Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 69.3% → 99.8%

Time: 11.8s

Alternatives: 21

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x \cdot \frac{\frac{y}{y + x}}{y + \left(x + 1\right)}}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (* x (/ (/ y (+ y x)) (+ y (+ x 1.0)))) (+ y x)))

C

assert(x < y);
double code(double x, double y) {
	return (x * ((y / (y + x)) / (y + (x + 1.0)))) / (y + x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((y / (y + x)) / (y + (x + 1.0d0)))) / (y + x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (x * ((y / (y + x)) / (y + (x + 1.0)))) / (y + x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x * ((y / (y + x)) / (y + (x + 1.0)))) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(y + Float64(x + 1.0)))) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x * ((y / (y + x)) / (y + (x + 1.0)))) / (y + x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 71.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 71.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   10. associate-+r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

   13. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

6. Applied egg-rr 99.9%

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

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1} \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(y + x\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(y + \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   10. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{x \cdot \frac{\frac{y}{y + x}}{y + \left(x + 1\right)}}{y + x} \]
10. Add Preprocessing

Alternative 2: 90.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot t\_0}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{t\_0}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 3.5e-143)
     (/ (/ y (+ (+ y x) 1.0)) (+ y x))
     (if (<= y 5.5e-9)
       (/ (* x t_0) (+ y x))
       (/ (* x (/ t_0 (+ y x))) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 3.5e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (y <= 5.5e-9) {
		tmp = (x * t_0) / (y + x);
	} else {
		tmp = (x * (t_0 / (y + x))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 3.5d-143) then
        tmp = (y / ((y + x) + 1.0d0)) / (y + x)
    else if (y <= 5.5d-9) then
        tmp = (x * t_0) / (y + x)
    else
        tmp = (x * (t_0 / (y + x))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 3.5e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (y <= 5.5e-9) {
		tmp = (x * t_0) / (y + x);
	} else {
		tmp = (x * (t_0 / (y + x))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 3.5e-143:
		tmp = (y / ((y + x) + 1.0)) / (y + x)
	elif y <= 5.5e-9:
		tmp = (x * t_0) / (y + x)
	else:
		tmp = (x * (t_0 / (y + x))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 3.5e-143)
		tmp = Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	elseif (y <= 5.5e-9)
		tmp = Float64(Float64(x * t_0) / Float64(y + x));
	else
		tmp = Float64(Float64(x * Float64(t_0 / Float64(y + x))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 3.5e-143)
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	elseif (y <= 5.5e-9)
		tmp = (x * t_0) / (y + x);
	else
		tmp = (x * (t_0 / (y + x))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.5e-143], N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-9], N[(N[(x * t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.50000000000000005e-143

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 61.4%

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

   if 3.50000000000000005e-143 < y < 5.4999999999999996e-9

   1. Initial program 88.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{y}, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 77.1%

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

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \color{blue}{1}\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 77.1%

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

   if 5.4999999999999996e-9 < y

   1. Initial program 61.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 61.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1} \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(y + x\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(y + \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      10. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 98.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{y + x}}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-143)
   (/ (/ y (+ (+ y x) 1.0)) (+ y x))
   (if (<= y 3e-15)
     (/ (* x (/ y (+ y x))) (+ y x))
     (/ (* x (/ 1.0 (+ y (+ x 1.0)))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (y <= 3e-15) {
		tmp = (x * (y / (y + x))) / (y + x);
	} else {
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d-143) then
        tmp = (y / ((y + x) + 1.0d0)) / (y + x)
    else if (y <= 3d-15) then
        tmp = (x * (y / (y + x))) / (y + x)
    else
        tmp = (x * (1.0d0 / (y + (x + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (y <= 3e-15) {
		tmp = (x * (y / (y + x))) / (y + x);
	} else {
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 5.2e-143:
		tmp = (y / ((y + x) + 1.0)) / (y + x)
	elif y <= 3e-15:
		tmp = (x * (y / (y + x))) / (y + x)
	else:
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-143)
		tmp = Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	elseif (y <= 3e-15)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(y + x));
	else
		tmp = Float64(Float64(x * Float64(1.0 / Float64(y + Float64(x + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-143)
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	elseif (y <= 3e-15)
		tmp = (x * (y / (y + x))) / (y + x);
	else
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 5.2e-143], N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-15], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.19999999999999974e-143

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 61.4%

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

   if 5.19999999999999974e-143 < y < 3e-15

   1. Initial program 88.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{y}, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 77.1%

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

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \color{blue}{1}\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 77.1%

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

   if 3e-15 < y

   1. Initial program 61.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 61.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1} \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(y + x\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(y + \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      10. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 74.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -280000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -280000.0)
   (/ (/ y x) x)
   (if (<= x -5e-14)
     (/ x (* y y))
     (if (<= x -1.8e-52) (/ y (+ y x)) (/ (/ x y) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -280000.0) {
		tmp = (y / x) / x;
	} else if (x <= -5e-14) {
		tmp = x / (y * y);
	} else if (x <= -1.8e-52) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-280000.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-5d-14)) then
        tmp = x / (y * y)
    else if (x <= (-1.8d-52)) then
        tmp = y / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -280000.0) {
		tmp = (y / x) / x;
	} else if (x <= -5e-14) {
		tmp = x / (y * y);
	} else if (x <= -1.8e-52) {
		tmp = y / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -280000.0:
		tmp = (y / x) / x
	elif x <= -5e-14:
		tmp = x / (y * y)
	elif x <= -1.8e-52:
		tmp = y / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -280000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5e-14)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -1.8e-52)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -280000.0)
		tmp = (y / x) / x;
	elseif (x <= -5e-14)
		tmp = x / (y * y);
	elseif (x <= -1.8e-52)
		tmp = y / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -280000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5e-14], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.8e-52], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.8e5

   1. Initial program 54.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 54.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 75.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \color{blue}{x}\right) \]
   9. Step-by-step derivation

   10. Simplified 75.1%

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

   if -2.8e5 < x < -5.0000000000000002e-14

   1. Initial program 68.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 68.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 100.0%

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

   if -5.0000000000000002e-14 < x < -1.79999999999999994e-52

   1. Initial program 88.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{1 + x}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(1 + x\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. Simplified 69.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(x, y\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 69.8%

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

   if -1.79999999999999994e-52 < x

   1. Initial program 76.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 58.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 58.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(y + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\color{blue}{y} + 1\right)\right) \]

      4. +-lowering-+.f64 58.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right) \]

   9. Applied egg-rr 58.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -280000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -320000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -320000.0)
   (/ (/ y x) x)
   (if (<= x -4.8e-9)
     (/ x (* y y))
     (if (<= x -8.2e-53) (/ y (+ y x)) (/ x (* y (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -320000.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.8e-9) {
		tmp = x / (y * y);
	} else if (x <= -8.2e-53) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-320000.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-4.8d-9)) then
        tmp = x / (y * y)
    else if (x <= (-8.2d-53)) then
        tmp = y / (y + x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -320000.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.8e-9) {
		tmp = x / (y * y);
	} else if (x <= -8.2e-53) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -320000.0:
		tmp = (y / x) / x
	elif x <= -4.8e-9:
		tmp = x / (y * y)
	elif x <= -8.2e-53:
		tmp = y / (y + x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -320000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.8e-9)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -8.2e-53)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -320000.0)
		tmp = (y / x) / x;
	elseif (x <= -4.8e-9)
		tmp = x / (y * y);
	elseif (x <= -8.2e-53)
		tmp = y / (y + x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -320000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.8e-9], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-53], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.2e5

   1. Initial program 54.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 54.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 75.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \color{blue}{x}\right) \]
   9. Step-by-step derivation

   10. Simplified 75.1%

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

   if -3.2e5 < x < -4.8e-9

   1. Initial program 68.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 68.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 100.0%

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

   if -4.8e-9 < x < -8.2000000000000001e-53

   1. Initial program 88.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 88.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{1 + x}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(1 + x\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 69.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. Simplified 69.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(x, y\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 69.8%

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

   if -8.2000000000000001e-53 < x

   1. Initial program 76.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 76.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 58.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 58.8%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -320000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-195)
   (/ (/ y x) x)
   (if (<= y 3.1e-130) (/ y x) (if (<= y 1.0) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-195) {
		tmp = (y / x) / x;
	} else if (y <= 3.1e-130) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.1d-195)) then
        tmp = (y / x) / x
    else if (y <= 3.1d-130) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-195) {
		tmp = (y / x) / x;
	} else if (y <= 3.1e-130) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2.1e-195:
		tmp = (y / x) / x
	elif y <= 3.1e-130:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-195)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.1e-130)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.1e-195)
		tmp = (y / x) / x;
	elseif (y <= 3.1e-130)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.1e-195], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.1e-130], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.1e-195

   1. Initial program 73.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 73.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 40.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 40.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \color{blue}{x}\right) \]
   9. Step-by-step derivation

   10. Simplified 36.0%

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

   if -2.1e-195 < y < 3.10000000000000011e-130

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 89.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 89.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 68.6%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 68.6%

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

   if 3.10000000000000011e-130 < y < 1

   1. Initial program 90.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 91.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 45.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 45.5%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 43.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 43.6%

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

   if 1 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 76.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right) \]

   9. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -5.5e-194)
   (/ y (* x x))
   (if (<= y 2.45e-129) (/ y x) (if (<= y 1.0) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-194) {
		tmp = y / (x * x);
	} else if (y <= 2.45e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.5d-194)) then
        tmp = y / (x * x)
    else if (y <= 2.45d-129) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -5.5e-194) {
		tmp = y / (x * x);
	} else if (y <= 2.45e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -5.5e-194:
		tmp = y / (x * x)
	elif y <= 2.45e-129:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -5.5e-194)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.45e-129)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.5e-194)
		tmp = y / (x * x);
	elseif (y <= 2.45e-129)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -5.5e-194], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e-129], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.49999999999999941e-194

   1. Initial program 73.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 73.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 35.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 35.7%

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

   if -5.49999999999999941e-194 < y < 2.45000000000000001e-129

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 89.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 89.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 68.6%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 68.6%

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

   if 2.45000000000000001e-129 < y < 1

   1. Initial program 90.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 91.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 45.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 45.5%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 43.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 43.6%

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

   if 1 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 76.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{y}\right) \]

      3. /-lowering-/.f64 74.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), y\right) \]

   9. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-194}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -5.2e-194)
   (/ y (* x x))
   (if (<= y 2.35e-129) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -5.2e-194) {
		tmp = y / (x * x);
	} else if (y <= 2.35e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-5.2d-194)) then
        tmp = y / (x * x)
    else if (y <= 2.35d-129) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -5.2e-194) {
		tmp = y / (x * x);
	} else if (y <= 2.35e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -5.2e-194:
		tmp = y / (x * x)
	elif y <= 2.35e-129:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -5.2e-194)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.35e-129)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.2e-194)
		tmp = y / (x * x);
	elseif (y <= 2.35e-129)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -5.2e-194], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.35e-129], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.20000000000000003e-194

   1. Initial program 73.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 73.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 35.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 35.7%

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

   if -5.20000000000000003e-194 < y < 2.3500000000000001e-129

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 89.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 89.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 68.6%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 68.6%

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

   if 2.3500000000000001e-129 < y < 1

   1. Initial program 90.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 91.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 45.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 45.5%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 43.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 43.6%

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

   if 1 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 76.0%

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

Alternative 9: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \frac{t\_0}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot t\_0}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.0)
     (/ (* x (/ t_0 (+ y x))) (+ y x))
     (/ (/ (* x t_0) (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 / (y + x))) / (y + x);
	} else {
		tmp = ((x * t_0) / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (x <= (-1.0d0)) then
        tmp = (x * (t_0 / (y + x))) / (y + x)
    else
        tmp = ((x * t_0) / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 / (y + x))) / (y + x);
	} else {
		tmp = ((x * t_0) / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if x <= -1.0:
		tmp = (x * (t_0 / (y + x))) / (y + x)
	else:
		tmp = ((x * t_0) / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(t_0 / Float64(y + x))) / Float64(y + x));
	else
		tmp = Float64(Float64(Float64(x * t_0) / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * (t_0 / (y + x))) / (y + x);
	else
		tmp = ((x * t_0) / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(x * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * t$95$0), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 55.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 55.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1} \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(y + x\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(y + \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      10. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \color{blue}{x}\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 99.8%

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

   if -1 < x

   1. Initial program 76.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 76.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 76.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{y}, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 82.6%

      \[\leadsto \frac{\frac{x \cdot \frac{y}{x + y}}{\color{blue}{y} + 1}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{y + x}}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{y + x}}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-143)
   (/ (/ y (+ (+ y x) 1.0)) (+ y x))
   (/ (* x (/ 1.0 (+ y (+ x 1.0)))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else {
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d-143) then
        tmp = (y / ((y + x) + 1.0d0)) / (y + x)
    else
        tmp = (x * (1.0d0 / (y + (x + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else {
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.5e-143:
		tmp = (y / ((y + x) + 1.0)) / (y + x)
	else:
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-143)
		tmp = Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x * Float64(1.0 / Float64(y + Float64(x + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-143)
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	else
		tmp = (x * (1.0 / (y + (x + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.5e-143], N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000072e-143

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 61.4%

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

   if 8.50000000000000072e-143 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1} \cdot x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{y}{x + y}}{\left(x + y\right) + 1}\right), x\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(x + y\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(\left(y + x\right) + 1\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(y + \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \left(x + 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      10. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), x\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 65.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{\left(y + x\right) + 1} \cdot \frac{\frac{y}{y + x}}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (* (/ x (+ (+ y x) 1.0)) (/ (/ y (+ y x)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	return (x / ((y + x) + 1.0)) * ((y / (y + x)) / (y + x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / ((y + x) + 1.0d0)) * ((y / (y + x)) / (y + x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (x / ((y + x) + 1.0)) * ((y / (y + x)) / (y + x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / ((y + x) + 1.0)) * ((y / (y + x)) / (y + x))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(Float64(y + x) + 1.0)) * Float64(Float64(y / Float64(y + x)) / Float64(y + x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / ((y + x) + 1.0)) * ((y / (y + x)) / (y + x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 71.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 71.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{x + \left(y + 1\right)}\right), \color{blue}{\left(\frac{\frac{y}{x + y}}{x + y}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(x + \left(y + 1\right)\right)\right), \left(\frac{\color{blue}{\frac{y}{x + y}}}{x + y}\right)\right) \]

   7. associate-+r+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\left(x + y\right) + 1\right)\right), \left(\frac{\frac{y}{\color{blue}{x + y}}}{x + y}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(\frac{\frac{y}{\color{blue}{x + y}}}{x + y}\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(\frac{\frac{y}{\color{blue}{x} + y}}{x + y}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{/.f64}\left(\left(\frac{y}{x + y}\right), \color{blue}{\left(x + y\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + y\right)\right), \left(\color{blue}{x} + y\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \left(x + y\right)\right)\right) \]

   13. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{x}{\left(y + x\right) + 1} \cdot \frac{\frac{y}{y + x}}{y + x} \]
8. Add Preprocessing

Alternative 12: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 8.5e-143) (/ (/ y t_0) (+ y x)) (/ (/ x t_0) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 8.5d-143) then
        tmp = (y / t_0) / (y + x)
    else
        tmp = (x / t_0) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 8.5e-143:
		tmp = (y / t_0) / (y + x)
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 8.5e-143)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 8.5e-143)
		tmp = (y / t_0) / (y + x);
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 8.5e-143], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000072e-143

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{y}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 61.4%

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

   if 8.50000000000000072e-143 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 65.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-143) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ (+ y x) 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / ((y + x) + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d-143) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / ((y + x) + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-143) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / ((y + x) + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 8.5e-143:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / ((y + x) + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-143)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-143)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / ((y + x) + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 8.5e-143], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000072e-143

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 60.4%

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

   if 8.50000000000000072e-143 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 65.8%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-129) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-129) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-129) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.4e-129:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-129)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-129)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.4e-129], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.39999999999999989e-129

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 60.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 60.5%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 35.7%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 35.7%

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

   if 2.39999999999999989e-129 < y < 1

   1. Initial program 90.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 91.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 45.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 45.5%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 43.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 43.6%

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

   if 1 < y

   1. Initial program 60.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 60.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   7. Simplified 76.0%

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

Alternative 15: 82.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.45e-129) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-129) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.45d-129) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.45e-129) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.45e-129:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.45e-129)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.45e-129)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.45e-129], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.45000000000000001e-129

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{1 + x}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(1 + x\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 61.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. Simplified 61.0%

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

   if 2.45000000000000001e-129 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{1 + y}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(1 + y\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 82.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e-129) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-129) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.65d-129) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-129) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.65e-129:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.65e-129)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e-129)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.65e-129], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.64999999999999987e-129

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 60.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 60.5%

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

   if 2.64999999999999987e-129 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      10. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

      13. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{1 + y}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(1 + y\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   9. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 82.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e-129) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-129) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.65d-129) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-129) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.65e-129:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.65e-129)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.65e-129)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2.65e-129], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.64999999999999987e-129

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 72.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 72.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 60.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 60.5%

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

   if 2.64999999999999987e-129 < y

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 69.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 66.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 66.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y}\right), \color{blue}{\left(y + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \left(\color{blue}{y} + 1\right)\right) \]

      4. +-lowering-+.f64 65.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, y\right), \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right) \]

   9. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -1.75e-128) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-128) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.75d-128)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.75e-128) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.75e-128:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.75e-128)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.75e-128)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.75e-128], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-128

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 64.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(1 + x\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{1} + x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(x + \color{blue}{1}\right)\right) \]

      5. +-lowering-+.f64 65.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Simplified 65.2%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 28.2%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   10. Simplified 28.2%

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

   if -1.75e-128 < x

   1. Initial program 75.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

      8. associate-+l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

      10. +-lowering-+.f64 75.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

      4. +-lowering-+.f64 57.7%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   7. Simplified 57.7%

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

   8. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 39.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   10. Simplified 39.4%

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

Alternative 19: 25.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 71.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 71.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + y\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + y\right)}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \left(y + \color{blue}{1}\right)\right)\right) \]

   4. +-lowering-+.f64 50.3%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

7. Simplified 50.3%

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

8. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 28.9%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

10. Simplified 28.9%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
11. Add Preprocessing

Alternative 20: 4.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 71.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 71.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \color{blue}{\left({x}^{2}\right)}\right)\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   2. *-lowering-*.f64 30.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

7. Simplified 30.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.2%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{x}\right) \]

10. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 21: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(x + y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} \cdot \left(\left(x + y\right) + 1\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(\left(x + y\right) + 1\right)}\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{\left(x + y\right)} + 1\right)\right)\right)\right) \]

   8. associate-+l+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(x + \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \color{blue}{\left(y + 1\right)}\right)\right)\right)\right) \]

   10. +-lowering-+.f64 71.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right)\right)\right) \]

3. Simplified 71.5%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{x + \left(y + 1\right)}\right), \color{blue}{\left(x + y\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(\color{blue}{x} + y\right)\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \frac{y}{x + y}\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{y}{x + y}\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \left(x + y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   10. associate-+r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\left(x + y\right), 1\right)\right), \left(x + y\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \left(x + y\right)\right) \]

   13. +-lowering-+.f64 99.9%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

6. Applied egg-rr 99.9%

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

7. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{1 + x}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
8. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(1 + x\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   3. +-lowering-+.f64 51.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

9. Simplified 51.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1} \]
11. Step-by-step derivation

12. Simplified 3.6%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))