_divideComplex, real part

Percentage Accurate: 60.8% → 90.1%

Time: 13.7s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.8% 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: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)\\ t_1 := \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{t\_0 \cdot t\_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma x.re (/ y.re y.im) x.im)) (t_1 (/ y.im (hypot y.re y.im))))
   (if (<= y.im -1.3e-67)
     (/ (* t_0 t_1) (hypot y.re y.im))
     (if (<= y.im 1.95e-42)
       (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
       (* t_1 (/ t_0 (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_re, (y_46_re / y_46_im), x_46_im);
	double t_1 = y_46_im / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.3e-67) {
		tmp = (t_0 * t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 1.95e-42) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_1 * (t_0 / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im)
	t_1 = Float64(y_46_im / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.3e-67)
		tmp = Float64(Float64(t_0 * t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 1.95e-42)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(t_1 * Float64(t_0 / hypot(y_46_re, y_46_im)));
	end
	return 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$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.3e-67], N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.95e-42], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(t$95$1 * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2999999999999999e-67

   1. Initial program 56.1%

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

   3. Taylor expanded in y.im around inf 56.1%

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

      1. associate-/l* 56.0%

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

   5. Simplified 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 56.0%

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

      3. hypot-undefine 56.0%

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

      4. hypot-undefine 56.0%

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

      5. times-frac 94.0%

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

      6. +-commutative 94.0%

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

      7. fma-define 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. associate-*l/ 94.1%

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

   9. Applied egg-rr 94.1%

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

   if -1.2999999999999999e-67 < y.im < 1.9500000000000001e-42

   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. Add Preprocessing

   3. Taylor expanded in y.re around inf 88.4%

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

      1. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

   if 1.9500000000000001e-42 < y.im

   1. Initial program 52.1%

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

   3. Taylor expanded in y.im around inf 52.1%

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

      1. associate-/l* 52.1%

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

   5. Simplified 52.1%

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

      1. *-commutative 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. hypot-undefine 52.1%

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

      4. hypot-undefine 52.1%

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

      5. times-frac 96.8%

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

      6. +-commutative 96.8%

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

      7. fma-define 96.8%

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right) \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-66} \lor \neg \left(y.im \leq 1.95 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.65e-66) (not (<= y.im 1.95e-41)))
   (*
    (/ y.im (hypot y.re y.im))
    (/ (fma x.re (/ y.re y.im) x.im) (hypot y.re y.im)))
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.65e-66) || !(y_46_im <= 1.95e-41)) {
		tmp = (y_46_im / hypot(y_46_re, y_46_im)) * (fma(x_46_re, (y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.65e-66) || !(y_46_im <= 1.95e-41))
		tmp = Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.65e-66], N[Not[LessEqual[y$46$im, 1.95e-41]], $MachinePrecision]], N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.6499999999999999e-66 or 1.94999999999999995e-41 < y.im

   1. Initial program 54.3%

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

   3. Taylor expanded in y.im around inf 54.3%

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

      1. associate-/l* 54.3%

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

   5. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. add-sqr-sqrt 54.3%

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

      3. hypot-undefine 54.3%

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

      4. hypot-undefine 54.3%

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

      5. times-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. fma-define 95.2%

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

   7. Applied egg-rr 95.2%

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

   if -1.6499999999999999e-66 < y.im < 1.94999999999999995e-41

   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. Add Preprocessing

   3. Taylor expanded in y.re around inf 88.4%

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

      1. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-66} \lor \neg \left(y.im \leq 1.95 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + y.im \cdot x.im\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* y.im x.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (y_46_im * x_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= math.inf:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (Float64(t_0 / 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(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= Inf)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / 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[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $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 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. associate-*r/ 75.5%

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

      3. fma-define 75.5%

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

      4. add-sqr-sqrt 75.5%

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

      5. times-frac 75.4%

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

      6. fma-define 75.4%

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

      7. hypot-define 75.4%

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

      8. fma-define 75.4%

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

      9. fma-define 75.5%

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

      10. hypot-define 94.5%

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

   4. Applied egg-rr 94.5%

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

      1. fma-define 94.5%

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

   6. Applied egg-rr 94.5%

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

   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. Add Preprocessing

   3. Taylor expanded in y.im around inf 44.6%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

4. Final simplification 88.4%

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

Alternative 4: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -9.5e+103)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (if (<= y.im -7.2e-67)
       (/ (+ (* x.re y.re) (* y.im x.im)) t_0)
       (if (<= y.im 1.05e-45)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 4.1e+148)
           (/ (* y.im (+ x.im (* y.re (/ x.re y.im)))) t_0)
           (/ (fma y.re (/ x.re y.im) x.im) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -9.5e+103) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -7.2e-67) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	} else if (y_46_im <= 1.05e-45) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 4.1e+148) {
		tmp = (y_46_im * (x_46_im + (y_46_re * (x_46_re / y_46_im)))) / t_0;
	} else {
		tmp = fma(y_46_re, (x_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -9.5e+103)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= -7.2e-67)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / t_0);
	elseif (y_46_im <= 1.05e-45)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 4.1e+148)
		tmp = Float64(Float64(y_46_im * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im)))) / t_0);
	else
		tmp = Float64(fma(y_46_re, Float64(x_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+103], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.2e-67], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.05e-45], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.1e+148], N[(N[(y$46$im * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.49999999999999922e103

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around inf 82.7%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -9.49999999999999922e103 < y.im < -7.19999999999999998e-67

   1. Initial program 80.1%

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

   if -7.19999999999999998e-67 < y.im < 1.04999999999999998e-45

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

   if 1.04999999999999998e-45 < y.im < 4.0999999999999998e148

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x.re around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r/ 78.1%

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

   8. Simplified 78.1%

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

   if 4.0999999999999998e148 < y.im

   1. Initial program 24.3%

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

   3. Taylor expanded in y.im around inf 24.3%

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

      1. associate-/l* 24.3%

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

   5. Simplified 24.3%

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

      1. *-commutative 24.3%

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

      2. add-sqr-sqrt 24.3%

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

      3. hypot-undefine 24.3%

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

      4. hypot-undefine 24.3%

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

      5. times-frac 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-define 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in y.re around 0 73.0%

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

       1. +-commutative 73.0%

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

       2. *-commutative 73.0%

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

       3. associate-*r/ 86.1%

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

       4. fma-define 86.1%

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

   12. Simplified 86.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := y.re \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + t\_1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \left(t\_1 - x.im \cdot \left(\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}\right)\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))) (t_1 (* y.re (/ x.re y.im))))
   (if (<= y.im -1.3e+106)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (if (<= y.im -7.5e-67)
       (/ (+ (* x.re y.re) (* y.im x.im)) t_0)
       (if (<= y.im 1.6e-45)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 3.1e+146)
           (/ (* y.im (+ x.im t_1)) t_0)
           (/
            (+ x.im (- t_1 (* x.im (* (/ y.re y.im) (/ y.re y.im)))))
            y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = y_46_re * (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.3e+106) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -7.5e-67) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	} else if (y_46_im <= 1.6e-45) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+146) {
		tmp = (y_46_im * (x_46_im + t_1)) / t_0;
	} else {
		tmp = (x_46_im + (t_1 - (x_46_im * ((y_46_re / y_46_im) * (y_46_re / y_46_im))))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = y_46re * (x_46re / y_46im)
    if (y_46im <= (-1.3d+106)) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else if (y_46im <= (-7.5d-67)) then
        tmp = ((x_46re * y_46re) + (y_46im * x_46im)) / t_0
    else if (y_46im <= 1.6d-45) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 3.1d+146) then
        tmp = (y_46im * (x_46im + t_1)) / t_0
    else
        tmp = (x_46im + (t_1 - (x_46im * ((y_46re / y_46im) * (y_46re / y_46im))))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = y_46_re * (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.3e+106) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -7.5e-67) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	} else if (y_46_im <= 1.6e-45) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+146) {
		tmp = (y_46_im * (x_46_im + t_1)) / t_0;
	} else {
		tmp = (x_46_im + (t_1 - (x_46_im * ((y_46_re / y_46_im) * (y_46_re / y_46_im))))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = y_46_re * (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.3e+106:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	elif y_46_im <= -7.5e-67:
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0
	elif y_46_im <= 1.6e-45:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.1e+146:
		tmp = (y_46_im * (x_46_im + t_1)) / t_0
	else:
		tmp = (x_46_im + (t_1 - (x_46_im * ((y_46_re / y_46_im) * (y_46_re / y_46_im))))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(y_46_re * Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.3e+106)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= -7.5e-67)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / t_0);
	elseif (y_46_im <= 1.6e-45)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.1e+146)
		tmp = Float64(Float64(y_46_im * Float64(x_46_im + t_1)) / t_0);
	else
		tmp = Float64(Float64(x_46_im + Float64(t_1 - Float64(x_46_im * Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / y_46_im))))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = y_46_re * (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.3e+106)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	elseif (y_46_im <= -7.5e-67)
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	elseif (y_46_im <= 1.6e-45)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.1e+146)
		tmp = (y_46_im * (x_46_im + t_1)) / t_0;
	else
		tmp = (x_46_im + (t_1 - (x_46_im * ((y_46_re / y_46_im) * (y_46_re / y_46_im))))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.3e+106], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.5e-67], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.6e-45], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+146], N[(N[(y$46$im * N[(x$46$im + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x$46$im + N[(t$95$1 - N[(x$46$im * N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.3000000000000001e106

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around inf 82.7%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -1.3000000000000001e106 < y.im < -7.5000000000000005e-67

   1. Initial program 80.1%

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

   if -7.5000000000000005e-67 < y.im < 1.60000000000000004e-45

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

   if 1.60000000000000004e-45 < y.im < 3.1000000000000002e146

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x.re around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r/ 78.1%

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

   8. Simplified 78.1%

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

   if 3.1000000000000002e146 < y.im

   1. Initial program 24.3%

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

   3. Taylor expanded in y.im around inf 58.1%

      \[\leadsto \color{blue}{\frac{x.im + \left(-1 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}} + \frac{x.re \cdot y.re}{y.im}\right)}{y.im}} \]
   4. Step-by-step derivation

      1. +-commutative 58.1%

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

      2. mul-1-neg 58.1%

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

      3. unsub-neg 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-/l* 61.2%

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

      6. associate-/l* 70.9%

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

   5. Simplified 70.9%

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

      1. pow2 70.9%

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

      2. pow2 70.9%

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

      3. frac-times 86.0%

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

   7. Applied egg-rr 86.0%

      \[\leadsto \frac{x.im + \left(y.re \cdot \frac{x.re}{y.im} - x.im \cdot \color{blue}{\left(\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}\right)}\right)}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \left(y.re \cdot \frac{x.re}{y.im} - x.im \cdot \left(\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}\right)\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.7 \cdot 10^{+145}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -2.1e+105)
     t_1
     (if (<= y.im -9.6e-68)
       (/ (+ (* x.re y.re) (* y.im x.im)) t_0)
       (if (<= y.im 2.05e-44)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 9.7e+145)
           (/ (* y.im (+ x.im (* y.re (/ x.re y.im)))) t_0)
           t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.1e+105) {
		tmp = t_1;
	} else if (y_46_im <= -9.6e-68) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	} else if (y_46_im <= 2.05e-44) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.7e+145) {
		tmp = (y_46_im * (x_46_im + (y_46_re * (x_46_re / y_46_im)))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-2.1d+105)) then
        tmp = t_1
    else if (y_46im <= (-9.6d-68)) then
        tmp = ((x_46re * y_46re) + (y_46im * x_46im)) / t_0
    else if (y_46im <= 2.05d-44) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 9.7d+145) then
        tmp = (y_46im * (x_46im + (y_46re * (x_46re / y_46im)))) / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.1e+105) {
		tmp = t_1;
	} else if (y_46_im <= -9.6e-68) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	} else if (y_46_im <= 2.05e-44) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.7e+145) {
		tmp = (y_46_im * (x_46_im + (y_46_re * (x_46_re / y_46_im)))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -2.1e+105:
		tmp = t_1
	elif y_46_im <= -9.6e-68:
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0
	elif y_46_im <= 2.05e-44:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 9.7e+145:
		tmp = (y_46_im * (x_46_im + (y_46_re * (x_46_re / y_46_im)))) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.1e+105)
		tmp = t_1;
	elseif (y_46_im <= -9.6e-68)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / t_0);
	elseif (y_46_im <= 2.05e-44)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 9.7e+145)
		tmp = Float64(Float64(y_46_im * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im)))) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.1e+105)
		tmp = t_1;
	elseif (y_46_im <= -9.6e-68)
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / t_0;
	elseif (y_46_im <= 2.05e-44)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 9.7e+145)
		tmp = (y_46_im * (x_46_im + (y_46_re * (x_46_re / y_46_im)))) / t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.1e+105], t$95$1, If[LessEqual[y$46$im, -9.6e-68], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 2.05e-44], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 9.7e+145], N[(N[(y$46$im * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.1000000000000001e105 or 9.69999999999999987e145 < y.im

   1. Initial program 30.8%

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

   3. Taylor expanded in y.im around inf 78.7%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if -2.1000000000000001e105 < y.im < -9.59999999999999965e-68

   1. Initial program 80.1%

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

   if -9.59999999999999965e-68 < y.im < 2.04999999999999996e-44

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

   if 2.04999999999999996e-44 < y.im < 9.69999999999999987e145

   1. Initial program 78.0%

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

   3. Taylor expanded in y.im around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in x.re around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r/ 78.1%

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

   8. Simplified 78.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -9.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.7 \cdot 10^{+145}:\\ \;\;\;\;\frac{y.im \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -1.05e+104)
     t_1
     (if (<= y.im -2.5e-68)
       t_0
       (if (<= y.im 3.1e-44)
         (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
         (if (<= y.im 9.5e+145) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.05e+104) {
		tmp = t_1;
	} else if (y_46_im <= -2.5e-68) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-44) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.5e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-1.05d+104)) then
        tmp = t_1
    else if (y_46im <= (-2.5d-68)) then
        tmp = t_0
    else if (y_46im <= 3.1d-44) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 9.5d+145) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.05e+104) {
		tmp = t_1;
	} else if (y_46_im <= -2.5e-68) {
		tmp = t_0;
	} else if (y_46_im <= 3.1e-44) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.5e+145) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -1.05e+104:
		tmp = t_1
	elif y_46_im <= -2.5e-68:
		tmp = t_0
	elif y_46_im <= 3.1e-44:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 9.5e+145:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.05e+104)
		tmp = t_1;
	elseif (y_46_im <= -2.5e-68)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-44)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 9.5e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.05e+104)
		tmp = t_1;
	elseif (y_46_im <= -2.5e-68)
		tmp = t_0;
	elseif (y_46_im <= 3.1e-44)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 9.5e+145)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+104], t$95$1, If[LessEqual[y$46$im, -2.5e-68], t$95$0, If[LessEqual[y$46$im, 3.1e-44], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 9.5e+145], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.0499999999999999e104 or 9.49999999999999948e145 < y.im

   1. Initial program 30.8%

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

   3. Taylor expanded in y.im around inf 78.7%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if -1.0499999999999999e104 < y.im < -2.49999999999999986e-68 or 3.09999999999999984e-44 < y.im < 9.49999999999999948e145

   1. Initial program 79.1%

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

   if -2.49999999999999986e-68 < y.im < 3.09999999999999984e-44

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.62 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.85 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -2.8e+85)
     t_0
     (if (<= y.im -1.62e+22)
       (/ (* y.im x.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im -2.85e-20)
         t_0
         (if (<= y.im 2.6e-31)
           (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
           (/ (+ x.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 t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.8e+85) {
		tmp = t_0;
	} else if (y_46_im <= -1.62e+22) {
		tmp = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -2.85e-20) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-31) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-2.8d+85)) then
        tmp = t_0
    else if (y_46im <= (-1.62d+22)) then
        tmp = (y_46im * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= (-2.85d-20)) then
        tmp = t_0
    else if (y_46im <= 2.6d-31) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_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 t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.8e+85) {
		tmp = t_0;
	} else if (y_46_im <= -1.62e+22) {
		tmp = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= -2.85e-20) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-31) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_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):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -2.8e+85:
		tmp = t_0
	elif y_46_im <= -1.62e+22:
		tmp = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= -2.85e-20:
		tmp = t_0
	elif y_46_im <= 2.6e-31:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (x_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)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.8e+85)
		tmp = t_0;
	elseif (y_46_im <= -1.62e+22)
		tmp = Float64(Float64(y_46_im * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= -2.85e-20)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-31)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + 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)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.8e+85)
		tmp = t_0;
	elseif (y_46_im <= -1.62e+22)
		tmp = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= -2.85e-20)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-31)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_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_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.8e+85], t$95$0, If[LessEqual[y$46$im, -1.62e+22], N[(N[(y$46$im * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.85e-20], t$95$0, If[LessEqual[y$46$im, 2.6e-31], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.7999999999999999e85 or -1.62e22 < y.im < -2.85000000000000012e-20

   1. Initial program 45.9%

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

   3. Taylor expanded in y.im around inf 77.7%

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

      1. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   if -2.7999999999999999e85 < y.im < -1.62e22

   1. Initial program 88.7%

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

   3. Taylor expanded in x.re around 0 83.1%

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

   if -2.85000000000000012e-20 < y.im < 2.59999999999999995e-31

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if 2.59999999999999995e-31 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.im around inf 56.7%

      \[\leadsto \color{blue}{\frac{x.im + \left(-1 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}} + \frac{x.re \cdot y.re}{y.im}\right)}{y.im}} \]
   4. Step-by-step derivation

      1. +-commutative 56.7%

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

      2. mul-1-neg 56.7%

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

      3. unsub-neg 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-/l* 58.2%

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

      6. associate-/l* 62.9%

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

   5. Simplified 62.9%

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

      1. pow2 62.9%

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

      2. pow2 62.9%

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

      3. frac-times 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in y.re around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*r/ 70.7%

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

   10. Simplified 70.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.62 \cdot 10^{+22}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.85 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.32 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t\_0}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.re \cdot y.re}{t\_0}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.im -1.32e+85)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (if (<= y.im -4.6e-7)
       (/ (* y.im x.im) t_0)
       (if (<= y.im -4.2e-20)
         (/ (* x.re y.re) t_0)
         (if (<= y.im 2.7e-30)
           (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
           (/ (+ x.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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -1.32e+85) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -4.6e-7) {
		tmp = (y_46_im * x_46_im) / t_0;
	} else if (y_46_im <= -4.2e-20) {
		tmp = (x_46_re * y_46_re) / t_0;
	} else if (y_46_im <= 2.7e-30) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_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) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46im <= (-1.32d+85)) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else if (y_46im <= (-4.6d-7)) then
        tmp = (y_46im * x_46im) / t_0
    else if (y_46im <= (-4.2d-20)) then
        tmp = (x_46re * y_46re) / t_0
    else if (y_46im <= 2.7d-30) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -1.32e+85) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -4.6e-7) {
		tmp = (y_46_im * x_46_im) / t_0;
	} else if (y_46_im <= -4.2e-20) {
		tmp = (x_46_re * y_46_re) / t_0;
	} else if (y_46_im <= 2.7e-30) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_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):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_im <= -1.32e+85:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	elif y_46_im <= -4.6e-7:
		tmp = (y_46_im * x_46_im) / t_0
	elif y_46_im <= -4.2e-20:
		tmp = (x_46_re * y_46_re) / t_0
	elif y_46_im <= 2.7e-30:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (x_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)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.32e+85)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= -4.6e-7)
		tmp = Float64(Float64(y_46_im * x_46_im) / t_0);
	elseif (y_46_im <= -4.2e-20)
		tmp = Float64(Float64(x_46_re * y_46_re) / t_0);
	elseif (y_46_im <= 2.7e-30)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + 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)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.32e+85)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	elseif (y_46_im <= -4.6e-7)
		tmp = (y_46_im * x_46_im) / t_0;
	elseif (y_46_im <= -4.2e-20)
		tmp = (x_46_re * y_46_re) / t_0;
	elseif (y_46_im <= 2.7e-30)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_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_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.32e+85], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.6e-7], N[(N[(y$46$im * x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, -4.2e-20], N[(N[(x$46$re * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 2.7e-30], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.32000000000000007e85

   1. Initial program 36.6%

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

   3. Taylor expanded in y.im around inf 80.2%

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

      1. associate-/l* 88.2%

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

   5. Simplified 88.2%

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

   if -1.32000000000000007e85 < y.im < -4.5999999999999999e-7

   1. Initial program 87.8%

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

   3. Taylor expanded in x.re around 0 75.8%

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

   if -4.5999999999999999e-7 < y.im < -4.1999999999999998e-20

   1. Initial program 81.6%

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

   3. Taylor expanded in x.re around inf 81.6%

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

      1. *-commutative 81.6%

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

   5. Simplified 81.6%

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

   if -4.1999999999999998e-20 < y.im < 2.69999999999999987e-30

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if 2.69999999999999987e-30 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.im around inf 56.7%

      \[\leadsto \color{blue}{\frac{x.im + \left(-1 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}} + \frac{x.re \cdot y.re}{y.im}\right)}{y.im}} \]
   4. Step-by-step derivation

      1. +-commutative 56.7%

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

      2. mul-1-neg 56.7%

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

      3. unsub-neg 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-/l* 58.2%

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

      6. associate-/l* 62.9%

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

   5. Simplified 62.9%

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

      1. pow2 62.9%

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

      2. pow2 62.9%

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

      3. frac-times 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in y.re around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*r/ 70.7%

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

   10. Simplified 70.7%

       \[\leadsto \frac{x.im + \color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.32 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.3e+17)
   (/ x.im y.im)
   (if (<= y.im 1.4e-206)
     (/ x.re y.re)
     (if (<= y.im 6.3e-183)
       (/ (* x.im (/ y.im y.re)) y.re)
       (if (<= y.im 1.7e-41) (/ 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_im <= -4.3e+17) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.4e-206) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 6.3e-183) {
		tmp = (x_46_im * (y_46_im / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.7e-41) {
		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_46im <= (-4.3d+17)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.4d-206) then
        tmp = x_46re / y_46re
    else if (y_46im <= 6.3d-183) then
        tmp = (x_46im * (y_46im / y_46re)) / y_46re
    else if (y_46im <= 1.7d-41) 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_im <= -4.3e+17) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.4e-206) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 6.3e-183) {
		tmp = (x_46_im * (y_46_im / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.7e-41) {
		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_im <= -4.3e+17:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.4e-206:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 6.3e-183:
		tmp = (x_46_im * (y_46_im / y_46_re)) / y_46_re
	elif y_46_im <= 1.7e-41:
		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_im <= -4.3e+17)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.4e-206)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 6.3e-183)
		tmp = Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.7e-41)
		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_im <= -4.3e+17)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.4e-206)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 6.3e-183)
		tmp = (x_46_im * (y_46_im / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.7e-41)
		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[LessEqual[y$46$im, -4.3e+17], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.4e-206], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.3e-183], N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.7e-41], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.3e17 or 1.6999999999999999e-41 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.re around 0 68.3%

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

   if -4.3e17 < y.im < 1.4000000000000001e-206 or 6.2999999999999998e-183 < y.im < 1.6999999999999999e-41

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf 68.1%

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

   if 1.4000000000000001e-206 < y.im < 6.2999999999999998e-183

   1. Initial program 80.4%

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

   3. Taylor expanded in y.re around inf 81.1%

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

      1. associate-/l* 81.1%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in y.im around inf 81.0%

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

      1. +-commutative 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in y.im around inf 80.0%

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

       1. associate-/l* 80.0%

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

   11. Simplified 80.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-206}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+68} \lor \neg \left(y.re \leq 3.95 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.7e+68) (not (<= y.re 3.95e+68)))
   (/ x.re y.re)
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.7e+68) || !(y_46_re <= 3.95e+68)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.7d+68)) .or. (.not. (y_46re <= 3.95d+68))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.7e+68) || !(y_46_re <= 3.95e+68)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.7e+68) or not (y_46_re <= 3.95e+68):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.7e+68) || !(y_46_re <= 3.95e+68))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.7e+68) || ~((y_46_re <= 3.95e+68)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.7e+68], N[Not[LessEqual[y$46$re, 3.95e+68]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.69999999999999991e68 or 3.95e68 < y.re

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around inf 75.3%

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

   if -2.69999999999999991e68 < y.re < 3.95e68

   1. Initial program 71.4%

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

   3. Taylor expanded in y.im around inf 70.3%

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

      1. associate-/l* 72.7%

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

   5. Simplified 72.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+68} \lor \neg \left(y.re \leq 3.95 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.1e-20)
   (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
   (if (<= y.im 2.1e-30)
     (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
     (/ (+ x.im (* y.re (/ x.re y.im))) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.1e-20) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= 2.1e-30) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-4.1d-20)) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else if (y_46im <= 2.1d-30) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_46im + (y_46re * (x_46re / y_46im))) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.1e-20) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= 2.1e-30) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.1e-20:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	elif y_46_im <= 2.1e-30:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.1e-20)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= 2.1e-30)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.1e-20)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	elseif (y_46_im <= 2.1e-30)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.1e-20], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.1e-30], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.1000000000000001e-20

   1. Initial program 55.3%

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

   3. Taylor expanded in y.im around inf 73.5%

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

      1. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   if -4.1000000000000001e-20 < y.im < 2.1000000000000002e-30

   1. Initial program 69.1%

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

   3. Taylor expanded in y.re around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if 2.1000000000000002e-30 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.im around inf 56.7%

      \[\leadsto \color{blue}{\frac{x.im + \left(-1 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}} + \frac{x.re \cdot y.re}{y.im}\right)}{y.im}} \]
   4. Step-by-step derivation

      1. +-commutative 56.7%

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

      2. mul-1-neg 56.7%

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

      3. unsub-neg 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-/l* 58.2%

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

      6. associate-/l* 62.9%

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

   5. Simplified 62.9%

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

      1. pow2 62.9%

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

      2. pow2 62.9%

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

      3. frac-times 70.3%

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

   7. Applied egg-rr 70.3%

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

   8. Taylor expanded in y.re around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*r/ 70.7%

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

   10. Simplified 70.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15} \lor \neg \left(y.im \leq 2.26 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.5e+15) (not (<= y.im 2.26e-38)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e+15) || !(y_46_im <= 2.26e-38)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.5d+15)) .or. (.not. (y_46im <= 2.26d-38))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.5e+15) || !(y_46_im <= 2.26e-38)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.5e+15) or not (y_46_im <= 2.26e-38):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.5e+15) || !(y_46_im <= 2.26e-38))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.5e+15) || ~((y_46_im <= 2.26e-38)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.5e+15], N[Not[LessEqual[y$46$im, 2.26e-38]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5e15 or 2.26e-38 < y.im

   1. Initial program 51.4%

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

   3. Taylor expanded in y.re around 0 68.3%

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

   if -2.5e15 < y.im < 2.26e-38

   1. Initial program 70.0%

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

   3. Taylor expanded in y.re around inf 65.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+15} \lor \neg \left(y.im \leq 2.26 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.1% 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 60.4%

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

3. Taylor expanded in y.re around 0 42.5%

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

4. Final simplification 42.5%

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

Reproduce

herbie shell --seed 2024055 
(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))))