_divideComplex, real part

Percentage Accurate: 61.6% → 84.9%

Time: 11.9s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      INFINITY)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (* (/ y.re (hypot y.re y.im)) (/ x.re (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 tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_re / 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_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. add-sqr-sqrt 78.1%

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

      3. times-frac 78.0%

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

      4. hypot-define 78.0%

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

      5. fma-define 78.0%

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

      6. hypot-define 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. associate-*l/ 95.0%

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

      2. *-un-lft-identity 95.0%

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

   6. Applied egg-rr 95.0%

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

   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 x.re around inf 1.6%

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

      1. *-commutative 1.6%

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

   5. Simplified 1.6%

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

      1. add-sqr-sqrt 1.6%

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

      2. hypot-undefine 1.6%

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

      3. hypot-undefine 1.6%

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

      4. times-frac 53.2%

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

   7. Applied egg-rr 53.2%

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

4. Final simplification 86.7%

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

Alternative 2: 80.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.22e+88)
     (+ (/ y.re (* y.im (/ y.im x.re))) (/ x.im y.im))
     (if (<= y.im -7e-127)
       t_0
       (if (<= y.im 2.6e-130)
         (+ (/ x.re y.re) (* x.im (/ y.im (pow y.re 2.0))))
         (if (<= y.im 7.2e+75)
           t_0
           (/ (+ x.im (* x.re (/ y.re y.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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.22e+88) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	} else if (y_46_im <= -7e-127) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-130) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 7.2e+75) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.22e+88) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	} else if (y_46_im <= -7e-127) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-130) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 7.2e+75) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.22e+88:
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im)
	elif y_46_im <= -7e-127:
		tmp = t_0
	elif y_46_im <= 2.6e-130:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 7.2e+75:
		tmp = t_0
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / math.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(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.22e+88)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))) + Float64(x_46_im / y_46_im));
	elseif (y_46_im <= -7e-127)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-130)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 7.2e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.22e+88)
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	elseif (y_46_im <= -7e-127)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-130)
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 7.2e+75)
		tmp = t_0;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.22e+88], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7e-127], t$95$0, If[LessEqual[y$46$im, 2.6e-130], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.2e+75], t$95$0, N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.22e88

   1. Initial program 43.7%

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

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

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

      1. +-commutative 77.5%

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

      2. *-commutative 77.5%

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

      3. associate-/l* 78.2%

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

      4. fma-define 78.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, \frac{x.re}{{y.im}^{2}}, \frac{x.im}{y.im}\right)} \]

   5. Simplified 78.2%

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

      1. fma-undefine 78.2%

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

      2. pow2 78.2%

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

      3. clear-num 78.2%

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

      4. un-div-inv 78.2%

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

      5. pow2 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. unpow2 78.2%

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

      2. *-un-lft-identity 78.2%

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

      3. times-frac 81.9%

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

   9. Applied egg-rr 81.9%

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

   if -1.22e88 < y.im < -6.99999999999999979e-127 or 2.6000000000000001e-130 < y.im < 7.2e75

   1. Initial program 82.7%

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

   if -6.99999999999999979e-127 < y.im < 2.6000000000000001e-130

   1. Initial program 65.3%

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

   3. Taylor expanded in y.re around inf 90.1%

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

      1. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

   if 7.2e75 < y.im

   1. Initial program 39.0%

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

      1. *-un-lft-identity 39.0%

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

      2. add-sqr-sqrt 39.0%

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

      3. times-frac 39.0%

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

      4. hypot-define 39.0%

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

      5. fma-define 39.0%

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

      6. hypot-define 63.1%

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

   4. Applied egg-rr 63.1%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in y.re around 0 80.8%

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

      1. associate-/l* 83.9%

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

   9. Simplified 83.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.im + x.re \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{t\_1}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ x.im (* x.re (/ y.re y.im)))))
   (if (<= y.im -6.6e+88)
     (/ t_1 (- (hypot y.re y.im)))
     (if (<= y.im -2.4e-129)
       t_0
       (if (<= y.im 3.2e-136)
         (+ (/ x.re y.re) (* x.im (/ y.im (pow y.re 2.0))))
         (if (<= y.im 2.2e+74) t_0 (/ t_1 (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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_im + (x_46_re * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6.6e+88) {
		tmp = t_1 / -hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -2.4e-129) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-136) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 2.2e+74) {
		tmp = t_0;
	} else {
		tmp = t_1 / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_im + (x_46_re * (y_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -6.6e+88) {
		tmp = t_1 / -Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -2.4e-129) {
		tmp = t_0;
	} else if (y_46_im <= 3.2e-136) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 2.2e+74) {
		tmp = t_0;
	} else {
		tmp = t_1 / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_im + (x_46_re * (y_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -6.6e+88:
		tmp = t_1 / -math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -2.4e-129:
		tmp = t_0
	elif y_46_im <= 3.2e-136:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 2.2e+74:
		tmp = t_0
	else:
		tmp = t_1 / math.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(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)))
	t_1 = Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -6.6e+88)
		tmp = Float64(t_1 / Float64(-hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -2.4e-129)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-136)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 2.2e+74)
		tmp = t_0;
	else
		tmp = Float64(t_1 / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_im + (x_46_re * (y_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -6.6e+88)
		tmp = t_1 / -hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -2.4e-129)
		tmp = t_0;
	elseif (y_46_im <= 3.2e-136)
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 2.2e+74)
		tmp = t_0;
	else
		tmp = t_1 / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.6e+88], N[(t$95$1 / (-N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision])), $MachinePrecision], If[LessEqual[y$46$im, -2.4e-129], t$95$0, If[LessEqual[y$46$im, 3.2e-136], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.2e+74], t$95$0, N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.6000000000000006e88

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. times-frac 43.6%

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

      4. hypot-define 43.6%

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

      5. fma-define 43.6%

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

      6. hypot-define 62.4%

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

   4. Applied egg-rr 62.4%

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

      1. associate-*l/ 62.5%

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

      2. *-un-lft-identity 62.5%

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

   6. Applied egg-rr 62.5%

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

   7. Taylor expanded in y.im around -inf 79.8%

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

      1. distribute-lft-out 79.8%

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

      2. associate-/l* 85.6%

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

   9. Simplified 85.6%

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

   if -6.6000000000000006e88 < y.im < -2.39999999999999989e-129 or 3.19999999999999993e-136 < y.im < 2.2000000000000001e74

   1. Initial program 82.7%

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

   if -2.39999999999999989e-129 < y.im < 3.19999999999999993e-136

   1. Initial program 65.3%

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

   3. Taylor expanded in y.re around inf 90.1%

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

      1. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

   if 2.2000000000000001e74 < y.im

   1. Initial program 39.0%

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

      1. *-un-lft-identity 39.0%

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

      2. add-sqr-sqrt 39.0%

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

      3. times-frac 39.0%

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

      4. hypot-define 39.0%

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

      5. fma-define 39.0%

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

      6. hypot-define 63.1%

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

   4. Applied egg-rr 63.1%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in y.re around 0 80.8%

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

      1. associate-/l* 83.9%

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

   9. Simplified 83.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ y.re (* y.im (/ y.im x.re))) (/ x.im y.im))))
   (if (<= y.im -1.4e+89)
     t_1
     (if (<= y.im -3.4e-131)
       t_0
       (if (<= y.im 5e-132)
         (+ (/ x.re y.re) (* x.im (/ y.im (pow y.re 2.0))))
         (if (<= y.im 3.4e+61) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+89) {
		tmp = t_1;
	} else if (y_46_im <= -3.4e-131) {
		tmp = t_0;
	} else if (y_46_im <= 5e-132) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_im <= 3.4e+61) {
		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) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (y_46re / (y_46im * (y_46im / x_46re))) + (x_46im / y_46im)
    if (y_46im <= (-1.4d+89)) then
        tmp = t_1
    else if (y_46im <= (-3.4d-131)) then
        tmp = t_0
    else if (y_46im <= 5d-132) then
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re ** 2.0d0)))
    else if (y_46im <= 3.4d+61) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+89) {
		tmp = t_1;
	} else if (y_46_im <= -3.4e-131) {
		tmp = t_0;
	} else if (y_46_im <= 5e-132) {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_im <= 3.4e+61) {
		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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im)
	tmp = 0
	if y_46_im <= -1.4e+89:
		tmp = t_1
	elif y_46_im <= -3.4e-131:
		tmp = t_0
	elif y_46_im <= 5e-132:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_im <= 3.4e+61:
		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(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))) + Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.4e+89)
		tmp = t_1;
	elseif (y_46_im <= -3.4e-131)
		tmp = t_0;
	elseif (y_46_im <= 5e-132)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_im <= 3.4e+61)
		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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.4e+89)
		tmp = t_1;
	elseif (y_46_im <= -3.4e-131)
		tmp = t_0;
	elseif (y_46_im <= 5e-132)
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_im <= 3.4e+61)
		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[(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]}, Block[{t$95$1 = N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+89], t$95$1, If[LessEqual[y$46$im, -3.4e-131], t$95$0, If[LessEqual[y$46$im, 5e-132], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3999999999999999e89 or 3.40000000000000026e61 < y.im

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/l* 78.4%

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

      4. fma-define 78.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, \frac{x.re}{{y.im}^{2}}, \frac{x.im}{y.im}\right)} \]

   5. Simplified 78.4%

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

      1. fma-undefine 78.4%

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

      2. pow2 78.4%

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

      3. clear-num 78.4%

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

      4. un-div-inv 78.4%

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

      5. pow2 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. *-un-lft-identity 78.4%

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

      3. times-frac 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -1.3999999999999999e89 < y.im < -3.39999999999999995e-131 or 4.9999999999999999e-132 < y.im < 3.40000000000000026e61

   1. Initial program 83.0%

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

   if -3.39999999999999995e-131 < y.im < 4.9999999999999999e-132

   1. Initial program 65.3%

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

   3. Taylor expanded in y.re around inf 90.1%

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

      1. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ y.re (* y.im (/ y.im x.re))) (/ x.im y.im))))
   (if (<= y.im -4.1e+89)
     t_1
     (if (<= y.im -1.25e-232)
       t_0
       (if (<= y.im 1.4e-136) (/ x.re y.re) (if (<= y.im 2.4e+57) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -4.1e+89) {
		tmp = t_1;
	} else if (y_46_im <= -1.25e-232) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-136) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.4e+57) {
		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) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (y_46re / (y_46im * (y_46im / x_46re))) + (x_46im / y_46im)
    if (y_46im <= (-4.1d+89)) then
        tmp = t_1
    else if (y_46im <= (-1.25d-232)) then
        tmp = t_0
    else if (y_46im <= 1.4d-136) then
        tmp = x_46re / y_46re
    else if (y_46im <= 2.4d+57) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_im <= -4.1e+89) {
		tmp = t_1;
	} else if (y_46_im <= -1.25e-232) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-136) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.4e+57) {
		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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im)
	tmp = 0
	if y_46_im <= -4.1e+89:
		tmp = t_1
	elif y_46_im <= -1.25e-232:
		tmp = t_0
	elif y_46_im <= 1.4e-136:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 2.4e+57:
		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(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))) + Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.1e+89)
		tmp = t_1;
	elseif (y_46_im <= -1.25e-232)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-136)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 2.4e+57)
		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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4.1e+89)
		tmp = t_1;
	elseif (y_46_im <= -1.25e-232)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-136)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 2.4e+57)
		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[(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]}, Block[{t$95$1 = N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.1e+89], t$95$1, If[LessEqual[y$46$im, -1.25e-232], t$95$0, If[LessEqual[y$46$im, 1.4e-136], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+57], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.09999999999999985e89 or 2.40000000000000005e57 < y.im

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-/l* 78.4%

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

      4. fma-define 78.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, \frac{x.re}{{y.im}^{2}}, \frac{x.im}{y.im}\right)} \]

   5. Simplified 78.4%

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

      1. fma-undefine 78.4%

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

      2. pow2 78.4%

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

      3. clear-num 78.4%

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

      4. un-div-inv 78.4%

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

      5. pow2 78.4%

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

   7. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. *-un-lft-identity 78.4%

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

      3. times-frac 81.6%

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

   9. Applied egg-rr 81.6%

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

   if -4.09999999999999985e89 < y.im < -1.25e-232 or 1.4e-136 < y.im < 2.40000000000000005e57

   1. Initial program 81.4%

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

   if -1.25e-232 < y.im < 1.4e-136

   1. Initial program 62.2%

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

   3. Taylor expanded in y.re around inf 86.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-232}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -3.2e+88)
     (/ x.im y.im)
     (if (<= y.im -2.4e-179)
       t_0
       (if (<= y.im 7.6e-124)
         (/ x.re y.re)
         (if (<= y.im 2.3e+51) t_0 (/ x.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 * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3.2e+88) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -2.4e-179) {
		tmp = t_0;
	} else if (y_46_im <= 7.6e-124) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.3e+51) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-3.2d+88)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-2.4d-179)) then
        tmp = t_0
    else if (y_46im <= 7.6d-124) then
        tmp = x_46re / y_46re
    else if (y_46im <= 2.3d+51) then
        tmp = t_0
    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 t_0 = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3.2e+88) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -2.4e-179) {
		tmp = t_0;
	} else if (y_46_im <= 7.6e-124) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.3e+51) {
		tmp = t_0;
	} 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):
	t_0 = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -3.2e+88:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -2.4e-179:
		tmp = t_0
	elif y_46_im <= 7.6e-124:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 2.3e+51:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -3.2e+88)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -2.4e-179)
		tmp = t_0;
	elseif (y_46_im <= 7.6e-124)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 2.3e+51)
		tmp = t_0;
	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)
	t_0 = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3.2e+88)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -2.4e-179)
		tmp = t_0;
	elseif (y_46_im <= 7.6e-124)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 2.3e+51)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$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, -3.2e+88], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -2.4e-179], t$95$0, If[LessEqual[y$46$im, 7.6e-124], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.3e+51], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.1999999999999999e88 or 2.30000000000000005e51 < y.im

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around 0 72.3%

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

   if -3.1999999999999999e88 < y.im < -2.4e-179 or 7.60000000000000025e-124 < y.im < 2.30000000000000005e51

   1. Initial program 84.2%

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

   3. Taylor expanded in x.re around 0 56.4%

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

   if -2.4e-179 < y.im < 7.60000000000000025e-124

   1. Initial program 64.6%

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

   3. Taylor expanded in y.re around inf 80.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ y.re (* y.im (/ y.im x.re))) (/ x.im y.im))))
   (if (<= y.re -5.5e+86)
     (/ x.re y.re)
     (if (<= y.re 1.06e-23)
       t_0
       (if (<= y.re 1.2e+43)
         (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 6.2e+89) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_re <= -5.5e+86) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.06e-23) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e+43) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.2e+89) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y_46re / (y_46im * (y_46im / x_46re))) + (x_46im / y_46im)
    if (y_46re <= (-5.5d+86)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.06d-23) then
        tmp = t_0
    else if (y_46re <= 1.2d+43) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 6.2d+89) then
        tmp = t_0
    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 t_0 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	double tmp;
	if (y_46_re <= -5.5e+86) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.06e-23) {
		tmp = t_0;
	} else if (y_46_re <= 1.2e+43) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.2e+89) {
		tmp = t_0;
	} 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):
	t_0 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im)
	tmp = 0
	if y_46_re <= -5.5e+86:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.06e-23:
		tmp = t_0
	elif y_46_re <= 1.2e+43:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 6.2e+89:
		tmp = t_0
	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)
	t_0 = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))) + Float64(x_46_im / y_46_im))
	tmp = 0.0
	if (y_46_re <= -5.5e+86)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.06e-23)
		tmp = t_0;
	elseif (y_46_re <= 1.2e+43)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 6.2e+89)
		tmp = t_0;
	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)
	t_0 = (y_46_re / (y_46_im * (y_46_im / x_46_re))) + (x_46_im / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -5.5e+86)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.06e-23)
		tmp = t_0;
	elseif (y_46_re <= 1.2e+43)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 6.2e+89)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.5e+86], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.06e-23], t$95$0, If[LessEqual[y$46$re, 1.2e+43], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+89], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.5000000000000002e86 or 6.2e89 < y.re

   1. Initial program 40.4%

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

   3. Taylor expanded in y.re around inf 77.4%

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

   if -5.5000000000000002e86 < y.re < 1.05999999999999994e-23 or 1.20000000000000012e43 < y.re < 6.2e89

   1. Initial program 71.7%

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

   3. Taylor expanded in y.re around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. *-commutative 71.6%

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

      3. associate-/l* 69.9%

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

      4. fma-define 69.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y.re, \frac{x.re}{{y.im}^{2}}, \frac{x.im}{y.im}\right)} \]

   5. Simplified 69.9%

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

      1. fma-undefine 69.9%

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

      2. pow2 69.9%

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

      3. clear-num 69.9%

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

      4. un-div-inv 69.9%

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

      5. pow2 69.9%

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

   7. Applied egg-rr 69.9%

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

      1. unpow2 69.9%

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

      2. *-un-lft-identity 69.9%

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

      3. times-frac 72.5%

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

   9. Applied egg-rr 72.5%

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

   if 1.05999999999999994e-23 < y.re < 1.20000000000000012e43

   1. Initial program 93.7%

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

   3. Taylor expanded in x.re around inf 60.0%

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

      1. *-commutative 60.0%

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

   5. Simplified 60.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-23}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.re}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+87} \lor \neg \left(y.im \leq 1.7 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.7e+87) (not (<= y.im 1.7e-96)))
   (/ 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 <= -3.7e+87) || !(y_46_im <= 1.7e-96)) {
		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 <= (-3.7d+87)) .or. (.not. (y_46im <= 1.7d-96))) 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 <= -3.7e+87) || !(y_46_im <= 1.7e-96)) {
		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 <= -3.7e+87) or not (y_46_im <= 1.7e-96):
		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 <= -3.7e+87) || !(y_46_im <= 1.7e-96))
		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 <= -3.7e+87) || ~((y_46_im <= 1.7e-96)))
		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, -3.7e+87], N[Not[LessEqual[y$46$im, 1.7e-96]], $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 < -3.70000000000000003e87 or 1.7e-96 < y.im

   1. Initial program 53.2%

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

   3. Taylor expanded in y.re around 0 64.9%

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

   if -3.70000000000000003e87 < y.im < 1.7e-96

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf 62.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+87} \lor \neg \left(y.im \leq 1.7 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+165}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e+165) (/ x.im y.re) (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e+165) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.2d+165)) then
        tmp = x_46im / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e+165) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.2e+165:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e+165)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.2e+165)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.2e+165], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.2000000000000001e165

   1. Initial program 35.9%

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

      1. *-un-lft-identity 35.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. times-frac 35.9%

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

      4. hypot-define 35.9%

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

      5. fma-define 35.9%

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

      6. hypot-define 48.4%

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

   4. Applied egg-rr 48.4%

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

   5. Taylor expanded in y.im around -inf 21.9%

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

      1. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

   8. Taylor expanded in y.re around -inf 21.9%

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

   if -4.2000000000000001e165 < y.re

   1. Initial program 66.6%

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

   3. Taylor expanded in y.re around 0 50.0%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+165}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.8% 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 62.5%

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

3. Taylor expanded in y.re around 0 44.6%

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

4. Final simplification 44.6%

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

Reproduce

herbie shell --seed 2024043 
(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))))