_divideComplex, real part

Percentage Accurate: 61.5% → 85.1%

Time: 14.2s

Alternatives: 17

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+294)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+294) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+294)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+294], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.00000000000000013e294

   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. Add Preprocessing
   3. 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-define 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-define 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-define 98.1%

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

   4. 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 2.00000000000000013e294 < (/.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 11.9%

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

   3. Taylor expanded in y.re around inf 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. pow2 40.5%

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

      3. times-frac 45.6%

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

   5. Applied egg-rr 45.6%

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

      1. +-commutative 45.6%

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

      2. *-commutative 45.6%

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

      3. div-inv 45.5%

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

      4. distribute-rgt-out 47.3%

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

      5. associate-/l* 61.1%

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

   7. Applied egg-rr 61.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im + x.re \cdot \frac{y.re}{y.im}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.4e+27)
   (* (/ y.im (hypot y.re y.im)) (/ x.im (hypot y.re y.im)))
   (if (<= y.im 1.7e-51)
     (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))
     (if (<= y.im 3e+136)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ 1.0 (/ (hypot y.re y.im) (+ x.im (* x.re (/ y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -7.4e+27) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 1.7e-51) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 3e+136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -7.4e+27) {
		tmp = (y_46_im / Math.hypot(y_46_re, y_46_im)) * (x_46_im / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 1.7e-51) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 3e+136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = 1.0 / (Math.hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -7.4e+27:
		tmp = (y_46_im / math.hypot(y_46_re, y_46_im)) * (x_46_im / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= 1.7e-51:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	elif y_46_im <= 3e+136:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = 1.0 / (math.hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.4e+27)
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(x_46_im / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 1.7e-51)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 3e+136)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -7.4e+27)
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (x_46_im / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 1.7e-51)
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	elseif (y_46_im <= 3e+136)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -7.4e+27], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.7e-51], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3e+136], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.40000000000000004e27

   1. Initial program 36.7%

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

   3. Taylor expanded in x.re around 0 36.8%

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

      1. *-commutative 36.8%

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

      2. add-sqr-sqrt 36.8%

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

      3. hypot-undefine 36.8%

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

      4. hypot-undefine 36.8%

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

      5. times-frac 83.8%

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

   5. Applied egg-rr 83.8%

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

   if -7.40000000000000004e27 < y.im < 1.70000000000000001e-51

   1. Initial program 75.2%

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

   3. Taylor expanded in y.re around inf 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. pow2 78.9%

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

      3. times-frac 85.0%

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

   5. Applied egg-rr 85.0%

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

      1. +-commutative 85.0%

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

      2. *-commutative 85.0%

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

      3. div-inv 84.8%

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

      4. distribute-rgt-out 86.3%

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

      5. associate-/l* 85.5%

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

   7. Applied egg-rr 85.5%

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

   if 1.70000000000000001e-51 < y.im < 2.99999999999999979e136

   1. Initial program 85.5%

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

   if 2.99999999999999979e136 < y.im

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. add-sqr-sqrt 40.6%

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

      3. times-frac 40.6%

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

      4. hypot-define 40.6%

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

      5. fma-define 40.6%

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

      6. hypot-define 80.4%

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

   4. Applied egg-rr 80.4%

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

      1. associate-*l/ 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. clear-num 78.3%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in y.re around 0 82.1%

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

      1. associate-/l* 91.2%

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

   9. Simplified 91.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im + x.re \cdot \frac{y.re}{y.im}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.8e+38)
   (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re)))
   (if (<= y.im 1.4e-53)
     (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))
     (if (<= y.im 2.2e+136)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (* (/ 1.0 (hypot y.re y.im)) (+ x.im (* x.re (/ y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.8e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 1.4e-53) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 2.2e+136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.8e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 1.4e-53) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 2.2e+136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.8e+38:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re))
	elif y_46_im <= 1.4e-53:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	elif y_46_im <= 2.2e+136:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re * (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.8e+38)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	elseif (y_46_im <= 1.4e-53)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 2.2e+136)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.8e+38)
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	elseif (y_46_im <= 1.4e-53)
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	elseif (y_46_im <= 2.2e+136)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.8e+38], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e-53], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.2e+136], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.80000000000000035e38

   1. Initial program 37.3%

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

   3. Taylor expanded in y.re around 0 69.0%

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

      1. associate-/l* 74.1%

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

   5. Simplified 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. associate-*r* 79.1%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.1%

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

      4. un-div-inv 79.1%

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

   9. Applied egg-rr 79.1%

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

   if -4.80000000000000035e38 < y.im < 1.39999999999999993e-53

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 1.39999999999999993e-53 < y.im < 2.1999999999999999e136

   1. Initial program 85.5%

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

   if 2.1999999999999999e136 < y.im

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. add-sqr-sqrt 40.6%

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

      3. times-frac 40.6%

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

      4. hypot-define 40.6%

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

      5. fma-define 40.6%

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

      6. hypot-define 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in y.re around 0 81.5%

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

      1. associate-/l* 91.2%

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

   7. Simplified 91.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im + x.re \cdot \frac{y.re}{y.im}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.1e+38)
   (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re)))
   (if (<= y.im 5.1e-45)
     (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))
     (if (<= y.im 1.05e+138)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ 1.0 (/ (hypot y.re y.im) (+ x.im (* x.re (/ y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.1e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 5.1e-45) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 1.05e+138) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.1e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 5.1e-45) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 1.05e+138) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = 1.0 / (Math.hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6.1e+38:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re))
	elif y_46_im <= 5.1e-45:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	elif y_46_im <= 1.05e+138:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = 1.0 / (math.hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.1e+38)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	elseif (y_46_im <= 5.1e-45)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 1.05e+138)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6.1e+38)
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	elseif (y_46_im <= 5.1e-45)
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	elseif (y_46_im <= 1.05e+138)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (x_46_im + (x_46_re * (y_46_re / y_46_im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6.1e+38], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.1e-45], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.05e+138], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.0999999999999999e38

   1. Initial program 37.3%

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

   3. Taylor expanded in y.re around 0 69.0%

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

      1. associate-/l* 74.1%

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

   5. Simplified 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. associate-*r* 79.1%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.1%

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

      4. un-div-inv 79.1%

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

   9. Applied egg-rr 79.1%

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

   if -6.0999999999999999e38 < y.im < 5.0999999999999997e-45

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 5.0999999999999997e-45 < y.im < 1.05000000000000003e138

   1. Initial program 85.5%

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

   if 1.05000000000000003e138 < y.im

   1. Initial program 40.6%

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

      1. *-un-lft-identity 40.6%

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

      2. add-sqr-sqrt 40.6%

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

      3. times-frac 40.6%

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

      4. hypot-define 40.6%

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

      5. fma-define 40.6%

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

      6. hypot-define 80.4%

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

   4. Applied egg-rr 80.4%

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

      1. associate-*l/ 80.5%

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

      2. *-un-lft-identity 80.5%

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

      3. clear-num 78.3%

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in y.re around 0 82.1%

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

      1. associate-/l* 91.2%

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

   9. Simplified 91.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im + x.re \cdot \frac{y.re}{y.im}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.2e+125)
   (+ (/ x.re y.re) (* (/ 1.0 y.re) (* y.im (/ x.im y.re))))
   (if (<= y.re -1.06e-132)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 4.4e-131)
       (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))
       (+ (/ x.re y.re) (/ 1.0 (/ y.re (* x.im (/ y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.2e+125) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -1.06e-132) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3.2d+125)) then
        tmp = (x_46re / y_46re) + ((1.0d0 / y_46re) * (y_46im * (x_46im / y_46re)))
    else if (y_46re <= (-1.06d-132)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 4.4d-131) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    else
        tmp = (x_46re / y_46re) + (1.0d0 / (y_46re / (x_46im * (y_46im / y_46re))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.2e+125) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= -1.06e-132) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.2e+125:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= -1.06e-132:
		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))
	elif y_46_re <= 4.4e-131:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.2e+125)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= -1.06e-132)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 4.4e-131)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.2e+125)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= -1.06e-132)
		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));
	elseif (y_46_re <= 4.4e-131)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.2e+125], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.06e-132], 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, 4.4e-131], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.19999999999999983e125

   1. Initial program 38.1%

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

   3. Taylor expanded in y.re around inf 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. pow2 74.2%

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

      3. times-frac 74.4%

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

   5. Applied egg-rr 74.4%

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

      1. *-commutative 74.4%

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

      2. *-un-lft-identity 74.4%

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

      3. times-frac 78.6%

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

   7. Applied egg-rr 78.6%

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

   if -3.19999999999999983e125 < y.re < -1.05999999999999997e-132

   1. Initial program 81.6%

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

   if -1.05999999999999997e-132 < y.re < 4.3999999999999999e-131

   1. Initial program 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. add-sqr-sqrt 74.6%

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

      3. times-frac 74.5%

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

      4. hypot-define 74.6%

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

      5. fma-define 74.6%

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

      6. hypot-define 87.1%

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

   4. Applied egg-rr 87.1%

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

   5. Taylor expanded in y.re around 0 53.4%

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

      1. associate-/l* 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in y.re around 0 95.7%

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

   if 4.3999999999999999e-131 < y.re

   1. Initial program 62.6%

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

   3. Taylor expanded in y.re around inf 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. pow2 72.5%

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

      3. times-frac 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. associate-*l/ 75.7%

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

      2. *-un-lft-identity 75.7%

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

      3. clear-num 76.0%

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

      4. associate-/l* 81.9%

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

   7. Applied egg-rr 81.9%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.06 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1600000 \lor \neg \left(y.re \leq 4.4 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1600000.0) (not (<= y.re 4.4e-131)))
   (+ (/ x.re y.re) (/ 1.0 (/ y.re (* x.im (/ y.im y.re)))))
   (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1600000.0) || !(y_46_re <= 4.4e-131)) {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1600000.0d0)) .or. (.not. (y_46re <= 4.4d-131))) then
        tmp = (x_46re / y_46re) + (1.0d0 / (y_46re / (x_46im * (y_46im / y_46re))))
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1600000.0) || !(y_46_re <= 4.4e-131)) {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1600000.0) or not (y_46_re <= 4.4e-131):
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))))
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1600000.0) || !(y_46_re <= 4.4e-131))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im * Float64(y_46_im / y_46_re)))));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1600000.0) || ~((y_46_re <= 4.4e-131)))
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1600000.0], N[Not[LessEqual[y$46$re, 4.4e-131]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.6e6 or 4.3999999999999999e-131 < y.re

   1. Initial program 57.8%

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

   3. Taylor expanded in y.re around inf 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. pow2 70.5%

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

      3. times-frac 72.1%

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

   5. Applied egg-rr 72.1%

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

      1. associate-*l/ 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. clear-num 72.3%

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

      4. associate-/l* 75.9%

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

   7. Applied egg-rr 75.9%

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

   if -1.6e6 < y.re < 4.3999999999999999e-131

   1. Initial program 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 78.9%

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

      4. hypot-define 78.9%

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

      5. fma-define 78.9%

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

      6. hypot-define 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in y.re around 0 52.5%

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

      1. associate-/l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in y.re around 0 89.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1600000 \lor \neg \left(y.re \leq 4.4 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5e-6)
   (+ (/ x.re y.re) (* (/ 1.0 y.re) (* y.im (/ x.im y.re))))
   (if (<= y.re 4.4e-131)
     (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))
     (+ (/ x.re y.re) (/ 1.0 (/ y.re (* x.im (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5e-6) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-5d-6)) then
        tmp = (x_46re / y_46re) + ((1.0d0 / y_46re) * (y_46im * (x_46im / y_46re)))
    else if (y_46re <= 4.4d-131) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    else
        tmp = (x_46re / y_46re) + (1.0d0 / (y_46re / (x_46im * (y_46im / y_46re))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5e-6) {
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5e-6:
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)))
	elif y_46_re <= 4.4e-131:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5e-6)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_re <= 4.4e-131)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -5e-6)
		tmp = (x_46_re / y_46_re) + ((1.0 / y_46_re) * (y_46_im * (x_46_im / y_46_re)));
	elseif (y_46_re <= 4.4e-131)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im * (y_46_im / y_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5e-6], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.4e-131], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.00000000000000041e-6

   1. Initial program 52.9%

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

   3. Taylor expanded in y.re around inf 68.5%

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

      1. *-un-lft-identity 68.5%

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

      2. pow2 68.5%

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

      3. times-frac 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. *-commutative 68.6%

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

      2. *-un-lft-identity 68.6%

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

      3. times-frac 71.0%

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

   7. Applied egg-rr 71.0%

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

   if -5.00000000000000041e-6 < y.re < 4.3999999999999999e-131

   1. Initial program 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 78.9%

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

      4. hypot-define 78.9%

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

      5. fma-define 78.9%

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

      6. hypot-define 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in y.re around 0 52.5%

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

      1. associate-/l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in y.re around 0 89.0%

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

   if 4.3999999999999999e-131 < y.re

   1. Initial program 62.6%

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

   3. Taylor expanded in y.re around inf 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. pow2 72.5%

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

      3. times-frac 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. associate-*l/ 75.7%

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

      2. *-un-lft-identity 75.7%

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

      3. clear-num 76.0%

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

      4. associate-/l* 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{y.re} \cdot \left(y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{x.im \cdot \frac{y.im}{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+123} \lor \neg \left(y.re \leq 4.4 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.2e+123) (not (<= y.re 4.4e-131)))
   (/ x.re y.re)
   (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.2e+123) || !(y_46_re <= 4.4e-131)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.2d+123)) .or. (.not. (y_46re <= 4.4d-131))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.2e+123) || !(y_46_re <= 4.4e-131)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.2e+123) or not (y_46_re <= 4.4e-131):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.2e+123) || !(y_46_re <= 4.4e-131))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4.2e+123) || ~((y_46_re <= 4.4e-131)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.2e+123], N[Not[LessEqual[y$46$re, 4.4e-131]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.19999999999999988e123 or 4.3999999999999999e-131 < y.re

   1. Initial program 53.6%

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

   3. Taylor expanded in y.re around inf 69.4%

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

   if -4.19999999999999988e123 < y.re < 4.3999999999999999e-131

   1. Initial program 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. add-sqr-sqrt 77.6%

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

      3. times-frac 77.6%

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

      4. hypot-define 77.6%

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

      5. fma-define 77.6%

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

      6. hypot-define 87.8%

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

   4. Applied egg-rr 87.8%

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

   5. Taylor expanded in y.re around 0 46.9%

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

      1. associate-/l* 47.6%

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

   7. Simplified 47.6%

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

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

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+123} \lor \neg \left(y.re \leq 4.4 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+38} \lor \neg \left(y.im \leq 2.15 \cdot 10^{-33}\right):\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.8e+38) (not (<= y.im 2.15e-33)))
   (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))
   (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.8e+38) || !(y_46_im <= 2.15e-33)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.8d+38)) .or. (.not. (y_46im <= 2.15d-33))) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    else
        tmp = (1.0d0 / y_46re) * (x_46re + (x_46im * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.8e+38) || !(y_46_im <= 2.15e-33)) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.8e+38) or not (y_46_im <= 2.15e-33):
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	else:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.8e+38) || !(y_46_im <= 2.15e-33))
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.8e+38) || ~((y_46_im <= 2.15e-33)))
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	else
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.8e+38], N[Not[LessEqual[y$46$im, 2.15e-33]], $MachinePrecision]], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.80000000000000013e38 or 2.15000000000000015e-33 < y.im

   1. Initial program 54.5%

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

      1. *-un-lft-identity 54.5%

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

      2. add-sqr-sqrt 54.5%

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

      3. times-frac 54.4%

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

      4. hypot-define 54.4%

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

      5. fma-define 54.4%

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

      6. hypot-define 71.0%

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

   4. Applied egg-rr 71.0%

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

   5. Taylor expanded in y.re around 0 53.9%

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

      1. associate-/l* 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in y.re around 0 76.6%

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

   if -5.80000000000000013e38 < y.im < 2.15000000000000015e-33

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+38} \lor \neg \left(y.im \leq 2.15 \cdot 10^{-33}\right):\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+38} \lor \neg \left(y.im \leq 4.8 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.8e+38) (not (<= y.im 4.8e-33)))
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.8e+38) || !(y_46_im <= 4.8e-33)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.8d+38)) .or. (.not. (y_46im <= 4.8d-33))) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (1.0d0 / y_46re) * (x_46re + (x_46im * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.8e+38) || !(y_46_im <= 4.8e-33)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.8e+38) or not (y_46_im <= 4.8e-33):
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.8e+38) || !(y_46_im <= 4.8e-33))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.8e+38) || ~((y_46_im <= 4.8e-33)))
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.8e+38], N[Not[LessEqual[y$46$im, 4.8e-33]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.80000000000000035e38 or 4.8e-33 < y.im

   1. Initial program 54.5%

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

   3. Taylor expanded in y.re around 0 67.8%

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

      1. associate-/l* 71.2%

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

   5. Simplified 71.2%

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

      1. *-un-lft-identity 71.2%

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

      2. pow2 71.2%

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

      3. times-frac 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. associate-*r* 76.8%

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

      2. associate-*r/ 76.8%

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

      3. un-div-inv 76.8%

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

   9. Applied egg-rr 76.8%

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

   if -4.80000000000000035e38 < y.im < 4.8e-33

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+38} \lor \neg \left(y.im \leq 4.8 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+38}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.4e+38)
   (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))
   (if (<= y.im 6e-33)
     (* (/ 1.0 y.re) (+ x.re (* x.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_im <= -5.4e+38) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else if (y_46_im <= 6e-33) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_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_46im <= (-5.4d+38)) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    else if (y_46im <= 6d-33) then
        tmp = (1.0d0 / y_46re) * (x_46re + (x_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_im <= -5.4e+38) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else if (y_46_im <= 6e-33) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_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_im <= -5.4e+38:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	elif y_46_im <= 6e-33:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_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_im <= -5.4e+38)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	elseif (y_46_im <= 6e-33)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(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_im <= -5.4e+38)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	elseif (y_46_im <= 6e-33)
		tmp = (1.0 / y_46_re) * (x_46_re + (x_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[LessEqual[y$46$im, -5.4e+38], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6e-33], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.39999999999999992e38

   1. Initial program 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. add-sqr-sqrt 37.3%

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

      3. times-frac 37.4%

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

      4. hypot-define 37.4%

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

      5. fma-define 37.4%

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

      6. hypot-define 49.5%

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

   4. Applied egg-rr 49.5%

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

   5. Taylor expanded in y.re around 0 27.6%

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

      1. associate-/l* 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in y.re around 0 78.7%

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

   if -5.39999999999999992e38 < y.im < 6.0000000000000003e-33

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 6.0000000000000003e-33 < y.im

   1. Initial program 66.5%

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

   3. Taylor expanded in y.re around 0 67.0%

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

      1. associate-/l* 69.2%

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

   5. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. pow2 69.2%

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

      3. times-frac 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. associate-*l/ 75.2%

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

      2. *-un-lft-identity 75.2%

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

   9. Applied egg-rr 75.2%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+38}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6e+38)
   (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re)))
   (if (<= y.im 1.6e-43)
     (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re))))
     (+ (/ x.im y.im) (/ (* y.re (/ x.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_im <= -6e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 1.6e-43) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_46im <= (-6d+38)) then
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) / (y_46im / y_46re))
    else if (y_46im <= 1.6d-43) then
        tmp = (1.0d0 / y_46re) * (x_46re + (x_46im * (y_46im / y_46re)))
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_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_im <= -6e+38) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_im <= 1.6e-43) {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_im <= -6e+38:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re))
	elif y_46_im <= 1.6e-43:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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_im <= -6e+38)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	elseif (y_46_im <= 1.6e-43)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_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_im <= -6e+38)
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	elseif (y_46_im <= 1.6e-43)
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_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[LessEqual[y$46$im, -6e+38], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e-43], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$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 * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.0000000000000002e38

   1. Initial program 37.3%

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

   3. Taylor expanded in y.re around 0 69.0%

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

      1. associate-/l* 74.1%

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

   5. Simplified 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 77.3%

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

   7. Applied egg-rr 77.3%

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

      1. associate-*r* 79.1%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.1%

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

      4. un-div-inv 79.1%

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

   9. Applied egg-rr 79.1%

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

   if -6.0000000000000002e38 < y.im < 1.59999999999999992e-43

   1. Initial program 74.7%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. pow2 78.4%

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

      3. times-frac 84.4%

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

   5. Applied egg-rr 84.4%

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

      1. +-commutative 84.4%

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

      2. *-commutative 84.4%

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

      3. div-inv 84.2%

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

      4. distribute-rgt-out 85.7%

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

      5. associate-/l* 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 1.59999999999999992e-43 < y.im

   1. Initial program 66.5%

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

   3. Taylor expanded in y.re around 0 67.0%

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

      1. associate-/l* 69.2%

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

   5. Simplified 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. pow2 69.2%

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

      3. times-frac 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. associate-*r* 75.2%

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

      2. associate-*r/ 75.3%

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

      3. un-div-inv 75.3%

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

   9. Applied egg-rr 75.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -320000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -320000000.0)
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
   (if (<= y.re 4.4e-131)
     (* (+ x.im (* x.re (/ y.re y.im))) (/ 1.0 y.im))
     (* (/ 1.0 y.re) (+ x.re (* x.im (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -320000000.0) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-320000000.0d0)) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else if (y_46re <= 4.4d-131) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) * (1.0d0 / y_46im)
    else
        tmp = (1.0d0 / y_46re) * (x_46re + (x_46im * (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -320000000.0) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= 4.4e-131) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	} else {
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -320000000.0:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	elif y_46_re <= 4.4e-131:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im)
	else:
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -320000000.0)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= 4.4e-131)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(1.0 / y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -320000000.0)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	elseif (y_46_re <= 4.4e-131)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (1.0 / y_46_im);
	else
		tmp = (1.0 / y_46_re) * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -320000000.0], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.4e-131], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.2e8

   1. Initial program 52.9%

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

   3. Taylor expanded in y.re around inf 68.5%

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

      1. *-commutative 68.5%

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

      2. pow2 68.5%

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

      3. times-frac 69.8%

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

   5. Applied egg-rr 69.8%

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

   if -3.2e8 < y.re < 4.3999999999999999e-131

   1. Initial program 79.0%

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

      1. *-un-lft-identity 79.0%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 78.9%

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

      4. hypot-define 78.9%

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

      5. fma-define 78.9%

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

      6. hypot-define 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in y.re around 0 52.5%

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

      1. associate-/l* 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in y.re around 0 89.0%

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

   if 4.3999999999999999e-131 < y.re

   1. Initial program 62.6%

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

   3. Taylor expanded in y.re around inf 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. pow2 72.5%

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

      3. times-frac 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. +-commutative 75.6%

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

      2. *-commutative 75.6%

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

      3. div-inv 75.4%

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

      4. distribute-rgt-out 75.5%

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

      5. associate-/l* 81.3%

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

   7. Applied egg-rr 81.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -320000000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-131}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.05e+29)
   (/ x.im y.im)
   (if (<= y.im 3.8e-29) (/ x.re y.re) (/ 1.0 (/ y.im x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.05e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.8e-29) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = 1.0 / (y_46_im / x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.05d+29)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 3.8d-29) then
        tmp = x_46re / y_46re
    else
        tmp = 1.0d0 / (y_46im / x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.05e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.8e-29) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = 1.0 / (y_46_im / x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.05e+29:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 3.8e-29:
		tmp = x_46_re / y_46_re
	else:
		tmp = 1.0 / (y_46_im / x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.05e+29)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 3.8e-29)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(1.0 / Float64(y_46_im / x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.05e+29)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 3.8e-29)
		tmp = x_46_re / y_46_re;
	else
		tmp = 1.0 / (y_46_im / x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.05e+29], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.8e-29], N[(x$46$re / y$46$re), $MachinePrecision], N[(1.0 / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.0500000000000001e29

   1. Initial program 36.7%

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

   3. Taylor expanded in y.re around 0 70.9%

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

   if -1.0500000000000001e29 < y.im < 3.79999999999999976e-29

   1. Initial program 75.7%

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

   3. Taylor expanded in y.re around inf 68.8%

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

   if 3.79999999999999976e-29 < y.im

   1. Initial program 65.1%

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

      1. *-un-lft-identity 65.1%

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

      2. add-sqr-sqrt 65.1%

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

      3. times-frac 65.0%

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

      4. hypot-define 65.0%

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

      5. fma-define 65.0%

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

      6. hypot-define 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. associate-*l/ 85.6%

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

      2. *-un-lft-identity 85.6%

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

      3. clear-num 84.6%

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

   6. Applied egg-rr 84.6%

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

   7. Taylor expanded in y.re around 0 61.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+35} \lor \neg \left(y.im \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.5e+35) (not (<= y.im 5.2e-29)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.5e+35) || !(y_46_im <= 5.2e-29)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.5d+35)) .or. (.not. (y_46im <= 5.2d-29))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.5e+35) || !(y_46_im <= 5.2e-29)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.5e+35) or not (y_46_im <= 5.2e-29):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.5e+35) || !(y_46_im <= 5.2e-29))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.5e+35) || ~((y_46_im <= 5.2e-29)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.5e+35], N[Not[LessEqual[y$46$im, 5.2e-29]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.5000000000000003e35 or 5.2000000000000004e-29 < y.im

   1. Initial program 52.9%

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

   3. Taylor expanded in y.re around 0 65.4%

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

   if -6.5000000000000003e35 < y.im < 5.2000000000000004e-29

   1. Initial program 75.7%

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

   3. Taylor expanded in y.re around inf 68.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+35} \lor \neg \left(y.im \leq 5.2 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.75 \cdot 10^{+194}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.75e+194) (/ x.im y.re) (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.75e+194) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.75d+194)) then
        tmp = x_46im / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.75e+194) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.75e+194:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.75e+194)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.75e+194)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.75e+194], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.75e194

   1. Initial program 40.7%

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

      1. *-un-lft-identity 40.7%

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

      2. add-sqr-sqrt 40.7%

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

      3. times-frac 40.7%

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

      4. hypot-define 40.7%

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

      5. fma-define 40.7%

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

      6. hypot-define 51.2%

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

   4. Applied egg-rr 51.2%

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

   5. Taylor expanded in y.re around 0 17.5%

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

      1. associate-/l* 20.6%

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

   7. Simplified 20.6%

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

   8. Taylor expanded in y.re around inf 17.7%

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

      1. +-commutative 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in x.re around 0 28.6%

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

   if -2.75e194 < y.re

   1. Initial program 68.1%

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

   3. Taylor expanded in y.re around 0 40.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.75 \cdot 10^{+194}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

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

3. Taylor expanded in y.re around 0 37.6%

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

4. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2024062 
(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))))