_divideComplex, real part

Percentage Accurate: 61.6% → 83.9%

Time: 8.2s

Alternatives: 12

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: 61.6% 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: 83.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

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)))
      2e+255)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.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))) <= 2e+255) {
		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_im / y_46_im) + ((y_46_re / y_46_im) * (x_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))) <= 2e+255)
		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_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_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], 2e+255], 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$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $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))) < 1.99999999999999998e255

   1. Initial program 79.9%

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

      1. *-un-lft-identity 79.9%

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

      2. add-sqr-sqrt 79.9%

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

      3. times-frac 79.9%

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

      4. hypot-def 79.9%

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

      5. fma-def 79.9%

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

      6. hypot-def 95.5%

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

   3. Applied egg-rr 95.5%

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

   if 1.99999999999999998e255 < (/.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 9.4%

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

   2. Taylor expanded in y.re around 0 50.3%

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

      1. +-commutative 50.3%

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

      2. *-commutative 50.3%

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

      3. unpow2 50.3%

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

      4. times-frac 65.7%

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

   4. Simplified 65.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+255}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

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.im -1.45e+67)
     (/ x.im y.im)
     (if (<= y.im -4.8e-156)
       t_0
       (if (<= y.im 6.2e-145)
         (+ (/ x.re y.re) (/ x.im (/ y.re (/ y.im y.re))))
         (if (<= y.im 5.6e+112)
           t_0
           (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.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_im <= -1.45e+67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4.8e-156) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-145) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_46_im / y_46_re)));
	} else if (y_46_im <= 5.6e+112) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_46im <= (-1.45d+67)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-4.8d-156)) then
        tmp = t_0
    else if (y_46im <= 6.2d-145) then
        tmp = (x_46re / y_46re) + (x_46im / (y_46re / (y_46im / y_46re)))
    else if (y_46im <= 5.6d+112) then
        tmp = t_0
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_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_im <= -1.45e+67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4.8e-156) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-145) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_46_im / y_46_re)));
	} else if (y_46_im <= 5.6e+112) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_im <= -1.45e+67:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -4.8e-156:
		tmp = t_0
	elif y_46_im <= 6.2e-145:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_46_im / y_46_re)))
	elif y_46_im <= 5.6e+112:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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_im <= -1.45e+67)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -4.8e-156)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-145)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 5.6e+112)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_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_im <= -1.45e+67)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -4.8e-156)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-145)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_46_im / y_46_re)));
	elseif (y_46_im <= 5.6e+112)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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$46$im, -1.45e+67], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.8e-156], t$95$0, If[LessEqual[y$46$im, 6.2e-145], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.6e+112], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.45000000000000012e67

   1. Initial program 38.2%

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

   2. Taylor expanded in y.re around 0 90.5%

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

   if -1.45000000000000012e67 < y.im < -4.8e-156 or 6.20000000000000001e-145 < y.im < 5.6000000000000003e112

   1. Initial program 83.5%

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

   if -4.8e-156 < y.im < 6.20000000000000001e-145

   1. Initial program 64.0%

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

   2. Taylor expanded in y.re around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. unpow2 85.9%

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

      3. associate-/l* 87.2%

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

   4. Simplified 87.2%

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

   5. Taylor expanded in y.re around 0 87.2%

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

      1. unpow2 87.2%

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

      2. associate-*l/ 92.6%

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

      3. associate-/r/ 92.6%

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

   7. Simplified 92.6%

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

   if 5.6000000000000003e112 < y.im

   1. Initial program 40.6%

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

   2. Taylor expanded in y.re around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. *-commutative 77.2%

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

      3. unpow2 77.2%

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

      4. times-frac 86.7%

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

   4. Simplified 86.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -190000000000 \lor \neg \left(y.re \leq 3.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -190000000000.0) (not (<= y.re 3.4e-32)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.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 <= -190000000000.0) || !(y_46_re <= 3.4e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= (-190000000000.0d0)) .or. (.not. (y_46re <= 3.4d-32))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_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 <= -190000000000.0) || !(y_46_re <= 3.4e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= -190000000000.0) or not (y_46_re <= 3.4e-32):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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 <= -190000000000.0) || !(y_46_re <= 3.4e-32))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_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 <= -190000000000.0) || ~((y_46_re <= 3.4e-32)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_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$46$re, -190000000000.0], N[Not[LessEqual[y$46$re, 3.4e-32]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.9e11 or 3.39999999999999978e-32 < y.re

   1. Initial program 59.5%

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

   2. Taylor expanded in y.re around inf 67.5%

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

   if -1.9e11 < y.re < 3.39999999999999978e-32

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. unpow2 76.1%

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

      4. times-frac 81.7%

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

   4. Simplified 81.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -190000000000 \lor \neg \left(y.re \leq 3.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -240000000000 \lor \neg \left(y.re \leq 8.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -240000000000.0) (not (<= y.re 8.8e-32)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (/ (* x.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) {
	double tmp;
	if ((y_46_re <= -240000000000.0) || !(y_46_re <= 8.8e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= (-240000000000.0d0)) .or. (.not. (y_46re <= 8.8d-32))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_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 <= -240000000000.0) || !(y_46_re <= 8.8e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= -240000000000.0) or not (y_46_re <= 8.8e-32):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= -240000000000.0) || !(y_46_re <= 8.8e-32))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_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 <= -240000000000.0) || ~((y_46_re <= 8.8e-32)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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, -240000000000.0], N[Not[LessEqual[y$46$re, 8.8e-32]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.4e11 or 8.7999999999999999e-32 < y.re

   1. Initial program 59.5%

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

   2. Taylor expanded in y.re around inf 67.5%

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

   if -2.4e11 < y.re < 8.7999999999999999e-32

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. unpow2 76.1%

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

      4. times-frac 81.7%

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

   4. Simplified 81.7%

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

      1. associate-*r/ 82.4%

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

   6. Applied egg-rr 82.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -240000000000 \lor \neg \left(y.re \leq 8.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 5: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -25000000000 \lor \neg \left(y.re \leq 3.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -25000000000.0) (not (<= y.re 3.4e-32)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (/ (/ x.re (/ y.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_re <= -25000000000.0) || !(y_46_re <= 3.4e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_46re <= (-25000000000.0d0)) .or. (.not. (y_46re <= 3.4d-32))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re / (y_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_re <= -25000000000.0) || !(y_46_re <= 3.4e-32)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_re <= -25000000000.0) or not (y_46_re <= 3.4e-32):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_re <= -25000000000.0) || !(y_46_re <= 3.4e-32))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / Float64(y_46_im / 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 <= -25000000000.0) || ~((y_46_re <= 3.4e-32)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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$46$re, -25000000000.0], N[Not[LessEqual[y$46$re, 3.4e-32]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.5e10 or 3.39999999999999978e-32 < y.re

   1. Initial program 59.5%

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

   2. Taylor expanded in y.re around inf 67.5%

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

   if -2.5e10 < y.re < 3.39999999999999978e-32

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. unpow2 76.1%

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

      4. times-frac 81.7%

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

   4. Simplified 81.7%

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

      1. associate-*r/ 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. *-commutative 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

   8. Applied egg-rr 82.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -25000000000 \lor \neg \left(y.re \leq 3.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]

Alternative 6: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -195000 \lor \neg \left(y.re \leq 2.25 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -195000.0) (not (<= y.re 2.25e-21)))
   (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
   (+ (/ x.im y.im) (/ (/ x.re (/ y.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_re <= -195000.0) || !(y_46_re <= 2.25e-21)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_46re <= (-195000.0d0)) .or. (.not. (y_46re <= 2.25d-21))) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else
        tmp = (x_46im / y_46im) + ((x_46re / (y_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_re <= -195000.0) || !(y_46_re <= 2.25e-21)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_re <= -195000.0) or not (y_46_re <= 2.25e-21):
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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_re <= -195000.0) || !(y_46_re <= 2.25e-21))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / Float64(y_46_im / 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 <= -195000.0) || ~((y_46_re <= 2.25e-21)))
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_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$46$re, -195000.0], N[Not[LessEqual[y$46$re, 2.25e-21]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -195000 or 2.24999999999999984e-21 < y.re

   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. Taylor expanded in y.re around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. associate-/l* 78.3%

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

   4. Simplified 78.3%

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

      1. associate-/r/ 78.3%

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

   6. Applied egg-rr 78.3%

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

   if -195000 < y.re < 2.24999999999999984e-21

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. *-commutative 75.4%

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

      3. unpow2 75.4%

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

      4. times-frac 80.8%

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

   4. Simplified 80.8%

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

      1. associate-*r/ 81.5%

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

   6. Applied egg-rr 81.5%

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

      1. *-commutative 81.5%

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

      2. clear-num 81.5%

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

      3. un-div-inv 81.6%

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

   8. Applied egg-rr 81.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -195000 \lor \neg \left(y.re \leq 2.25 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]

Alternative 7: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -14000 \lor \neg \left(y.re \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -14000.0) (not (<= y.re 2.3e-68)))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (+ (/ x.im y.im) (/ (* x.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) {
	double tmp;
	if ((y_46_re <= -14000.0) || !(y_46_re <= 2.3e-68)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= (-14000.0d0)) .or. (.not. (y_46re <= 2.3d-68))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_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 <= -14000.0) || !(y_46_re <= 2.3e-68)) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= -14000.0) or not (y_46_re <= 2.3e-68):
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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 <= -14000.0) || !(y_46_re <= 2.3e-68))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_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 <= -14000.0) || ~((y_46_re <= 2.3e-68)))
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_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, -14000.0], N[Not[LessEqual[y$46$re, 2.3e-68]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -14000 or 2.29999999999999997e-68 < y.re

   1. Initial program 60.7%

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

   2. Taylor expanded in y.re around inf 75.1%

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

      1. unpow2 75.1%

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

      2. times-frac 79.7%

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

   4. Simplified 79.7%

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

   if -14000 < y.re < 2.29999999999999997e-68

   1. Initial program 66.9%

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

   2. Taylor expanded in y.re around 0 77.5%

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

      1. +-commutative 77.5%

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

      2. *-commutative 77.5%

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

      3. unpow2 77.5%

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

      4. times-frac 83.6%

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

   4. Simplified 83.6%

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

      1. associate-*r/ 84.3%

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

   6. Applied egg-rr 84.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -14000 \lor \neg \left(y.re \leq 2.3 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6200000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6200000000.0)
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (if (<= y.re 8.2e-32)
     (+ (/ x.im y.im) (/ (/ x.re (/ y.im y.re)) y.im))
     (+ (/ x.re y.re) (/ x.im (/ y.re (/ y.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 <= -6200000000.0) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= 8.2e-32) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= (-6200000000.0d0)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else if (y_46re <= 8.2d-32) then
        tmp = (x_46im / y_46im) + ((x_46re / (y_46im / y_46re)) / y_46im)
    else
        tmp = (x_46re / y_46re) + (x_46im / (y_46re / (y_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 <= -6200000000.0) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= 8.2e-32) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= -6200000000.0:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_re <= 8.2e-32:
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= -6200000000.0)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= 8.2e-32)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / Float64(y_46_im / y_46_re)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re / Float64(y_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 <= -6200000000.0)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_re <= 8.2e-32)
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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, -6200000000.0], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.2e-32], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.2e9

   1. Initial program 57.6%

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

   2. Taylor expanded in y.re around inf 76.2%

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

      1. unpow2 76.2%

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

      2. times-frac 85.0%

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

   4. Simplified 85.0%

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

   if -6.2e9 < y.re < 8.1999999999999995e-32

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. unpow2 76.1%

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

      4. times-frac 81.7%

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

   4. Simplified 81.7%

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

      1. associate-*r/ 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. *-commutative 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

   8. Applied egg-rr 82.5%

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

   if 8.1999999999999995e-32 < y.re

   1. Initial program 61.1%

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

   2. Taylor expanded in y.re around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. associate-/l* 78.4%

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

   4. Simplified 78.4%

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

   5. Taylor expanded in y.re around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*l/ 78.3%

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

      3. associate-/r/ 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6200000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5600000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5600000.0)
   (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
   (if (<= y.re 4.2e-32)
     (+ (/ x.im y.im) (/ (/ x.re (/ y.im y.re)) y.im))
     (+ (/ x.re y.re) (/ x.im (/ y.re (/ y.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 <= -5600000.0) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 4.2e-32) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= (-5600000.0d0)) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else if (y_46re <= 4.2d-32) then
        tmp = (x_46im / y_46im) + ((x_46re / (y_46im / y_46re)) / y_46im)
    else
        tmp = (x_46re / y_46re) + (x_46im / (y_46re / (y_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 <= -5600000.0) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 4.2e-32) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= -5600000.0:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 4.2e-32:
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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 <= -5600000.0)
		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 <= 4.2e-32)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / Float64(y_46_im / y_46_re)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re / Float64(y_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 <= -5600000.0)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 4.2e-32)
		tmp = (x_46_im / y_46_im) + ((x_46_re / (y_46_im / y_46_re)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re / (y_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, -5600000.0], 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, 4.2e-32], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.6e6

   1. Initial program 57.6%

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

   2. Taylor expanded in y.re around inf 76.2%

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

      1. unpow2 76.2%

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

      2. times-frac 85.0%

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

   4. Simplified 85.0%

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

      1. associate-*r/ 85.0%

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

   6. Applied egg-rr 85.0%

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

   if -5.6e6 < y.re < 4.1999999999999998e-32

   1. Initial program 67.6%

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

   2. Taylor expanded in y.re around 0 76.1%

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

      1. +-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. unpow2 76.1%

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

      4. times-frac 81.7%

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

   4. Simplified 81.7%

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

      1. associate-*r/ 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. *-commutative 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

   8. Applied egg-rr 82.5%

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

   if 4.1999999999999998e-32 < y.re

   1. Initial program 61.1%

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

   2. Taylor expanded in y.re around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. associate-/l* 78.4%

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

   4. Simplified 78.4%

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

   5. Taylor expanded in y.re around 0 78.4%

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

      1. unpow2 78.4%

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

      2. associate-*l/ 78.3%

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

      3. associate-/r/ 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5600000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 10: 63.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.5e-45)
   (/ x.re y.re)
   (if (<= y.re 1.4e-38) (/ 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_re <= -4.5e-45) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.4e-38) {
		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_46re <= (-4.5d-45)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.4d-38) 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_re <= -4.5e-45) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.4e-38) {
		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_re <= -4.5e-45:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.4e-38:
		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_re <= -4.5e-45)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.4e-38)
		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_re <= -4.5e-45)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.4e-38)
		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[LessEqual[y$46$re, -4.5e-45], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.4e-38], 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.re < -4.4999999999999999e-45 or 1.4e-38 < y.re

   1. Initial program 60.6%

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

   2. Taylor expanded in y.re around inf 65.2%

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

   if -4.4999999999999999e-45 < y.re < 1.4e-38

   1. Initial program 67.2%

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

   2. Taylor expanded in y.re around 0 72.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 43.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.6e+173) (/ x.im 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 <= -5.6e+173) {
		tmp = x_46_im / 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 <= (-5.6d+173)) then
        tmp = x_46im / 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 <= -5.6e+173) {
		tmp = x_46_im / 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 <= -5.6e+173:
		tmp = x_46_im / 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 <= -5.6e+173)
		tmp = Float64(x_46_im / 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 <= -5.6e+173)
		tmp = x_46_im / 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[LessEqual[y$46$re, -5.6e+173], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.59999999999999964e173

   1. Initial program 42.5%

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

      1. *-un-lft-identity 42.5%

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

      2. add-sqr-sqrt 42.5%

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

      3. times-frac 42.5%

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

      4. hypot-def 42.5%

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

      5. fma-def 42.5%

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

      6. hypot-def 73.7%

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

   3. Applied egg-rr 73.7%

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

   4. Taylor expanded in y.re around 0 43.3%

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

   5. Taylor expanded in y.re around inf 39.7%

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

   if -5.59999999999999964e173 < y.re

   1. Initial program 66.0%

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

   2. Taylor expanded in y.re around 0 49.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.6 \cdot 10^{+173}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 42.7% 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 63.7%

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

2. Taylor expanded in y.re around 0 46.8%

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

3. Final simplification 46.8%

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

Reproduce

herbie shell --seed 2023178 
(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))))