or 3.8e-86 < y.re < 7.20000000000000015e47

   1. Initial program 87.3%

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

   if -8.8000000000000004e-147 < y.re < 3.8e-86

   1. Initial program 73.5%

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

   2. Taylor expanded in y.re around 0 88.7%

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

      1. associate-/l* 86.4%

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

      2. associate-/r/ 86.4%

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

   4. Simplified 86.4%

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

   if 7.20000000000000015e47 < y.re

   1. Initial program 48.3%

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

      1. *-un-lft-identity 48.3%

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

      2. add-sqr-sqrt 48.3%

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

      3. times-frac 48.2%

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

      4. hypot-def 48.2%

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

      5. fma-def 48.2%

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

      6. hypot-def 72.7%

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

   3. Applied egg-rr 72.7%

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

   4. Taylor expanded in y.re around -inf 21.1%

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

      1. neg-mul-1 21.1%

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

      2. +-commutative 21.1%

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

      3. unsub-neg 21.1%

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

      4. mul-1-neg 21.1%

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

      5. associate-/l* 21.2%

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

      6. distribute-neg-frac 21.2%

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

   6. Simplified 21.2%

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

   7. Taylor expanded in y.re around -inf 92.8%

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

   8. Taylor expanded in x.im around 0 90.2%

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

      1. mul-1-neg 90.2%

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

      2. associate-*l/ 94.3%

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

      3. distribute-rgt-neg-out 94.3%

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

   10. Simplified 94.3%

       \[\leadsto \frac{-1}{y.re} \cdot \left(\color{blue}{\frac{x.im}{y.re} \cdot \left(-y.im\right)} - x.re\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+66}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -8.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \]

Alternative 4: 78.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))))
        (t_1 (* (/ -1.0 y.re) (- (* y.im (/ (- x.im) y.re)) x.re))))
   (if (<= y.re -7.2e+81)
     t_1
     (if (<= y.re -3e-257)
       t_0
       (if (<= y.re 7e-105)
         (/ x.im (+ y.im (* y.re (* y.re (/ 1.0 y.im)))))
         (if (<= y.re 7.2e+47) t_0 t_1))))))

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 t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+81) {
		tmp = t_1;
	} else if (y_46_re <= -3e-257) {
		tmp = t_0;
	} else if (y_46_re <= 7e-105) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((-1.0d0) / y_46re) * ((y_46im * (-x_46im / y_46re)) - x_46re)
    if (y_46re <= (-7.2d+81)) then
        tmp = t_1
    else if (y_46re <= (-3d-257)) then
        tmp = t_0
    else if (y_46re <= 7d-105) then
        tmp = x_46im / (y_46im + (y_46re * (y_46re * (1.0d0 / y_46im))))
    else if (y_46re <= 7.2d+47) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+81) {
		tmp = t_1;
	} else if (y_46_re <= -3e-257) {
		tmp = t_0;
	} else if (y_46_re <= 7e-105) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else if (y_46_re <= 7.2e+47) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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))
	t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re)
	tmp = 0
	if y_46_re <= -7.2e+81:
		tmp = t_1
	elif y_46_re <= -3e-257:
		tmp = t_0
	elif y_46_re <= 7e-105:
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))))
	elif y_46_re <= 7.2e+47:
		tmp = t_0
	else:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(y_46_im * Float64(Float64(-x_46_im) / y_46_re)) - x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.2e+81)
		tmp = t_1;
	elseif (y_46_re <= -3e-257)
		tmp = t_0;
	elseif (y_46_re <= 7e-105)
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_1;
	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));
	t_1 = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / y_46_re)) - x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7.2e+81)
		tmp = t_1;
	elseif (y_46_re <= -3e-257)
		tmp = t_0;
	elseif (y_46_re <= 7e-105)
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	elseif (y_46_re <= 7.2e+47)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(y$46$im * N[((-x$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e+81], t$95$1, If[LessEqual[y$46$re, -3e-257], t$95$0, If[LessEqual[y$46$re, 7e-105], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+47], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.20000000000000011e81 or 7.20000000000000015e47 < y.re

   1. Initial program 45.0%

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

      1. *-un-lft-identity 45.0%

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

      2. add-sqr-sqrt 45.0%

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

      3. times-frac 44.9%

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

      4. hypot-def 44.9%

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

      5. fma-def 44.9%

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

      6. hypot-def 69.3%

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

   3. Applied egg-rr 69.3%

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

   4. Taylor expanded in y.re around -inf 48.8%

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

      1. neg-mul-1 48.8%

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

      2. +-commutative 48.8%

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

      3. unsub-neg 48.8%

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

      4. mul-1-neg 48.8%

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

      5. associate-/l* 48.9%

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

      6. distribute-neg-frac 48.9%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in y.re around -inf 90.5%

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

   8. Taylor expanded in x.im around 0 88.9%

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

      1. mul-1-neg 88.9%

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

      2. associate-*l/ 92.2%

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

      3. distribute-rgt-neg-out 92.2%

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

   10. Simplified 92.2%

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

   if -7.20000000000000011e81 < y.re < -2.9999999999999999e-257 or 7e-105 < y.re < 7.20000000000000015e47

   1. Initial program 85.3%

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

   if -2.9999999999999999e-257 < y.re < 7e-105

   1. Initial program 69.7%

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

      1. *-un-lft-identity 69.7%

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

      2. add-sqr-sqrt 69.7%

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

      3. times-frac 69.8%

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

      4. hypot-def 69.8%

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

      5. fma-def 69.8%

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

      6. hypot-def 86.6%

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

   3. Applied egg-rr 86.6%

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

   4. Taylor expanded in x.re around 0 60.1%

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

      1. associate-/l* 60.8%

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

      2. +-commutative 60.8%

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

      3. unpow2 60.8%

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

      4. fma-def 60.8%

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

   6. Simplified 60.8%

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

   7. Taylor expanded in y.re around 0 82.7%

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

      1. pow2 82.7%

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

      2. div-inv 82.7%

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

      3. associate-*l* 84.9%

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

   9. Applied egg-rr 84.9%

      \[\leadsto \frac{x.im}{y.im + \color{blue}{y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-257}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \]

Alternative 5: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+28} \lor \neg \left(y.im \leq 4.3 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.6e+28) (not (<= y.im 4.3e+30)))
   (/ x.im (+ y.im (* y.re (* y.re (/ 1.0 y.im)))))
   (* (/ -1.0 y.re) (- (* y.im (/ (- x.im) 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_im <= -7.6e+28) || !(y_46_im <= 4.3e+30)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else {
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / 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_46im <= (-7.6d+28)) .or. (.not. (y_46im <= 4.3d+30))) then
        tmp = x_46im / (y_46im + (y_46re * (y_46re * (1.0d0 / y_46im))))
    else
        tmp = ((-1.0d0) / y_46re) * ((y_46im * (-x_46im / 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_im <= -7.6e+28) || !(y_46_im <= 4.3e+30)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else {
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / 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_im <= -7.6e+28) or not (y_46_im <= 4.3e+30):
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))))
	else:
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / 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_im <= -7.6e+28) || !(y_46_im <= 4.3e+30))
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	else
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(y_46_im * Float64(Float64(-x_46_im) / 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_im <= -7.6e+28) || ~((y_46_im <= 4.3e+30)))
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	else
		tmp = (-1.0 / y_46_re) * ((y_46_im * (-x_46_im / 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[Or[LessEqual[y$46$im, -7.6e+28], N[Not[LessEqual[y$46$im, 4.3e+30]], $MachinePrecision]], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[(N[(y$46$im * N[((-x$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5999999999999998e28 or 4.3e30 < y.im

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. add-sqr-sqrt 58.2%

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

      3. times-frac 58.2%

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

      4. hypot-def 58.2%

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

      5. fma-def 58.2%

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

      6. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in x.re around 0 50.0%

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

      1. associate-/l* 52.4%

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

      2. +-commutative 52.4%

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

      3. unpow2 52.4%

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

      4. fma-def 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in y.re around 0 75.2%

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

      1. pow2 75.2%

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

      2. div-inv 75.2%

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

      3. associate-*l* 77.8%

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

   9. Applied egg-rr 77.8%

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

   if -7.5999999999999998e28 < y.im < 4.3e30

   1. Initial program 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. add-sqr-sqrt 71.2%

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

      3. times-frac 71.2%

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

      4. hypot-def 71.2%

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

      5. fma-def 71.2%

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

      6. hypot-def 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in y.re around -inf 47.6%

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

      1. neg-mul-1 47.6%

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

      2. +-commutative 47.6%

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

      3. unsub-neg 47.6%

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

      4. mul-1-neg 47.6%

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

      5. associate-/l* 46.9%

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

      6. distribute-neg-frac 46.9%

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

   6. Simplified 46.9%

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

   7. Taylor expanded in y.re around -inf 86.4%

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

   8. Taylor expanded in x.im around 0 87.6%

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

      1. mul-1-neg 87.6%

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

      2. associate-*l/ 84.4%

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

      3. distribute-rgt-neg-out 84.4%

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

   10. Simplified 84.4%

       \[\leadsto \frac{-1}{y.re} \cdot \left(\color{blue}{\frac{x.im}{y.re} \cdot \left(-y.im\right)} - x.re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+28} \lor \neg \left(y.im \leq 4.3 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(y.im \cdot \frac{-x.im}{y.re} - x.re\right)\\ \end{array} \]

Alternative 6: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+27} \lor \neg \left(y.im \leq 6.5 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.8e+27) (not (<= y.im 6.5e+28)))
   (/ x.im (+ y.im (* y.re (* y.re (/ 1.0 y.im)))))
   (* (/ -1.0 y.re) (- (- x.re) (/ x.im (/ 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_im <= -3.8e+27) || !(y_46_im <= 6.5e+28)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (x_46_im / (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_46im <= (-3.8d+27)) .or. (.not. (y_46im <= 6.5d+28))) then
        tmp = x_46im / (y_46im + (y_46re * (y_46re * (1.0d0 / y_46im))))
    else
        tmp = ((-1.0d0) / y_46re) * (-x_46re - (x_46im / (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_im <= -3.8e+27) || !(y_46_im <= 6.5e+28)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	} else {
		tmp = (-1.0 / y_46_re) * (-x_46_re - (x_46_im / (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_im <= -3.8e+27) or not (y_46_im <= 6.5e+28):
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))))
	else:
		tmp = (-1.0 / y_46_re) * (-x_46_re - (x_46_im / (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_im <= -3.8e+27) || !(y_46_im <= 6.5e+28))
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re * Float64(1.0 / y_46_im)))));
	else
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(x_46_im / 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_im <= -3.8e+27) || ~((y_46_im <= 6.5e+28)))
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / y_46_im))));
	else
		tmp = (-1.0 / y_46_re) * (-x_46_re - (x_46_im / (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[Or[LessEqual[y$46$im, -3.8e+27], N[Not[LessEqual[y$46$im, 6.5e+28]], $MachinePrecision]], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.80000000000000022e27 or 6.5000000000000001e28 < y.im

   1. Initial program 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. add-sqr-sqrt 58.2%

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

      3. times-frac 58.2%

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

      4. hypot-def 58.2%

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

      5. fma-def 58.2%

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

      6. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in x.re around 0 50.0%

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

      1. associate-/l* 52.4%

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

      2. +-commutative 52.4%

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

      3. unpow2 52.4%

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

      4. fma-def 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in y.re around 0 75.2%

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

      1. pow2 75.2%

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

      2. div-inv 75.2%

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

      3. associate-*l* 77.8%

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

   9. Applied egg-rr 77.8%

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

   if -3.80000000000000022e27 < y.im < 6.5000000000000001e28

   1. Initial program 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. add-sqr-sqrt 71.2%

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

      3. times-frac 71.2%

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

      4. hypot-def 71.2%

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

      5. fma-def 71.2%

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

      6. hypot-def 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in y.re around -inf 47.6%

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

      1. neg-mul-1 47.6%

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

      2. +-commutative 47.6%

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

      3. unsub-neg 47.6%

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

      4. mul-1-neg 47.6%

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

      5. associate-/l* 46.9%

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

      6. distribute-neg-frac 46.9%

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

   6. Simplified 46.9%

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

   7. Taylor expanded in y.re around -inf 86.4%

      \[\leadsto \color{blue}{\frac{-1}{y.re}} \cdot \left(\frac{-x.im}{\frac{y.re}{y.im}} - x.re\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+27} \lor \neg \left(y.im \leq 6.5 \cdot 10^{+28}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}\right)\\ \end{array} \]

Alternative 7: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-9} \lor \neg \left(y.re \leq 75000000\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1e-9) (not (<= y.re 75000000.0)))
   (/ x.re y.re)
   (/ x.im (+ y.im (* y.re (* y.re (/ 1.0 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 <= -1e-9) || !(y_46_re <= 75000000.0)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / 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 <= (-1d-9)) .or. (.not. (y_46re <= 75000000.0d0))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / (y_46im + (y_46re * (y_46re * (1.0d0 / 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 <= -1e-9) || !(y_46_re <= 75000000.0)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / 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 <= -1e-9) or not (y_46_re <= 75000000.0):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / 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 <= -1e-9) || !(y_46_re <= 75000000.0))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re * Float64(1.0 / 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 <= -1e-9) || ~((y_46_re <= 75000000.0)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re * (1.0 / 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, -1e-9], N[Not[LessEqual[y$46$re, 75000000.0]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.00000000000000006e-9 or 7.5e7 < y.re

   1. Initial program 53.2%

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

   2. Taylor expanded in y.re around inf 74.1%

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

   if -1.00000000000000006e-9 < y.re < 7.5e7

   1. Initial program 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. add-sqr-sqrt 79.2%

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

      3. times-frac 79.2%

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

      4. hypot-def 79.2%

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

      5. fma-def 79.2%

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

      6. hypot-def 90.1%

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

   3. Applied egg-rr 90.1%

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

   4. Taylor expanded in x.re around 0 57.8%

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

      1. associate-/l* 59.9%

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

      2. +-commutative 59.9%

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

      3. unpow2 59.9%

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

      4. fma-def 59.9%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in y.re around 0 74.5%

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

      1. pow2 74.5%

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

      2. div-inv 74.5%

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

      3. associate-*l* 75.4%

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

   9. Applied egg-rr 75.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-9} \lor \neg \left(y.re \leq 75000000\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \left(y.re \cdot \frac{1}{y.im}\right)}\\ \end{array} \]

Alternative 8: 63.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.08 \cdot 10^{+89} \lor \neg \left(y.im \leq 6.2 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.08e+89) (not (<= y.im 6.2e+31)))
   (/ x.im y.im)
   (/ x.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 <= -1.08e+89) || !(y_46_im <= 6.2e+31)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_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 <= (-1.08d+89)) .or. (.not. (y_46im <= 6.2d+31))) then
        tmp = x_46im / y_46im
    else
        tmp = x_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 <= -1.08e+89) || !(y_46_im <= 6.2e+31)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_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 <= -1.08e+89) or not (y_46_im <= 6.2e+31):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_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 <= -1.08e+89) || !(y_46_im <= 6.2e+31))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_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 <= -1.08e+89) || ~((y_46_im <= 6.2e+31)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_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, -1.08e+89], N[Not[LessEqual[y$46$im, 6.2e+31]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.08000000000000006e89 or 6.2000000000000004e31 < y.im

   1. Initial program 56.3%

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

   2. Taylor expanded in y.re around 0 76.0%

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

   if -1.08000000000000006e89 < y.im < 6.2000000000000004e31

   1. Initial program 71.1%

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

   2. Taylor expanded in y.re around inf 67.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.08 \cdot 10^{+89} \lor \neg \left(y.im \leq 6.2 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 2.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 2.5e+143) (/ x.im y.im) (/ x.im 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_re <= 2.5e+143) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / 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_46re <= 2.5d+143) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / 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_re <= 2.5e+143) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / 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_re <= 2.5e+143:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / 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_re <= 2.5e+143)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / 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_re <= 2.5e+143)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_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, 2.5e+143], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 2.50000000000000006e143

   1. Initial program 71.7%

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

   2. Taylor expanded in y.re around 0 44.7%

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

   if 2.50000000000000006e143 < y.re

   1. Initial program 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.4%

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

      4. hypot-def 31.4%

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

      5. fma-def 31.4%

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

      6. hypot-def 67.4%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in y.im around -inf 19.2%

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

      1. neg-mul-1 19.2%

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

   6. Simplified 19.2%

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

   7. Taylor expanded in y.re around -inf 19.2%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 2.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10: 42.8% 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 65.6%

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

2. Taylor expanded in y.re around 0 39.1%

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

3. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2023334 
(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))))