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

Percentage Accurate: 69.5% → 99.8%

Time: 13.4s

Alternatives: 17

Speedup: 1.1×

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: 69.5% 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

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return (x / ((x + y) + 1.0)) * ((y / (x + y)) / (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 / ((x + y) + 1.0d0)) * ((y / (x + y)) / (x + y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 2: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(x + y\right) + \left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.4e-105)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.8e+159)
     (/ x (+ (+ x y) (* (+ x y) (+ x y))))
     (* (/ (/ y (+ x y)) (+ x y)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+159) {
		tmp = x / ((x + y) + ((x + y) * (x + y)));
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (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 (y <= 6.4d-105) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.8d+159) then
        tmp = x / ((x + y) + ((x + y) * (x + y)))
    else
        tmp = ((y / (x + y)) / (x + y)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.4e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+159) {
		tmp = x / ((x + y) + ((x + y) * (x + y)));
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.4e-105:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.8e+159:
		tmp = x / ((x + y) + ((x + y) * (x + y)))
	else:
		tmp = ((y / (x + y)) / (x + y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.4e-105)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.8e+159)
		tmp = Float64(x / Float64(Float64(x + y) + Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) / Float64(x + y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.4e-105)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.8e+159)
		tmp = x / ((x + y) + ((x + y) * (x + y)));
	else
		tmp = ((y / (x + y)) / (x + y)) * (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[y, 6.4e-105], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+159], N[(x / N[(N[(x + y), $MachinePrecision] + N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.39999999999999962e-105

   1. Initial program 74.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 74.7%

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

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 59.6%

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

   7. Simplified 59.6%

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

   if 6.39999999999999962e-105 < y < 1.80000000000000018e159

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 68.8%

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

   3. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 66.8%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 82.6%

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

   11. Applied egg-rr 82.6%

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

       8. +-lowering-+.f64 82.6%

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

   13. Applied egg-rr 82.6%

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

   if 1.80000000000000018e159 < y

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.1%

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

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. 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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 93.0%

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

   9. Simplified 93.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(x + y\right) + \left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 6.1e-105)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.8e+159)
     (/ x (* (+ x y) (+ (+ x y) 1.0)))
     (* (/ (/ y (+ x y)) (+ x y)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.1e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+159) {
		tmp = x / ((x + y) * ((x + y) + 1.0));
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (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 (y <= 6.1d-105) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.8d+159) then
        tmp = x / ((x + y) * ((x + y) + 1.0d0))
    else
        tmp = ((y / (x + y)) / (x + y)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.1e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+159) {
		tmp = x / ((x + y) * ((x + y) + 1.0));
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.1e-105:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.8e+159:
		tmp = x / ((x + y) * ((x + y) + 1.0))
	else:
		tmp = ((y / (x + y)) / (x + y)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.1e-105)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.8e+159)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) / Float64(x + y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.1e-105)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.8e+159)
		tmp = x / ((x + y) * ((x + y) + 1.0));
	else
		tmp = ((y / (x + y)) / (x + y)) * (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[y, 6.1e-105], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+159], N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.09999999999999985e-105

   1. Initial program 74.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 74.7%

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

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 59.6%

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

   7. Simplified 59.6%

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

   if 6.09999999999999985e-105 < y < 1.80000000000000018e159

   1. Initial program 68.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 68.8%

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

   3. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 66.8%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 82.6%

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

   11. Applied egg-rr 82.6%

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

   if 1.80000000000000018e159 < y

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.1%

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

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. 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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 93.0%

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

   9. Simplified 93.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.1 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-74)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 5.5e+15)
     (/ x (* y (+ y 1.0)))
     (if (<= y 1.1e+163) (/ x (* y (+ x y))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.5e+15) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 1.1e+163) {
		tmp = x / (y * (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 <= 4.8d-74) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 5.5d+15) then
        tmp = x / (y * (y + 1.0d0))
    else if (y <= 1.1d+163) then
        tmp = x / (y * (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 <= 4.8e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.5e+15) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 1.1e+163) {
		tmp = x / (y * (x + y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.8e-74:
		tmp = (y / x) / (x + 1.0)
	elif y <= 5.5e+15:
		tmp = x / (y * (y + 1.0))
	elif y <= 1.1e+163:
		tmp = x / (y * (x + y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-74)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 5.5e+15)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (y <= 1.1e+163)
		tmp = Float64(x / Float64(y * 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 <= 4.8e-74)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 5.5e+15)
		tmp = x / (y * (y + 1.0));
	elseif (y <= 1.1e+163)
		tmp = x / (y * (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, 4.8e-74], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+15], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+163], N[(x / N[(y * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.7999999999999998e-74

   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 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 59.6%

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

   7. Simplified 59.6%

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

   if 4.7999999999999998e-74 < y < 5.5e15

   1. Initial program 85.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.5%

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

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 75.6%

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

   7. Simplified 75.6%

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

   if 5.5e15 < y < 1.09999999999999993e163

   1. Initial program 55.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.8%

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

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 65.4%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 81.3%

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

   11. Applied egg-rr 81.3%

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

   12. Taylor expanded in y around inf

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

   14. Simplified 81.3%

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

   if 1.09999999999999993e163 < y

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y around 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 79.4%

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

   7. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 89.5%

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

   9. Applied egg-rr 89.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+163}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.7e-74)
   (/ y (* x (+ x 1.0)))
   (if (<= y 1.25e+14)
     (/ x (* y (+ y 1.0)))
     (if (<= y 2e+161) (/ x (* y (+ x y))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-74) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.25e+14) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 2e+161) {
		tmp = x / (y * (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 <= 4.7d-74) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 1.25d+14) then
        tmp = x / (y * (y + 1.0d0))
    else if (y <= 2d+161) then
        tmp = x / (y * (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 <= 4.7e-74) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.25e+14) {
		tmp = x / (y * (y + 1.0));
	} else if (y <= 2e+161) {
		tmp = x / (y * (x + y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.7e-74:
		tmp = y / (x * (x + 1.0))
	elif y <= 1.25e+14:
		tmp = x / (y * (y + 1.0))
	elif y <= 2e+161:
		tmp = x / (y * (x + y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.7e-74)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 1.25e+14)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (y <= 2e+161)
		tmp = Float64(x / Float64(y * 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 <= 4.7e-74)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 1.25e+14)
		tmp = x / (y * (y + 1.0));
	elseif (y <= 2e+161)
		tmp = x / (y * (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, 4.7e-74], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+14], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+161], N[(x / N[(y * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.7000000000000001e-74

   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. 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. Taylor expanded in y around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 57.7%

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

   9. Simplified 57.7%

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

   if 4.7000000000000001e-74 < y < 1.25e14

   1. Initial program 85.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.5%

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

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 75.6%

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

   7. Simplified 75.6%

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

   if 1.25e14 < y < 2.0000000000000001e161

   1. Initial program 55.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.8%

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

   3. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 65.4%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 81.3%

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

   11. Applied egg-rr 81.3%

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

   12. Taylor expanded in y around inf

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

   14. Simplified 81.3%

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

   if 2.0000000000000001e161 < y

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y around 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 79.4%

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

   7. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 89.5%

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

   9. Applied egg-rr 89.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t\_0} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y 4.8e-105)
     (/ (/ y x) (+ x 1.0))
     (if (<= y 2.2e+134) (/ x (* (+ x y) t_0)) (* (/ x t_0) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 4.8e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e+134) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (x / t_0) * (1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= 4.8d-105) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.2d+134) then
        tmp = x / ((x + y) * t_0)
    else
        tmp = (x / t_0) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 4.8e-105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e+134) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (x / t_0) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= 4.8e-105:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.2e+134:
		tmp = x / ((x + y) * t_0)
	else:
		tmp = (x / t_0) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= 4.8e-105)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.2e+134)
		tmp = Float64(x / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / t_0) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= 4.8e-105)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.2e+134)
		tmp = x / ((x + y) * t_0);
	else
		tmp = (x / t_0) * (1.0 / y);
	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[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 4.8e-105], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+134], N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.8000000000000003e-105

   1. Initial program 74.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 74.7%

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

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 59.6%

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

   7. Simplified 59.6%

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

   if 4.8000000000000003e-105 < y < 2.2e134

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 69.6%

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

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 65.2%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 81.6%

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

   11. Applied egg-rr 81.6%

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

   if 2.2e134 < y

   1. Initial program 60.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 60.1%

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

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. 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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 86.7%

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

   9. Simplified 86.7%

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

Alternative 7: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) + 1} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-74)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.2e+134)
     (/ x (* (+ x y) (+ y 1.0)))
     (* (/ x (+ (+ x y) 1.0)) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e+134) {
		tmp = x / ((x + y) * (y + 1.0));
	} else {
		tmp = (x / ((x + y) + 1.0)) * (1.0 / 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 <= 3.9d-74) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.2d+134) then
        tmp = x / ((x + y) * (y + 1.0d0))
    else
        tmp = (x / ((x + y) + 1.0d0)) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.2e+134) {
		tmp = x / ((x + y) * (y + 1.0));
	} else {
		tmp = (x / ((x + y) + 1.0)) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.9e-74:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.2e+134:
		tmp = x / ((x + y) * (y + 1.0))
	else:
		tmp = (x / ((x + y) + 1.0)) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-74)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.2e+134)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(Float64(x + y) + 1.0)) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-74)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.2e+134)
		tmp = x / ((x + y) * (y + 1.0));
	else
		tmp = (x / ((x + y) + 1.0)) * (1.0 / 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, 3.9e-74], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+134], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.9000000000000001e-74

   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 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 59.6%

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

   7. Simplified 59.6%

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

   if 3.9000000000000001e-74 < y < 2.2e134

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 67.1%

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

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 68.1%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 81.9%

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

   11. Applied egg-rr 81.9%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 77.8%

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

   if 2.2e134 < y

   1. Initial program 60.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 60.1%

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

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. 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 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 86.7%

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

   9. Simplified 86.7%

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

Alternative 8: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1550000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -2.35e-222)
     t_0
     (if (<= y 1.2e-226) (/ y x) (if (<= y 1550000.0) t_0 (/ (/ x y) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -2.35e-222) {
		tmp = t_0;
	} else if (y <= 1.2e-226) {
		tmp = y / x;
	} else if (y <= 1550000.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-2.35d-222)) then
        tmp = t_0
    else if (y <= 1.2d-226) then
        tmp = y / x
    else if (y <= 1550000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -2.35e-222) {
		tmp = t_0;
	} else if (y <= 1.2e-226) {
		tmp = y / x;
	} else if (y <= 1550000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -2.35e-222:
		tmp = t_0
	elif y <= 1.2e-226:
		tmp = y / x
	elif y <= 1550000.0:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -2.35e-222)
		tmp = t_0;
	elseif (y <= 1.2e-226)
		tmp = Float64(y / x);
	elseif (y <= 1550000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -2.35e-222)
		tmp = t_0;
	elseif (y <= 1.2e-226)
		tmp = y / x;
	elseif (y <= 1550000.0)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.35e-222], t$95$0, If[LessEqual[y, 1.2e-226], N[(y / x), $MachinePrecision], If[LessEqual[y, 1550000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3499999999999999e-222 or 1.2e-226 < y < 1.55e6

   1. Initial program 79.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in 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 35.4%

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

   7. Simplified 35.4%

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

   if -2.3499999999999999e-222 < y < 1.2e-226

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 60.2%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 96.0%

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

   10. Simplified 96.0%

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

   if 1.55e6 < y

   1. Initial program 58.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in y around 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 72.4%

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

   7. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 75.3%

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

   9. Applied egg-rr 75.3%

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

Alternative 9: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3500000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -2.35e-222)
     t_0
     (if (<= y 1.2e-226) (/ y x) (if (<= y 3500000.0) t_0 (/ x (* y y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -2.35e-222) {
		tmp = t_0;
	} else if (y <= 1.2e-226) {
		tmp = y / x;
	} else if (y <= 3500000.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-2.35d-222)) then
        tmp = t_0
    else if (y <= 1.2d-226) then
        tmp = y / x
    else if (y <= 3500000.0d0) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -2.35e-222) {
		tmp = t_0;
	} else if (y <= 1.2e-226) {
		tmp = y / x;
	} else if (y <= 3500000.0) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -2.35e-222:
		tmp = t_0
	elif y <= 1.2e-226:
		tmp = y / x
	elif y <= 3500000.0:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -2.35e-222)
		tmp = t_0;
	elseif (y <= 1.2e-226)
		tmp = Float64(y / x);
	elseif (y <= 3500000.0)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -2.35e-222)
		tmp = t_0;
	elseif (y <= 1.2e-226)
		tmp = y / x;
	elseif (y <= 3500000.0)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.35e-222], t$95$0, If[LessEqual[y, 1.2e-226], N[(y / x), $MachinePrecision], If[LessEqual[y, 3500000.0], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3499999999999999e-222 or 1.2e-226 < y < 3.5e6

   1. Initial program 79.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in 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 35.4%

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

   7. Simplified 35.4%

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

   if -2.3499999999999999e-222 < y < 1.2e-226

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 60.2%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in 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 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 96.0%

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

   10. Simplified 96.0%

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

   if 3.5e6 < y

   1. Initial program 58.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in y around 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 72.4%

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

   7. Simplified 72.4%

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

Alternative 10: 92.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{y}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -520000000.0)
   (/ (/ 1.0 (/ x y)) (+ x 1.0))
   (* (/ (/ y (+ x y)) (+ x y)) (/ x (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -520000000.0) {
		tmp = (1.0 / (x / y)) / (x + 1.0);
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (x / (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 <= (-520000000.0d0)) then
        tmp = (1.0d0 / (x / y)) / (x + 1.0d0)
    else
        tmp = ((y / (x + y)) / (x + y)) * (x / (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 <= -520000000.0) {
		tmp = (1.0 / (x / y)) / (x + 1.0);
	} else {
		tmp = ((y / (x + y)) / (x + y)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -520000000.0:
		tmp = (1.0 / (x / y)) / (x + 1.0)
	else:
		tmp = ((y / (x + y)) / (x + y)) * (x / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -520000000.0)
		tmp = Float64(Float64(1.0 / Float64(x / y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) / Float64(x + y)) * Float64(x / 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 <= -520000000.0)
		tmp = (1.0 / (x / y)) / (x + 1.0);
	else
		tmp = ((y / (x + y)) / (x + y)) * (x / (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, -520000000.0], N[(N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e8

   1. Initial program 64.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in y around 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 82.0%

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

   7. Simplified 82.0%

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

      1. clear-num N/A

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

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

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

      3. /-lowering-/.f64 82.0%

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

   9. Applied egg-rr 82.0%

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

   if -5.2e8 < x

   1. Initial program 73.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 73.5%

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

   3. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 88.3%

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

   9. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{x}{y + 1}} \cdot \frac{\frac{y}{x + y}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{y}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + y} \cdot \frac{x}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-74)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 5.8e+159) (/ x (* (+ x y) (+ y 1.0))) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.8e+159) {
		tmp = x / ((x + y) * (y + 1.0));
	} 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 <= 3.5d-74) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 5.8d+159) then
        tmp = x / ((x + y) * (y + 1.0d0))
    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 <= 3.5e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 5.8e+159) {
		tmp = x / ((x + y) * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.5e-74:
		tmp = (y / x) / (x + 1.0)
	elif y <= 5.8e+159:
		tmp = x / ((x + y) * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-74)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 5.8e+159)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + 1.0)));
	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 <= 3.5e-74)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 5.8e+159)
		tmp = x / ((x + y) * (y + 1.0));
	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, 3.5e-74], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e+159], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.50000000000000015e-74

   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 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 59.6%

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

   7. Simplified 59.6%

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

   if 3.50000000000000015e-74 < y < 5.80000000000000029e159

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 66.6%

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

   3. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

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

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

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

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

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

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

      14. +-lowering-+.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 69.3%

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

       1. clear-num N/A

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

       2. associate-/l/ N/A

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

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

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

       4. *-commutative N/A

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

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

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

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

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

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

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

       8. +-lowering-+.f64 82.9%

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

   11. Applied egg-rr 82.9%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 79.4%

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

   if 5.80000000000000029e159 < y

   1. Initial program 59.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y around 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 79.4%

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

   7. Simplified 79.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 89.5%

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

   9. Applied egg-rr 89.5%

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

Alternative 12: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -1.6e-91) (/ y x) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-91) {
		tmp = 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 <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-1.6d-91)) then
        tmp = 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 <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-91) {
		tmp = 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 <= -1.0:
		tmp = (y / x) / x
	elif x <= -1.6e-91:
		tmp = 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 <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.6e-91)
		tmp = 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 <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -1.6e-91)
		tmp = 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, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.6e-91], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 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 76.8%

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

   7. Simplified 76.8%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

   if -1 < x < -1.59999999999999998e-91

   1. Initial program 81.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 81.1%

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

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 51.4%

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 50.7%

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

   10. Simplified 50.7%

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

   if -1.59999999999999998e-91 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 73.4%

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

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 61.0%

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

   7. Simplified 61.0%

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

Alternative 13: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -4.5e-92) (/ y x) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.5e-92) {
		tmp = y / x;
	} else {
		tmp = (x / y) * (1.0 / 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 <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-4.5d-92)) then
        tmp = y / x
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -4.5e-92) {
		tmp = y / x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -4.5e-92:
		tmp = y / x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4.5e-92)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -4.5e-92)
		tmp = y / x;
	else
		tmp = (x / y) * (1.0 / 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, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.5e-92], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 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 76.8%

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

   7. Simplified 76.8%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

   if -1 < x < -4.5e-92

   1. Initial program 81.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 81.1%

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

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 51.4%

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 50.7%

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

   10. Simplified 50.7%

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

   if -4.5e-92 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 73.4%

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

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 45.1%

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

   7. Simplified 45.1%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 47.8%

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

   9. Applied egg-rr 47.8%

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

Alternative 14: 70.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ (/ y x) x) (if (<= x -1.22e-91) (/ y x) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-91) {
		tmp = y / x;
	} 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 (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-1.22d-91)) then
        tmp = y / x
    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 (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.22e-91) {
		tmp = y / x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -1.22e-91:
		tmp = y / x
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.22e-91)
		tmp = Float64(y / x);
	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 (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -1.22e-91)
		tmp = y / x;
	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[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.22e-91], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 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 76.8%

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

   7. Simplified 76.8%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 78.8%

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

   9. Applied egg-rr 78.8%

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

   if -1 < x < -1.21999999999999998e-91

   1. Initial program 81.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 81.1%

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

   3. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 51.4%

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 50.7%

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

   10. Simplified 50.7%

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

   if -1.21999999999999998e-91 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 73.4%

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

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 45.1%

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

   7. Simplified 45.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 47.8%

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

   9. Applied egg-rr 47.8%

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

Alternative 15: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.55 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 2.55e-35) (/ y x) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.55e-35) {
		tmp = y / x;
	} 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 <= 2.55d-35) then
        tmp = y / x
    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 <= 2.55e-35) {
		tmp = y / x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.55e-35:
		tmp = y / x
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.55e-35)
		tmp = Float64(y / x);
	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 <= 2.55e-35)
		tmp = y / x;
	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, 2.55e-35], N[(y / x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.54999999999999993e-35

   1. Initial program 75.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 75.3%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 37.7%

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

   10. Simplified 37.7%

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

   if 2.54999999999999993e-35 < y

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 62.7%

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

   3. Simplified 62.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]
   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 64.2%

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

   7. Simplified 64.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{x} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 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 = y / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y / 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 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 48.5%

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

7. Simplified 48.5%

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

8. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 27.4%

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

10. Simplified 27.4%

    \[\leadsto \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 17: 4.4% 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. 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-/r* N/A

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

   3. clear-num N/A

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

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

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

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

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

   6. associate-+r+ N/A

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

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

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

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

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

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

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

   10. associate-/l* N/A

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

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

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

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

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

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

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

   14. +-lowering-+.f64 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in x around 0

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

9. Simplified 53.5%

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

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Step-by-step derivation

    1. /-lowering-/.f64 4.0%

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

12. Simplified 4.0%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
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 2024160 
(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))))