_divideComplex, real part

Percentage Accurate: 62.0% → 85.8%

Time: 10.6s

Alternatives: 19

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. add-sqr-sqrt 75.3%

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

      3. times-frac 75.4%

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

      4. hypot-def 75.4%

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

      5. fma-def 75.4%

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

      6. hypot-def 92.4%

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

   3. Applied egg-rr 92.4%

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

   if +inf.0 < (/.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 0.0%

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

   2. Taylor expanded in y.re around 0 41.1%

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

      1. +-commutative 41.1%

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

      2. *-commutative 41.1%

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

      3. unpow2 41.1%

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

      4. times-frac 55.3%

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

   4. Simplified 55.3%

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

4. Final simplification 84.0%

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

Alternative 2: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{1}{\frac{y.re}{x.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+91)
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.re 1.25e-141)
     (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
     (if (<= y.re 3.3e+56)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (*
        (/ 1.0 (hypot y.re y.im))
        (+ x.re (* y.im (/ 1.0 (/ y.re 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_re <= -1.15e+91) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1.25e-141) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 3.3e+56) {
		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_re + (y_46_im * (1.0 / (y_46_re / x_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_re <= -1.15e+91) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1.25e-141) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 3.3e+56) {
		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_re + (y_46_im * (1.0 / (y_46_re / 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_re <= -1.15e+91:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= 1.25e-141:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 3.3e+56:
		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_re + (y_46_im * (1.0 / (y_46_re / 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_re <= -1.15e+91)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 1.25e-141)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 3.3e+56)
		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_re + Float64(y_46_im * Float64(1.0 / Float64(y_46_re / 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_re <= -1.15e+91)
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 1.25e-141)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 3.3e+56)
		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_re + (y_46_im * (1.0 / (y_46_re / 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$46$re, -1.15e+91], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e-141], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.3e+56], 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$46$re + N[(y$46$im * N[(1.0 / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.14999999999999996e91

   1. Initial program 35.6%

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

      1. *-un-lft-identity 35.6%

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

      2. add-sqr-sqrt 35.6%

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

      3. times-frac 35.6%

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

      4. hypot-def 35.6%

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

      5. fma-def 35.6%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.re around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. unsub-neg 64.7%

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

      3. mul-1-neg 64.7%

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

      4. associate-/l* 74.8%

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

      5. distribute-neg-frac 74.8%

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

   6. Simplified 74.8%

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

   if -1.14999999999999996e91 < y.re < 1.25e-141

   1. Initial program 66.8%

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

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

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

      1. +-commutative 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/l* 72.3%

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

   4. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. inv-pow 72.3%

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

      3. associate-/l* 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. unpow-1 78.7%

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

      2. associate-/l/ 81.2%

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

   8. Simplified 81.2%

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

   if 1.25e-141 < y.re < 3.30000000000000002e56

   1. Initial program 80.4%

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

   if 3.30000000000000002e56 < y.re

   1. Initial program 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. hypot-def 37.8%

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

      5. fma-def 37.8%

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

      6. hypot-def 54.9%

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

   3. Applied egg-rr 54.9%

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

   4. Taylor expanded in y.re around inf 73.9%

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

      1. associate-/l* 82.9%

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

   6. Simplified 82.9%

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

      1. div-inv 82.9%

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

   8. Applied egg-rr 82.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + y.im \cdot \frac{1}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 3: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.6e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 1.3e-140)
     (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
     (if (<= y.re 9.2e+55)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (* (/ 1.0 (hypot y.re y.im)) (+ x.re (/ y.im (/ y.re 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_re <= -2.6e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 1.3e-140) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 9.2e+55) {
		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_re + (y_46_im / (y_46_re / x_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_re <= -2.6e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 1.3e-140) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 9.2e+55) {
		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_re + (y_46_im / (y_46_re / 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_re <= -2.6e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 1.3e-140:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 9.2e+55:
		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_re + (y_46_im / (y_46_re / 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_re <= -2.6e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 1.3e-140)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 9.2e+55)
		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_re + Float64(y_46_im / Float64(y_46_re / 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_re <= -2.6e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 1.3e-140)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 9.2e+55)
		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_re + (y_46_im / (y_46_re / 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$46$re, -2.6e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e-140], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.2e+55], 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$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.6e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -2.6e91 < y.re < 1.2999999999999999e-140

   1. Initial program 66.8%

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

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

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

      1. +-commutative 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/l* 72.3%

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

   4. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. inv-pow 72.3%

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

      3. associate-/l* 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. unpow-1 78.7%

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

      2. associate-/l/ 81.2%

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

   8. Simplified 81.2%

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

   if 1.2999999999999999e-140 < y.re < 9.1999999999999995e55

   1. Initial program 80.4%

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

   if 9.1999999999999995e55 < y.re

   1. Initial program 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. hypot-def 37.8%

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

      5. fma-def 37.8%

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

      6. hypot-def 54.9%

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

   3. Applied egg-rr 54.9%

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

   4. Taylor expanded in y.re around inf 73.9%

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

      1. associate-/l* 82.9%

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

   6. Simplified 82.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 4: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 10^{+57}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ x.re (/ y.im (/ y.re x.im)))))
   (if (<= y.re -1.15e+91)
     (* t_0 (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re 1.25e-136)
       (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
       (if (<= y.re 1e+57)
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (* (/ 1.0 (hypot y.re y.im)) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = t_0 * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1.25e-136) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 1e+57) {
		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)) * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = t_0 * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1.25e-136) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 1e+57) {
		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)) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re + (y_46_im / (y_46_re / x_46_im))
	tmp = 0
	if y_46_re <= -1.15e+91:
		tmp = t_0 * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= 1.25e-136:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 1e+57:
		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)) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.15e+91)
		tmp = Float64(t_0 * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 1.25e-136)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 1e+57)
		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)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.15e+91)
		tmp = t_0 * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 1.25e-136)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 1e+57)
		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)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+91], N[(t$95$0 * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e-136], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e+57], 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] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.14999999999999996e91

   1. Initial program 35.6%

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

      1. *-un-lft-identity 35.6%

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

      2. add-sqr-sqrt 35.6%

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

      3. times-frac 35.6%

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

      4. hypot-def 35.6%

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

      5. fma-def 35.6%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.re around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. unsub-neg 64.7%

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

      3. mul-1-neg 64.7%

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

      4. associate-/l* 74.8%

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

      5. distribute-neg-frac 74.8%

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

   6. Simplified 74.8%

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

   if -1.14999999999999996e91 < y.re < 1.25e-136

   1. Initial program 66.8%

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

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

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

      1. +-commutative 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/l* 72.3%

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

   4. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. inv-pow 72.3%

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

      3. associate-/l* 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. unpow-1 78.7%

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

      2. associate-/l/ 81.2%

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

   8. Simplified 81.2%

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

   if 1.25e-136 < y.re < 1.00000000000000005e57

   1. Initial program 80.4%

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

   if 1.00000000000000005e57 < y.re

   1. Initial program 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. hypot-def 37.8%

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

      5. fma-def 37.8%

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

      6. hypot-def 54.9%

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

   3. Applied egg-rr 54.9%

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

   4. Taylor expanded in y.re around inf 73.9%

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

      1. associate-/l* 82.9%

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

   6. Simplified 82.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 10^{+57}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 5: 79.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.65e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 2.5e-137)
     (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
     (if (<= y.re 2.3e+55)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.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 <= -1.65e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 2.5e-137) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 2.3e+55) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_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 <= (-1.65d+91)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 2.5d-137) then
        tmp = (x_46im / y_46im) + (1.0d0 / (y_46im / (x_46re * (y_46re / y_46im))))
    else if (y_46re <= 2.3d+55) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_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 <= -1.65e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 2.5e-137) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else if (y_46_re <= 2.3e+55) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_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 <= -1.65e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 2.5e-137:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	elif y_46_re <= 2.3e+55:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_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 <= -1.65e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 2.5e-137)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	elseif (y_46_re <= 2.3e+55)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_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 <= -1.65e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 2.5e-137)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	elseif (y_46_re <= 2.3e+55)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_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, -1.65e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.5e-137], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+55], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.65000000000000009e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -1.65000000000000009e91 < y.re < 2.5e-137

   1. Initial program 66.8%

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

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

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

      1. +-commutative 70.9%

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

      2. unpow2 70.9%

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

      3. associate-/l* 72.3%

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

   4. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. inv-pow 72.3%

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

      3. associate-/l* 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. unpow-1 78.7%

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

      2. associate-/l/ 81.2%

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

   8. Simplified 81.2%

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

   if 2.5e-137 < y.re < 2.29999999999999987e55

   1. Initial program 80.4%

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

   if 2.29999999999999987e55 < y.re

   1. Initial program 37.8%

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

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

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

      1. unpow2 67.4%

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

      2. times-frac 82.6%

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

   4. Simplified 82.6%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 6: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 240000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.26e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 240000000.0)
     (+ (/ x.im y.im) (/ 1.0 (/ y.im (* x.re (/ y.re y.im)))))
     (+ (/ x.re y.re) (/ (/ y.im (/ y.re x.im)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.26e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 240000000.0) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.26d+91)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 240000000.0d0) then
        tmp = (x_46im / y_46im) + (1.0d0 / (y_46im / (x_46re * (y_46re / y_46im))))
    else
        tmp = (x_46re / y_46re) + ((y_46im / (y_46re / x_46im)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.26e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 240000000.0) {
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.26e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 240000000.0:
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))))
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.26e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 240000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im)))));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_im)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.26e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 240000000.0)
		tmp = (x_46_im / y_46_im) + (1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im))));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.26e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 240000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.26e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -1.26e91 < y.re < 2.4e8

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. associate-/l* 70.3%

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

   4. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

      3. associate-/l* 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. unpow-1 75.6%

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

      2. associate-/l/ 78.3%

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

   8. Simplified 78.3%

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

   if 2.4e8 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. associate-*l/ 81.4%

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

      2. clear-num 81.4%

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

      3. un-div-inv 81.5%

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

   6. Applied egg-rr 81.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.26 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 240000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 7: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e+91)
   (/ x.re y.re)
   (if (<= y.re 1.15e-144)
     (/ x.im y.im)
     (if (<= y.re 7.2e-110)
       (/ 1.0 (/ y.im (* x.re (/ y.re y.im))))
       (if (<= y.re 1.3e-50) (/ x.im y.im) (/ x.re y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e-144) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 7.2e-110) {
		tmp = 1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.3e-50) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.15d+91)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.15d-144) then
        tmp = x_46im / y_46im
    else if (y_46re <= 7.2d-110) then
        tmp = 1.0d0 / (y_46im / (x_46re * (y_46re / y_46im)))
    else if (y_46re <= 1.3d-50) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e+91) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e-144) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 7.2e-110) {
		tmp = 1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im)));
	} else if (y_46_re <= 1.3e-50) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.15e+91:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.15e-144:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 7.2e-110:
		tmp = 1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im)))
	elif y_46_re <= 1.3e-50:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e+91)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.15e-144)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 7.2e-110)
		tmp = Float64(1.0 / Float64(y_46_im / Float64(x_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_re <= 1.3e-50)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.15e+91)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.15e-144)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 7.2e-110)
		tmp = 1.0 / (y_46_im / (x_46_re * (y_46_re / y_46_im)));
	elseif (y_46_re <= 1.3e-50)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e+91], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.15e-144], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-110], N[(1.0 / N[(y$46$im / N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e-50], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.14999999999999996e91 or 1.3000000000000001e-50 < y.re

   1. Initial program 43.2%

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

   2. Taylor expanded in y.re around inf 60.1%

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

   if -1.14999999999999996e91 < y.re < 1.15e-144 or 7.1999999999999999e-110 < y.re < 1.3000000000000001e-50

   1. Initial program 68.4%

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

   2. Taylor expanded in y.re around 0 60.1%

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

   if 1.15e-144 < y.re < 7.1999999999999999e-110

   1. Initial program 99.7%

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

   2. Taylor expanded in y.re around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. associate-/l* 99.7%

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

   4. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. clear-num 99.7%

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

      3. clear-num 99.7%

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

      4. frac-add 80.4%

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

      5. *-un-lft-identity 80.4%

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

      6. associate-/l* 80.4%

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

      7. associate-/l* 80.7%

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

   6. Applied egg-rr 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 99.7%

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

      3. *-rgt-identity 99.7%

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

      4. +-commutative 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. associate-/l/ 99.7%

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

      7. *-rgt-identity 99.7%

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

      8. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y.im around 0 99.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.18 \cdot 10^{+91} \lor \neg \left(y.re \leq 360000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.18e+91) (not (<= y.re 360000000.0)))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 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_re <= -1.18e+91) || !(y_46_re <= 360000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_46re <= (-1.18d+91)) .or. (.not. (y_46re <= 360000000.0d0))) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / 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_re <= -1.18e+91) || !(y_46_re <= 360000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_re <= -1.18e+91) or not (y_46_re <= 360000000.0):
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_re <= -1.18e+91) || !(y_46_re <= 360000000.0))
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(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_re <= -1.18e+91) || ~((y_46_re <= 360000000.0)))
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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[Or[LessEqual[y$46$re, -1.18e+91], N[Not[LessEqual[y$46$re, 360000000.0]], $MachinePrecision]], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.18000000000000008e91 or 3.6e8 < y.re

   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. 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-def 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-def 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-def 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in y.re around inf 56.8%

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

      1. associate-/l* 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in y.re around inf 78.8%

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

   if -1.18000000000000008e91 < y.re < 3.6e8

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. associate-/l* 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in x.re around 0 69.2%

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

      1. unpow2 69.2%

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

      2. associate-*l/ 66.8%

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

      3. *-commutative 66.8%

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

      4. associate-/r* 71.3%

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

   7. Simplified 71.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.18 \cdot 10^{+91} \lor \neg \left(y.re \leq 360000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 9: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91} \lor \neg \left(y.re \leq 56000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.15e+91) (not (<= y.re 56000000.0)))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 y.re))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.15e+91) || !(y_46_re <= 56000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.15d+91)) .or. (.not. (y_46re <= 56000000.0d0))) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / y_46re)
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.15e+91) || !(y_46_re <= 56000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.15e+91) or not (y_46_re <= 56000000.0):
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.15e+91) || !(y_46_re <= 56000000.0))
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.15e+91) || ~((y_46_re <= 56000000.0)))
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.15e+91], N[Not[LessEqual[y$46$re, 56000000.0]], $MachinePrecision]], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.14999999999999996e91 or 5.6e7 < y.re

   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. 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-def 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-def 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-def 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in y.re around inf 56.8%

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

      1. associate-/l* 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in y.re around inf 78.8%

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

   if -1.14999999999999996e91 < y.re < 5.6e7

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91} \lor \neg \left(y.re \leq 56000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 10: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+91} \lor \neg \left(y.re \leq 6.8 \cdot 10^{-51}\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.8e+91) (not (<= y.re 6.8e-51)))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 y.re))
   (+ (/ x.im y.im) (/ x.re (* y.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 <= -5.8e+91) || !(y_46_re <= 6.8e-51)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_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 <= (-5.8d+91)) .or. (.not. (y_46re <= 6.8d-51))) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / y_46re)
    else
        tmp = (x_46im / y_46im) + (x_46re / (y_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 <= -5.8e+91) || !(y_46_re <= 6.8e-51)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_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 <= -5.8e+91) or not (y_46_re <= 6.8e-51):
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_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 <= -5.8e+91) || !(y_46_re <= 6.8e-51))
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_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 <= -5.8e+91) || ~((y_46_re <= 6.8e-51)))
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_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$46$re, -5.8e+91], N[Not[LessEqual[y$46$re, 6.8e-51]], $MachinePrecision]], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.80000000000000028e91 or 6.80000000000000005e-51 < y.re

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. times-frac 43.7%

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

      4. hypot-def 43.7%

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

      5. fma-def 43.7%

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

      6. hypot-def 61.9%

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

   3. Applied egg-rr 61.9%

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

   4. Taylor expanded in y.re around inf 56.2%

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

      1. associate-/l* 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in y.re around inf 75.4%

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

   if -5.80000000000000028e91 < y.re < 6.80000000000000005e-51

   1. Initial program 69.3%

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

   2. Taylor expanded in y.re around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. associate-/l* 72.7%

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

   4. Simplified 72.7%

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

   5. Taylor expanded in y.im around 0 72.7%

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

      1. unpow2 72.7%

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

      2. *-rgt-identity 72.7%

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

      3. associate-*r/ 72.7%

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

      4. associate-*l* 78.4%

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

      5. associate-*r/ 78.5%

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

      6. *-rgt-identity 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+91} \lor \neg \left(y.re \leq 6.8 \cdot 10^{-51}\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 11: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91} \lor \neg \left(y.re \leq 80000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.15e+91) (not (<= y.re 80000000.0)))
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 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_re <= -1.15e+91) || !(y_46_re <= 80000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_46re <= (-1.15d+91)) .or. (.not. (y_46re <= 80000000.0d0))) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / 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_re <= -1.15e+91) || !(y_46_re <= 80000000.0)) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_re <= -1.15e+91) or not (y_46_re <= 80000000.0):
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_re <= -1.15e+91) || !(y_46_re <= 80000000.0))
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / 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_re <= -1.15e+91) || ~((y_46_re <= 80000000.0)))
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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[Or[LessEqual[y$46$re, -1.15e+91], N[Not[LessEqual[y$46$re, 80000000.0]], $MachinePrecision]], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $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 2 regimes

2. if y.re < -1.14999999999999996e91 or 8e7 < y.re

   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. 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-def 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-def 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-def 59.4%

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

   3. Applied egg-rr 59.4%

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

   4. Taylor expanded in y.re around inf 56.8%

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

      1. associate-/l* 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in y.re around inf 78.8%

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

   if -1.14999999999999996e91 < y.re < 8e7

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*r/ 76.2%

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

   6. Applied egg-rr 76.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+91} \lor \neg \left(y.re \leq 80000000\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 12: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.5e+142)
   (/ x.im y.im)
   (if (<= y.im 7.2e+62)
     (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 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_im <= -6.5e+142) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7.2e+62) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_46im <= (-6.5d+142)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 7.2d+62) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / 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_im <= -6.5e+142) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7.2e+62) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_im <= -6.5e+142:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 7.2e+62:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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_im <= -6.5e+142)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 7.2e+62)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / 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_im <= -6.5e+142)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 7.2e+62)
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / 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$46$im, -6.5e+142], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+62], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.4999999999999997e142 or 7.2e62 < y.im

   1. Initial program 35.0%

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

   2. Taylor expanded in y.re around 0 68.9%

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

   if -6.4999999999999997e142 < y.im < 7.2e62

   1. Initial program 71.6%

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

      1. *-un-lft-identity 71.6%

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

      2. add-sqr-sqrt 71.6%

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

      3. times-frac 71.6%

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

      4. hypot-def 71.6%

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

      5. fma-def 71.6%

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

      6. hypot-def 83.1%

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

   3. Applied egg-rr 83.1%

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

   4. Taylor expanded in y.re around inf 40.9%

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

      1. associate-/l* 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in y.re around inf 66.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+62}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 265000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.2e+91)
   (* (+ x.re (/ y.im (/ y.re x.im))) (/ 1.0 y.re))
   (if (<= y.re 265000000.0)
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
     (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+91) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else if (y_46_re <= 265000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.2d+91)) then
        tmp = (x_46re + (y_46im / (y_46re / x_46im))) * (1.0d0 / y_46re)
    else if (y_46re <= 265000000.0d0) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+91) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	} else if (y_46_re <= 265000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.2e+91:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re)
	elif y_46_re <= 265000000.0:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.2e+91)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= 265000000.0)
		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(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.2e+91)
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (1.0 / y_46_re);
	elseif (y_46_re <= 265000000.0)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.2e+91], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 265000000.0], 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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.19999999999999991e91

   1. Initial program 35.6%

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

      1. *-un-lft-identity 35.6%

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

      2. add-sqr-sqrt 35.6%

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

      3. times-frac 35.6%

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

      4. hypot-def 35.6%

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

      5. fma-def 35.6%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.re around inf 27.3%

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

      1. associate-/l* 27.5%

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

   6. Simplified 27.5%

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

   7. Taylor expanded in y.re around inf 74.3%

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

   if -1.19999999999999991e91 < y.re < 2.65e8

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*r/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 2.65e8 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 265000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 14: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 60000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.2e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 60000000.0)
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
     (+ (/ x.re y.re) (* (/ x.im y.re) (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 60000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.2d+91)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 60000000.0d0) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((x_46im / y_46re) * (y_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.2e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 60000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.2e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 60000000.0:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.2e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 60000000.0)
		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(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) * Float64(y_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.2e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 60000000.0)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) * (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.2e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 60000000.0], 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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.19999999999999991e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -1.19999999999999991e91 < y.re < 6e7

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*r/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 6e7 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 60000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 15: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 30000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.4e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 30000000.0)
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
     (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re 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_re <= -1.4e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 30000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / 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_46re <= (-1.4d+91)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 30000000.0d0) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) / (y_46re / 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_re <= -1.4e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 30000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / 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_re <= -1.4e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 30000000.0:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / 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_re <= -1.4e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 30000000.0)
		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(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / 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_re <= -1.4e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 30000000.0)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / 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$46$re, -1.4e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 30000000.0], 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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.3999999999999999e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -1.3999999999999999e91 < y.re < 3e7

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*r/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 3e7 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. clear-num 81.5%

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

      2. un-div-inv 81.4%

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

   6. Applied egg-rr 81.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 30000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 16: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 48000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.65e+91)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 48000000.0)
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
     (+ (/ x.re y.re) (/ (/ y.im (/ y.re x.im)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.65e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 48000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.65d+91)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 48000000.0d0) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else
        tmp = (x_46re / y_46re) + ((y_46im / (y_46re / x_46im)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.65e+91) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 48000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.65e+91:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 48000000.0:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.65e+91)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 48000000.0)
		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(x_46_re / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_im)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.65e+91)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 48000000.0)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.65e+91], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 48000000.0], 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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.65000000000000009e91

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 64.1%

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

      1. unpow2 64.1%

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

      2. times-frac 71.9%

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

   4. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

   if -1.65000000000000009e91 < y.re < 4.8e7

   1. Initial program 68.9%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. *-commutative 69.2%

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

      3. unpow2 69.2%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*r/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 4.8e7 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. associate-*l/ 81.4%

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

      2. clear-num 81.4%

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

      3. un-div-inv 81.5%

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

   6. Applied egg-rr 81.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 48000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 17: 63.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.18 \cdot 10^{+91} \lor \neg \left(y.re \leq 1.2 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.18e+91) (not (<= y.re 1.2e-50)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.18e+91) || !(y_46_re <= 1.2e-50)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.18d+91)) .or. (.not. (y_46re <= 1.2d-50))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.18e+91) || !(y_46_re <= 1.2e-50)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.18e+91) or not (y_46_re <= 1.2e-50):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.18e+91) || !(y_46_re <= 1.2e-50))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.18e+91) || ~((y_46_re <= 1.2e-50)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.18e+91], N[Not[LessEqual[y$46$re, 1.2e-50]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.18000000000000008e91 or 1.20000000000000001e-50 < y.re

   1. Initial program 43.2%

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

   2. Taylor expanded in y.re around inf 60.1%

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

   if -1.18000000000000008e91 < y.re < 1.20000000000000001e-50

   1. Initial program 69.5%

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

   2. Taylor expanded in y.re around 0 58.2%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.18 \cdot 10^{+91} \lor \neg \left(y.re \leq 1.2 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 18: 43.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 8 \cdot 10^{+164}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 8e+164) (/ x.im y.im) (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 8e+164) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= 8d+164) then
        tmp = x_46im / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 8e+164) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 8e+164:
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 8e+164)
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 8e+164)
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 8e+164], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 8e164

   1. Initial program 63.2%

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

   2. Taylor expanded in y.re around 0 47.4%

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

   if 8e164 < y.re

   1. Initial program 31.1%

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

      1. *-un-lft-identity 31.1%

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

      2. add-sqr-sqrt 31.1%

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

      3. times-frac 31.1%

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

      4. hypot-def 31.1%

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

      5. fma-def 31.1%

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

      6. hypot-def 49.9%

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

   3. Applied egg-rr 49.9%

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

   4. Taylor expanded in y.re around inf 73.5%

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

      1. associate-/l* 83.5%

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

   6. Simplified 83.5%

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

   7. Taylor expanded in y.re around 0 26.9%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 8 \cdot 10^{+164}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 19: 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 58.3%

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

2. Taylor expanded in y.re around 0 42.0%

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

3. Final simplification 42.0%

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

Reproduce

herbie shell --seed 2023238 
(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))))