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

Percentage Accurate: 68.2% → 99.8%

Time: 12.3s

Alternatives: 20

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.2% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

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

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

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

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

   5. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   10. +-commutative N/A

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

   11. associate-+l+ N/A

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

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

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

   13. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

   1. div-inv N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l/ N/A

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

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

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

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

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

   6. clear-num N/A

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

   7. associate-*l/ N/A

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

   8. *-lft-identity N/A

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

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

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

   10. /-lowering-/.f64 N/A

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

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

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

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

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

   13. associate-+r+ N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

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

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

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

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

   18. +-lowering-+.f64 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1e+17)
   (/ (/ y x) x)
   (if (<= x -1.25e-150)
     (/ x (* y y))
     (if (<= x 1.3e-165) (/ x y) (* (/ x y) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1e+17) {
		tmp = (y / x) / x;
	} else if (x <= -1.25e-150) {
		tmp = x / (y * y);
	} else if (x <= 1.3e-165) {
		tmp = x / y;
	} 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 <= (-1d+17)) then
        tmp = (y / x) / x
    else if (x <= (-1.25d-150)) then
        tmp = x / (y * y)
    else if (x <= 1.3d-165) then
        tmp = x / y
    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 <= -1e+17) {
		tmp = (y / x) / x;
	} else if (x <= -1.25e-150) {
		tmp = x / (y * y);
	} else if (x <= 1.3e-165) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1e+17:
		tmp = (y / x) / x
	elif x <= -1.25e-150:
		tmp = x / (y * y)
	elif x <= 1.3e-165:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1e+17)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.25e-150)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= 1.3e-165)
		tmp = Float64(x / y);
	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 <= -1e+17)
		tmp = (y / x) / x;
	elseif (x <= -1.25e-150)
		tmp = x / (y * y);
	elseif (x <= 1.3e-165)
		tmp = x / y;
	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, -1e+17], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.25e-150], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-165], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1e17

   1. Initial program 64.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 86.4%

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

   4. Applied egg-rr 86.4%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 83.4%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 83.0%

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

   10. Simplified 83.0%

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

   if -1e17 < x < -1.24999999999999997e-150

   1. Initial program 84.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 52.2%

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

   5. Simplified 52.2%

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

   if -1.24999999999999997e-150 < x < 1.30000000000000004e-165

   1. Initial program 54.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 84.4%

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

   8. Simplified 84.4%

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

   if 1.30000000000000004e-165 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 34.9%

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

   5. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. 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 37.6%

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

   7. Applied egg-rr 37.6%

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

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-151}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.12e+15)
   (/ (/ y x) x)
   (if (<= x -1.25e-151)
     (/ x (* y y))
     (if (<= x 2.7e-164) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.12e+15) {
		tmp = (y / x) / x;
	} else if (x <= -1.25e-151) {
		tmp = x / (y * y);
	} else if (x <= 2.7e-164) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.12d+15)) then
        tmp = (y / x) / x
    else if (x <= (-1.25d-151)) then
        tmp = x / (y * y)
    else if (x <= 2.7d-164) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.12e+15) {
		tmp = (y / x) / x;
	} else if (x <= -1.25e-151) {
		tmp = x / (y * y);
	} else if (x <= 2.7e-164) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.12e+15:
		tmp = (y / x) / x
	elif x <= -1.25e-151:
		tmp = x / (y * y)
	elif x <= 2.7e-164:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.12e+15)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.25e-151)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= 2.7e-164)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.12e+15)
		tmp = (y / x) / x;
	elseif (x <= -1.25e-151)
		tmp = x / (y * y);
	elseif (x <= 2.7e-164)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.12e+15], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.25e-151], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-164], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.12e15

   1. Initial program 65.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 86.7%

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

   4. Applied egg-rr 86.7%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 81.8%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 81.5%

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

   10. Simplified 81.5%

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

   if -1.12e15 < x < -1.25000000000000001e-151

   1. Initial program 83.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 50.7%

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

   5. Simplified 50.7%

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

   if -1.25000000000000001e-151 < x < 2.7000000000000001e-164

   1. Initial program 54.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 84.4%

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

   8. Simplified 84.4%

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

   if 2.7000000000000001e-164 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 34.9%

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

   5. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. 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 37.6%

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

   7. Applied egg-rr 37.6%

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

Alternative 4: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -125000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -125000000000.0)
   (/ y (* x x))
   (if (<= x -3.5e-150)
     (/ x (* y y))
     (if (<= x 3.7e-164) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -125000000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.5e-150) {
		tmp = x / (y * y);
	} else if (x <= 3.7e-164) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-125000000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-3.5d-150)) then
        tmp = x / (y * y)
    else if (x <= 3.7d-164) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -125000000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.5e-150) {
		tmp = x / (y * y);
	} else if (x <= 3.7e-164) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -125000000000.0:
		tmp = y / (x * x)
	elif x <= -3.5e-150:
		tmp = x / (y * y)
	elif x <= 3.7e-164:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -125000000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.5e-150)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= 3.7e-164)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -125000000000.0)
		tmp = y / (x * x);
	elseif (x <= -3.5e-150)
		tmp = x / (y * y);
	elseif (x <= 3.7e-164)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -125000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-150], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e-164], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.25e11

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 74.0%

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

   5. Simplified 74.0%

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

   if -1.25e11 < x < -3.4999999999999998e-150

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 49.3%

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

   5. Simplified 49.3%

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

   if -3.4999999999999998e-150 < x < 3.6999999999999999e-164

   1. Initial program 54.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 84.4%

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

   8. Simplified 84.4%

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

   if 3.6999999999999999e-164 < x

   1. Initial program 72.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 34.9%

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

   5. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. 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 37.6%

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

   7. Applied egg-rr 37.6%

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

Alternative 5: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 y))))
   (if (<= x -1.9e+15)
     (/ y (* x x))
     (if (<= x -5.2e-152) t_0 (if (<= x 5.1e-164) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.9e+15) {
		tmp = y / (x * x);
	} else if (x <= -5.2e-152) {
		tmp = t_0;
	} else if (x <= 5.1e-164) {
		tmp = x / y;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-1.9d+15)) then
        tmp = y / (x * x)
    else if (x <= (-5.2d-152)) then
        tmp = t_0
    else if (x <= 5.1d-164) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.9e+15) {
		tmp = y / (x * x);
	} else if (x <= -5.2e-152) {
		tmp = t_0;
	} else if (x <= 5.1e-164) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -1.9e+15:
		tmp = y / (x * x)
	elif x <= -5.2e-152:
		tmp = t_0
	elif x <= 5.1e-164:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -1.9e+15)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -5.2e-152)
		tmp = t_0;
	elseif (x <= 5.1e-164)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -1.9e+15)
		tmp = y / (x * x);
	elseif (x <= -5.2e-152)
		tmp = t_0;
	elseif (x <= 5.1e-164)
		tmp = x / y;
	else
		tmp = t_0;
	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[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e+15], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-152], t$95$0, If[LessEqual[x, 5.1e-164], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9e15

   1. Initial program 65.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. 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 75.4%

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

   5. Simplified 75.4%

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

   if -1.9e15 < x < -5.20000000000000026e-152 or 5.10000000000000036e-164 < x

   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. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 39.0%

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

   5. Simplified 39.0%

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

   if -5.20000000000000026e-152 < x < 5.10000000000000036e-164

   1. Initial program 54.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 84.4%

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

   8. Simplified 84.4%

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

Alternative 6: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -4000000000:\\ \;\;\;\;\frac{y \cdot \frac{t\_0}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -4000000000.0)
     (/ (* y (/ t_0 (+ y x))) (+ y x))
     (* t_0 (/ (/ y (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4000000000.0) {
		tmp = (y * (t_0 / (y + x))) / (y + x);
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-4000000000.0d0)) then
        tmp = (y * (t_0 / (y + x))) / (y + x)
    else
        tmp = t_0 * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4000000000.0) {
		tmp = (y * (t_0 / (y + x))) / (y + x);
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -4000000000.0:
		tmp = (y * (t_0 / (y + x))) / (y + x)
	else:
		tmp = t_0 * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -4000000000.0)
		tmp = Float64(Float64(y * Float64(t_0 / Float64(y + x))) / Float64(y + x));
	else
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -4000000000.0)
		tmp = (y * (t_0 / (y + x))) / (y + x);
	else
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4000000000.0], N[(N[(y * N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e9

   1. Initial program 65.8%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

      13. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.8%

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

   if -4e9 < x

   1. Initial program 67.7%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000000000:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{y + x}}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -4000000000:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -4000000000.0)
     (* t_0 (/ (/ y (+ y x)) (+ y x)))
     (* t_0 (/ (/ y (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4000000000.0) {
		tmp = t_0 * ((y / (y + x)) / (y + x));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-4000000000.0d0)) then
        tmp = t_0 * ((y / (y + x)) / (y + x))
    else
        tmp = t_0 * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -4000000000.0) {
		tmp = t_0 * ((y / (y + x)) / (y + x));
	} else {
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -4000000000.0:
		tmp = t_0 * ((y / (y + x)) / (y + x))
	else:
		tmp = t_0 * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -4000000000.0)
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + x)) / Float64(y + x)));
	else
		tmp = Float64(t_0 * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -4000000000.0)
		tmp = t_0 * ((y / (y + x)) / (y + x));
	else
		tmp = t_0 * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4000000000.0], N[(t$95$0 * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e9

   1. Initial program 65.8%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-lowering-+.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.7%

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

   if -4e9 < x

   1. Initial program 67.7%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 85.2%

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

   7. Simplified 85.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000000000:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-5)
   (/ (/ y (+ y (+ x 1.0))) (+ x (* y 2.0)))
   (* (/ x (+ y x)) (/ (/ y (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-5) {
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d-5)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + (y * 2.0d0))
    else
        tmp = (x / (y + x)) * ((y / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-5) {
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	} else {
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e-5:
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0))
	else:
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-5)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-5)
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	else
		tmp = (x / (y + x)) * ((y / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e-5], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999984e-5

   1. Initial program 68.7%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \color{blue}{\left(x + 2 \cdot y\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{2}\right)\right)\right) \]

      3. *-lowering-*.f64 77.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

   7. Simplified 77.0%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot 2}} \]

   if -2.59999999999999984e-5 < x

   1. Initial program 66.9%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 85.7%

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

   7. Simplified 85.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{y}{y + \left(x + 1\right)}\\ \mathbf{if}\;y \leq 2.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{t\_0}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + x} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y (+ x 1.0)))))
   (if (<= y 2.4e-148) (/ t_0 (+ x (* y 2.0))) (* (/ t_0 (+ y x)) (/ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y / (y + (x + 1.0));
	double tmp;
	if (y <= 2.4e-148) {
		tmp = t_0 / (x + (y * 2.0));
	} else {
		tmp = (t_0 / (y + x)) * (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) :: t_0
    real(8) :: tmp
    t_0 = y / (y + (x + 1.0d0))
    if (y <= 2.4d-148) then
        tmp = t_0 / (x + (y * 2.0d0))
    else
        tmp = (t_0 / (y + x)) * (x / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y / (y + (x + 1.0));
	double tmp;
	if (y <= 2.4e-148) {
		tmp = t_0 / (x + (y * 2.0));
	} else {
		tmp = (t_0 / (y + x)) * (x / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y / (y + (x + 1.0))
	tmp = 0
	if y <= 2.4e-148:
		tmp = t_0 / (x + (y * 2.0))
	else:
		tmp = (t_0 / (y + x)) * (x / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 2.4e-148)
		tmp = Float64(t_0 / Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(x / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y / (y + (x + 1.0));
	tmp = 0.0;
	if (y <= 2.4e-148)
		tmp = t_0 / (x + (y * 2.0));
	else
		tmp = (t_0 / (y + x)) * (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_] := Block[{t$95$0 = N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.4e-148], N[(t$95$0 / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e-148

   1. Initial program 67.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \color{blue}{\left(x + 2 \cdot y\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{2}\right)\right)\right) \]

      3. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

   7. Simplified 55.2%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot 2}} \]

   if 2.4000000000000001e-148 < y

   1. Initial program 66.2%

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

      1. times-frac N/A

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

      2. associate-*l/ N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-lowering-+.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 75.0%

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

   7. Simplified 75.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-157)
   (/ (/ y (+ y (+ x 1.0))) (+ x (* y 2.0)))
   (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-157) {
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	} 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 <= (-1.35d-157)) then
        tmp = (y / (y + (x + 1.0d0))) / (x + (y * 2.0d0))
    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 <= -1.35e-157) {
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.35e-157:
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-157)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + Float64(y * 2.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 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 <= -1.35e-157)
		tmp = (y / (y + (x + 1.0))) / (x + (y * 2.0));
	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, -1.35e-157], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e-157

   1. Initial program 71.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 91.0%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \color{blue}{\left(x + 2 \cdot y\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{2}\right)\right)\right) \]

      3. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

   7. Simplified 68.3%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot 2}} \]

   if -1.35e-157 < x

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 59.0%

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

   5. Simplified 59.0%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-lowering-+.f64 60.6%

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

   7. Applied egg-rr 60.6%

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

Alternative 11: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-113}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+51)
   (/ (/ y x) x)
   (if (<= x -1.15e-113) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+51) {
		tmp = (y / x) / x;
	} else if (x <= -1.15e-113) {
		tmp = y / (x * (x + 1.0));
	} 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 <= (-4.2d+51)) then
        tmp = (y / x) / x
    else if (x <= (-1.15d-113)) then
        tmp = y / (x * (x + 1.0d0))
    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 <= -4.2e+51) {
		tmp = (y / x) / x;
	} else if (x <= -1.15e-113) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.2e+51:
		tmp = (y / x) / x
	elif x <= -1.15e-113:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+51)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.15e-113)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 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 <= -4.2e+51)
		tmp = (y / x) / x;
	elseif (x <= -1.15e-113)
		tmp = y / (x * (x + 1.0));
	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, -4.2e+51], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.15e-113], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2000000000000002e51

   1. Initial program 57.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 82.6%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 82.2%

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

   10. Simplified 82.2%

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

   if -4.2000000000000002e51 < x < -1.15000000000000004e-113

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 52.4%

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

   5. Simplified 52.4%

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

   if -1.15000000000000004e-113 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 59.1%

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

   5. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-lowering-+.f64 60.6%

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

   7. Applied egg-rr 60.6%

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

Alternative 12: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-114}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4e+51)
   (/ (/ y x) x)
   (if (<= x -1.55e-114) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e+51) {
		tmp = (y / x) / x;
	} else if (x <= -1.55e-114) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4d+51)) then
        tmp = (y / x) / x
    else if (x <= (-1.55d-114)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4e+51) {
		tmp = (y / x) / x;
	} else if (x <= -1.55e-114) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4e+51:
		tmp = (y / x) / x
	elif x <= -1.55e-114:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4e+51)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.55e-114)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	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 <= -4e+51)
		tmp = (y / x) / x;
	elseif (x <= -1.55e-114)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4e+51], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.55e-114], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4e51

   1. Initial program 57.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 82.6%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 82.2%

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

   10. Simplified 82.2%

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

   if -4e51 < x < -1.55e-114

   1. Initial program 88.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. 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 52.4%

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

   5. Simplified 52.4%

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

   if -1.55e-114 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 59.1%

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

   5. Simplified 59.1%

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

Alternative 13: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. times-frac N/A

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

   2. associate-*l/ N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

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

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

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

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

   13. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 14: 82.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e-112) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e-112) {
		tmp = (y / (x + 1.0)) / (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 <= (-5.5d-112)) then
        tmp = (y / (x + 1.0d0)) / (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 <= -5.5e-112) {
		tmp = (y / (x + 1.0)) / (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 <= -5.5e-112:
		tmp = (y / (x + 1.0)) / (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 <= -5.5e-112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 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 <= -5.5e-112)
		tmp = (y / (x + 1.0)) / (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, -5.5e-112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e-112

   1. Initial program 72.5%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

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

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

      13. +-lowering-+.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. div-inv N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

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

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

      10. /-lowering-/.f64 N/A

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

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

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

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

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. associate-+l+ N/A

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

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

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

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

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

      18. +-lowering-+.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

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

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

      2. +-lowering-+.f64 68.2%

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

   9. Simplified 68.2%

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

   if -5.5e-112 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 59.1%

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

   5. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-lowering-+.f64 60.6%

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

   7. Applied egg-rr 60.6%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e-112) (/ (/ y (+ x 1.0)) x) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-112) {
		tmp = (y / (x + 1.0)) / 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 <= (-6.2d-112)) then
        tmp = (y / (x + 1.0d0)) / 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 <= -6.2e-112) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -6.2e-112:
		tmp = (y / (x + 1.0)) / 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 <= -6.2e-112)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + 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 <= -6.2e-112)
		tmp = (y / (x + 1.0)) / 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, -6.2e-112], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999995e-112

   1. Initial program 72.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 91.5%

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

   4. Applied egg-rr 91.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 68.2%

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

   8. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 67.8%

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

   10. Simplified 67.8%

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

   if -6.1999999999999995e-112 < x

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 59.1%

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

   5. Simplified 59.1%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-lowering-+.f64 60.6%

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

   7. Applied egg-rr 60.6%

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

Alternative 16: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7000000000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7000000000000.0) (/ (/ y x) x) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7000000000000.0) {
		tmp = (y / x) / 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 <= (-7000000000000.0d0)) then
        tmp = (y / x) / 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 <= -7000000000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7000000000000.0:
		tmp = (y / x) / 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 <= -7000000000000.0)
		tmp = Float64(Float64(y / x) / 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 <= -7000000000000.0)
		tmp = (y / x) / 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, -7000000000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7e12

   1. Initial program 65.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 80.3%

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

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 80.0%

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

   10. Simplified 80.0%

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

   if -7e12 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 57.7%

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

   5. Simplified 57.7%

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

Alternative 17: 46.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. 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 44.4%

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

   5. Simplified 44.4%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 31.0%

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

   8. Simplified 31.0%

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

   if 1 < y

   1. Initial program 58.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 69.1%

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

   5. Simplified 69.1%

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

Alternative 18: 27.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -7500000000.0) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7500000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7500000000.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7500000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7500000000.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7500000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7500000000.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7500000000.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e9

   1. Initial program 65.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      14. +-lowering-+.f64 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 80.3%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 6.0%

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

   10. Simplified 6.0%

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

   if -7.5e9 < x

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 37.5%

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

   8. Simplified 37.5%

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

Alternative 19: 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 67.3%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 86.1%

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

4. Applied egg-rr 86.1%

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

5. Taylor expanded in x around inf

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

7. Simplified 49.9%

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

8. Taylor expanded in y around inf

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

   1. /-lowering-/.f64 4.0%

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

10. Simplified 4.0%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 20: 4.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 y))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / 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 = 0.5d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   14. +-lowering-+.f64 86.1%

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

4. Applied egg-rr 86.1%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \color{blue}{\left(x + 2 \cdot y\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{\left(2 \cdot y\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{2}\right)\right)\right) \]

   3. *-lowering-*.f64 51.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right)\right) \]

7. Simplified 51.6%

   \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot 2}} \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.0%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{y}\right) \]

10. Simplified 4.0%

    \[\leadsto \color{blue}{\frac{0.5}{y}} \]
11. 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 2024161 
(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))))