_divideComplex, real part

Percentage Accurate: 61.2% → 85.1%

Time: 13.5s

Alternatives: 11

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: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

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)))
      5e+305)
   (*
    (/ 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))) <= 5e+305) {
		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))) <= 5e+305)
		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], 5e+305], 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))) < 5.00000000000000009e305

   1. Initial program 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. times-frac 82.8%

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

      4. hypot-def 82.8%

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

      5. fma-def 82.8%

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

      6. hypot-def 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 5.00000000000000009e305 < (/.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 15.0%

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

   2. Taylor expanded in y.re around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in y.re around 0 50.4%

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

      1. unpow2 50.4%

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

      2. *-commutative 50.4%

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

      3. times-frac 60.1%

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

   7. Simplified 60.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{x.im \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

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 -2.7e+109)
     (- (/ x.re y.re) (/ (* x.im (/ y.im y.re)) (- y.re)))
     (if (<= y.re -1.5e-31)
       t_0
       (if (<= y.re 2.8e-149)
         (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
         (if (<= y.re 3.1e+91)
           t_0
           (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -2.7e+109) {
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	} else if (y_46_re <= -1.5e-31) {
		tmp = t_0;
	} else if (y_46_re <= 2.8e-149) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 3.1e+91) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.7d+109)) then
        tmp = (x_46re / y_46re) - ((x_46im * (y_46im / y_46re)) / -y_46re)
    else if (y_46re <= (-1.5d-31)) then
        tmp = t_0
    else if (y_46re <= 2.8d-149) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46re <= 3.1d+91) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 <= -2.7e+109) {
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	} else if (y_46_re <= -1.5e-31) {
		tmp = t_0;
	} else if (y_46_re <= 2.8e-149) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_re <= 3.1e+91) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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 <= -2.7e+109:
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re)
	elif y_46_re <= -1.5e-31:
		tmp = t_0
	elif y_46_re <= 2.8e-149:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_re <= 3.1e+91:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -2.7e+109)
		tmp = Float64(Float64(x_46_re / y_46_re) - Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / Float64(-y_46_re)));
	elseif (y_46_re <= -1.5e-31)
		tmp = t_0;
	elseif (y_46_re <= 2.8e-149)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_re <= 3.1e+91)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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 <= -2.7e+109)
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	elseif (y_46_re <= -1.5e-31)
		tmp = t_0;
	elseif (y_46_re <= 2.8e-149)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_re <= 3.1e+91)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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, -2.7e+109], N[(N[(x$46$re / y$46$re), $MachinePrecision] - N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.5e-31], t$95$0, If[LessEqual[y$46$re, 2.8e-149], 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], If[LessEqual[y$46$re, 3.1e+91], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.70000000000000001e109

   1. Initial program 44.7%

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

      1. *-un-lft-identity 44.7%

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

      2. add-sqr-sqrt 44.7%

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

      3. times-frac 44.6%

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

      4. hypot-def 44.6%

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

      5. fma-def 44.6%

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

      6. hypot-def 67.5%

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

   3. Applied egg-rr 67.5%

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

   4. Taylor expanded in y.re around inf 68.4%

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

      1. *-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. times-frac 86.8%

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

   6. Simplified 86.8%

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

      1. frac-2neg 86.8%

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

      2. associate-*r/ 86.9%

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

   8. Applied egg-rr 86.9%

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

   if -2.70000000000000001e109 < y.re < -1.49999999999999991e-31 or 2.7999999999999999e-149 < y.re < 3.09999999999999998e91

   1. Initial program 78.0%

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

   if -1.49999999999999991e-31 < y.re < 2.7999999999999999e-149

   1. Initial program 78.3%

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

   2. Taylor expanded in y.re around 0 86.9%

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

      1. *-commutative 86.9%

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

      2. unpow2 86.9%

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

   4. Simplified 86.9%

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

      1. associate-/l* 78.7%

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

      2. div-inv 78.6%

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

   6. Applied egg-rr 78.6%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 78.7%

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

      2. *-rgt-identity 78.7%

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

      3. associate-*r/ 81.9%

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

   8. Simplified 81.9%

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

      1. expm1-log1p-u 75.4%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 68.1%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 68.1%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 75.4%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 81.9%

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

       3. *-lft-identity 81.9%

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

       4. times-frac 90.0%

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

       5. associate-*l/ 90.0%

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

       6. *-lft-identity 90.0%

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

       7. associate-/r/ 91.0%

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

   12. Simplified 91.0%

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

   if 3.09999999999999998e91 < y.re

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. times-frac 43.6%

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

      4. hypot-def 43.6%

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

      5. fma-def 43.6%

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

      6. hypot-def 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in y.re around inf 79.9%

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

      1. *-commutative 79.9%

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

      2. unpow2 79.9%

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

      3. times-frac 85.5%

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

   6. Simplified 85.5%

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

      1. associate-*l/ 87.9%

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

   8. Applied egg-rr 87.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{x.im \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 3: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.25e-29) (not (<= y.re 2.8e+91)))
   (/ x.re (+ y.re (* y.im (/ y.im 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 <= -1.25e-29) || !(y_46_re <= 2.8e+91)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_46_im / 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 <= (-1.25d-29)) .or. (.not. (y_46re <= 2.8d+91))) then
        tmp = x_46re / (y_46re + (y_46im * (y_46im / 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 <= -1.25e-29) || !(y_46_re <= 2.8e+91)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_46_im / 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 <= -1.25e-29) or not (y_46_re <= 2.8e+91):
		tmp = x_46_re / (y_46_re + (y_46_im * (y_46_im / 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 <= -1.25e-29) || !(y_46_re <= 2.8e+91))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / 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 <= -1.25e-29) || ~((y_46_re <= 2.8e+91)))
		tmp = x_46_re / (y_46_re + (y_46_im * (y_46_im / 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, -1.25e-29], N[Not[LessEqual[y$46$re, 2.8e+91]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $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 y.re < -1.24999999999999996e-29 or 2.7999999999999999e91 < y.re

   1. Initial program 53.4%

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

   2. Taylor expanded in x.re around inf 45.8%

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

      1. associate-/l* 50.5%

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

      2. unpow2 50.5%

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

      3. unpow2 50.5%

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

      4. fma-udef 50.5%

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

   4. Simplified 50.5%

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

   5. Taylor expanded in y.im around 0 70.0%

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

      1. unpow2 70.0%

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

      2. associate-*r/ 73.9%

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

   7. Simplified 73.9%

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

   if -1.24999999999999996e-29 < y.re < 2.7999999999999999e91

   1. Initial program 77.7%

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

   2. Taylor expanded in y.re around 0 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

   4. Simplified 79.7%

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

   5. Taylor expanded in y.re around 0 79.7%

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

      1. unpow2 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.8 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.65 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1e-29) (not (<= y.re 2.65e+91)))
   (/ x.re (+ y.re (* y.im (/ y.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 <= -1e-29) || !(y_46_re <= 2.65e+91)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= (-1d-29)) .or. (.not. (y_46re <= 2.65d+91))) then
        tmp = x_46re / (y_46re + (y_46im * (y_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 <= -1e-29) || !(y_46_re <= 2.65e+91)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= -1e-29) or not (y_46_re <= 2.65e+91):
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= -1e-29) || !(y_46_re <= 2.65e+91))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_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 <= -1e-29) || ~((y_46_re <= 2.65e+91)))
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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, -1e-29], N[Not[LessEqual[y$46$re, 2.65e+91]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $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 < -9.99999999999999943e-30 or 2.64999999999999998e91 < y.re

   1. Initial program 53.4%

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

   2. Taylor expanded in x.re around inf 45.8%

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

      1. associate-/l* 50.5%

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

      2. unpow2 50.5%

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

      3. unpow2 50.5%

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

      4. fma-udef 50.5%

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

   4. Simplified 50.5%

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

   5. Taylor expanded in y.im around 0 70.0%

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

      1. unpow2 70.0%

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

      2. associate-*r/ 73.9%

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

   7. Simplified 73.9%

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

   if -9.99999999999999943e-30 < y.re < 2.64999999999999998e91

   1. Initial program 77.7%

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

   2. Taylor expanded in y.re around 0 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

   4. Simplified 79.7%

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

      1. associate-/l* 74.0%

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

      2. div-inv 73.4%

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

   6. Applied egg-rr 73.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 74.0%

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

      2. *-rgt-identity 74.0%

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

      3. associate-*r/ 76.6%

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

   8. Simplified 76.6%

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

      1. expm1-log1p-u 68.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 61.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 61.5%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 68.9%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 76.6%

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

       3. *-lft-identity 76.6%

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

       4. times-frac 82.2%

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

       5. associate-*l/ 82.2%

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

       6. *-lft-identity 82.2%

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

       7. associate-/r/ 82.9%

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

   12. Simplified 82.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-29} \lor \neg \left(y.re \leq 2.65 \cdot 10^{+91}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 5: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+39} \lor \neg \left(y.re \leq 5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.5e+39) (not (<= y.re 5e+50)))
   (+ (/ 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 <= -2.5e+39) || !(y_46_re <= 5e+50)) {
		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 <= (-2.5d+39)) .or. (.not. (y_46re <= 5d+50))) 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 <= -2.5e+39) || !(y_46_re <= 5e+50)) {
		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 <= -2.5e+39) or not (y_46_re <= 5e+50):
		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 <= -2.5e+39) || !(y_46_re <= 5e+50))
		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 <= -2.5e+39) || ~((y_46_re <= 5e+50)))
		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, -2.5e+39], N[Not[LessEqual[y$46$re, 5e+50]], $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 < -2.50000000000000008e39 or 5e50 < y.re

   1. Initial program 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. add-sqr-sqrt 53.7%

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

      3. times-frac 53.5%

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

      4. hypot-def 53.5%

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

      5. fma-def 53.5%

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

      6. hypot-def 68.7%

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

   3. Applied egg-rr 68.7%

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

   4. Taylor expanded in y.re around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. times-frac 81.5%

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

   6. Simplified 81.5%

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

   if -2.50000000000000008e39 < y.re < 5e50

   1. Initial program 77.1%

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

   2. Taylor expanded in y.re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

      1. associate-/l* 73.6%

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

      2. div-inv 72.9%

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

   6. Applied egg-rr 72.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 73.6%

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

      2. *-rgt-identity 73.6%

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

      3. associate-*r/ 76.1%

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

   8. Simplified 76.1%

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

      1. expm1-log1p-u 68.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 61.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 68.5%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 76.1%

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

       3. *-lft-identity 76.1%

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

       4. times-frac 81.7%

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

       5. associate-*l/ 81.7%

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

       6. *-lft-identity 81.7%

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

       7. associate-/r/ 82.5%

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

   12. Simplified 82.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+39} \lor \neg \left(y.re \leq 5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 6: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+39} \lor \neg \left(y.re \leq 3.8 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.6e+39) (not (<= y.re 3.8e+47)))
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.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 <= -2.6e+39) || !(y_46_re <= 3.8e+47)) {
		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_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 <= (-2.6d+39)) .or. (.not. (y_46re <= 3.8d+47))) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_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 <= -2.6e+39) || !(y_46_re <= 3.8e+47)) {
		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_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 <= -2.6e+39) or not (y_46_re <= 3.8e+47):
		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_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 <= -2.6e+39) || !(y_46_re <= 3.8e+47))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_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 <= -2.6e+39) || ~((y_46_re <= 3.8e+47)))
		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_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, -2.6e+39], N[Not[LessEqual[y$46$re, 3.8e+47]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $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 < -2.6e39 or 3.8000000000000003e47 < y.re

   1. Initial program 53.7%

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

      1. *-un-lft-identity 53.7%

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

      2. add-sqr-sqrt 53.7%

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

      3. times-frac 53.5%

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

      4. hypot-def 53.5%

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

      5. fma-def 53.5%

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

      6. hypot-def 68.7%

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

   3. Applied egg-rr 68.7%

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

   4. Taylor expanded in y.re around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. unpow2 72.7%

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

      3. times-frac 81.5%

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

   6. Simplified 81.5%

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

      1. associate-*l/ 82.5%

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

   8. Applied egg-rr 82.5%

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

   if -2.6e39 < y.re < 3.8000000000000003e47

   1. Initial program 77.1%

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

   2. Taylor expanded in y.re around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. unpow2 79.2%

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

   4. Simplified 79.2%

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

      1. associate-/l* 73.6%

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

      2. div-inv 72.9%

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

   6. Applied egg-rr 72.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 73.6%

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

      2. *-rgt-identity 73.6%

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

      3. associate-*r/ 76.1%

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

   8. Simplified 76.1%

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

      1. expm1-log1p-u 68.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 61.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 61.8%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 68.5%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 76.1%

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

       3. *-lft-identity 76.1%

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

       4. times-frac 81.7%

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

       5. associate-*l/ 81.7%

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

       6. *-lft-identity 81.7%

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

       7. associate-/r/ 82.5%

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

   12. Simplified 82.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+39} \lor \neg \left(y.re \leq 3.8 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 7: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.75 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.9e-9)
   (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
   (if (<= y.re 2.75e+51)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.9e-9) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 2.75e+51) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.9d-9)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / x_46im))
    else if (y_46re <= 2.75d+51) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.9e-9) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_re <= 2.75e+51) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.9e-9:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im))
	elif y_46_re <= 2.75e+51:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.9e-9)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_re <= 2.75e+51)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.9e-9)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	elseif (y_46_re <= 2.75e+51)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.9e-9], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.75e+51], 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], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.89999999999999991e-9

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. add-sqr-sqrt 59.9%

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

      3. times-frac 59.8%

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

      4. hypot-def 59.8%

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

      5. fma-def 59.8%

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

      6. hypot-def 73.8%

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

   3. Applied egg-rr 73.8%

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

   4. Taylor expanded in y.re around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. times-frac 77.3%

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

   6. Simplified 77.3%

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

      1. clear-num 77.4%

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

      2. un-div-inv 78.7%

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

   8. Applied egg-rr 78.7%

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

   if -2.89999999999999991e-9 < y.re < 2.75e51

   1. Initial program 77.4%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

   4. Simplified 81.1%

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

      1. associate-/l* 75.1%

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

      2. div-inv 74.4%

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

   6. Applied egg-rr 74.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 75.1%

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

      2. *-rgt-identity 75.1%

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

      3. associate-*r/ 77.8%

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

   8. Simplified 77.8%

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

      1. expm1-log1p-u 69.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 62.7%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 62.7%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 69.8%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 77.8%

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

       3. *-lft-identity 77.8%

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

       4. times-frac 83.6%

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

       5. associate-*l/ 83.6%

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

       6. *-lft-identity 83.6%

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

       7. associate-/r/ 84.4%

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

   12. Simplified 84.4%

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

   if 2.75e51 < 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 63.3%

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

   3. Applied egg-rr 63.3%

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

   4. Taylor expanded in y.re around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 80.8%

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

   6. Simplified 80.8%

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

      1. associate-*l/ 82.9%

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

   8. Applied egg-rr 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 2.75 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{x.im \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e-9)
   (- (/ x.re y.re) (/ (* x.im (/ y.im y.re)) (- y.re)))
   (if (<= y.re 2e+49)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-9) {
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	} else if (y_46_re <= 2e+49) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.2d-9)) then
        tmp = (x_46re / y_46re) - ((x_46im * (y_46im / y_46re)) / -y_46re)
    else if (y_46re <= 2d+49) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-9) {
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	} else if (y_46_re <= 2e+49) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.2e-9:
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re)
	elif y_46_re <= 2e+49:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e-9)
		tmp = Float64(Float64(x_46_re / y_46_re) - Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / Float64(-y_46_re)));
	elseif (y_46_re <= 2e+49)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.2e-9)
		tmp = (x_46_re / y_46_re) - ((x_46_im * (y_46_im / y_46_re)) / -y_46_re);
	elseif (y_46_re <= 2e+49)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.2e-9], 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, 2e+49], 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], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.20000000000000039e-9

   1. Initial program 59.9%

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

      1. *-un-lft-identity 59.9%

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

      2. add-sqr-sqrt 59.9%

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

      3. times-frac 59.8%

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

      4. hypot-def 59.8%

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

      5. fma-def 59.8%

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

      6. hypot-def 73.8%

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

   3. Applied egg-rr 73.8%

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

   4. Taylor expanded in y.re around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. times-frac 77.3%

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

   6. Simplified 77.3%

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

      1. frac-2neg 77.3%

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

      2. associate-*r/ 78.7%

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

   8. Applied egg-rr 78.7%

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

   if -4.20000000000000039e-9 < y.re < 1.99999999999999989e49

   1. Initial program 77.4%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

   4. Simplified 81.1%

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

      1. associate-/l* 75.1%

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

      2. div-inv 74.4%

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

   6. Applied egg-rr 74.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{y.re \cdot \frac{1}{\frac{y.im \cdot y.im}{x.re}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 75.1%

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

      2. *-rgt-identity 75.1%

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

      3. associate-*r/ 77.8%

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

   8. Simplified 77.8%

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

      1. expm1-log1p-u 69.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

      2. expm1-udef 62.7%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]

   10. Applied egg-rr 62.7%

       \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 69.8%

          \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\right)\right)} \]

       2. expm1-log1p 77.8%

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

       3. *-lft-identity 77.8%

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

       4. times-frac 83.6%

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

       5. associate-*l/ 83.6%

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

       6. *-lft-identity 83.6%

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

       7. associate-/r/ 84.4%

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

   12. Simplified 84.4%

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

   if 1.99999999999999989e49 < 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 63.3%

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

   3. Applied egg-rr 63.3%

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

   4. Taylor expanded in y.re around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 80.8%

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

   6. Simplified 80.8%

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

      1. associate-*l/ 82.9%

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

   8. Applied egg-rr 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{x.im \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{-31} \lor \neg \left(y.re \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -8.5e-31) (not (<= y.re 5e-35)))
   (/ x.re (+ y.re (* y.im (/ y.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 <= -8.5e-31) || !(y_46_re <= 5e-35)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= (-8.5d-31)) .or. (.not. (y_46re <= 5d-35))) then
        tmp = x_46re / (y_46re + (y_46im * (y_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 <= -8.5e-31) || !(y_46_re <= 5e-35)) {
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= -8.5e-31) or not (y_46_re <= 5e-35):
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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 <= -8.5e-31) || !(y_46_re <= 5e-35))
		tmp = Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_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 <= -8.5e-31) || ~((y_46_re <= 5e-35)))
		tmp = x_46_re / (y_46_re + (y_46_im * (y_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[Or[LessEqual[y$46$re, -8.5e-31], N[Not[LessEqual[y$46$re, 5e-35]], $MachinePrecision]], N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.5000000000000007e-31 or 4.99999999999999964e-35 < y.re

   1. Initial program 56.0%

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

   2. Taylor expanded in x.re around inf 47.4%

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

      1. associate-/l* 51.5%

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

      2. unpow2 51.5%

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

      3. unpow2 51.5%

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

      4. fma-udef 51.5%

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

   4. Simplified 51.5%

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

   5. Taylor expanded in y.im around 0 68.3%

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

      1. unpow2 68.3%

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

      2. associate-*r/ 71.7%

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

   7. Simplified 71.7%

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

   if -8.5000000000000007e-31 < y.re < 4.99999999999999964e-35

   1. Initial program 78.6%

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

   2. Taylor expanded in y.re around 0 65.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{-31} \lor \neg \left(y.re \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10: 63.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e-30)
   (/ x.re y.re)
   (if (<= y.re 6.4e+91) (/ 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 <= -2.1e-30) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.4e+91) {
		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 <= (-2.1d-30)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 6.4d+91) 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 <= -2.1e-30) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 6.4e+91) {
		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 <= -2.1e-30:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 6.4e+91:
		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 <= -2.1e-30)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 6.4e+91)
		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 <= -2.1e-30)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 6.4e+91)
		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, -2.1e-30], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 6.4e+91], 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 < -2.1000000000000002e-30 or 6.39999999999999979e91 < y.re

   1. Initial program 53.4%

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

   2. Taylor expanded in y.re around inf 67.9%

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

   if -2.1000000000000002e-30 < y.re < 6.39999999999999979e91

   1. Initial program 77.7%

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

   2. Taylor expanded in y.re around 0 63.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 42.6% 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 66.2%

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

2. Taylor expanded in y.re around 0 43.0%

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

3. Final simplification 43.0%

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

Reproduce

herbie shell --seed 2023275 
(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))))