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

Percentage Accurate: 69.1% → 99.8%

Time: 13.6s

Alternatives: 22

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.3%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. associate-/r* N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

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

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

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

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

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

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

   13. associate-+r+ N/A

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

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

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

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

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

   16. +-lowering-+.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000000:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x \cdot \left(1 + \frac{y}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -70000000000.0)
   (* (/ x (+ y x)) (/ (/ y (+ y x)) (+ y x)))
   (if (<= x -3.2e-236)
     (/ (/ y (+ x 1.0)) (* x (+ 1.0 (/ y x))))
     (/ (/ x (+ x (+ y 1.0))) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -70000000000.0) {
		tmp = (x / (y + x)) * ((y / (y + x)) / (y + x));
	} else if (x <= -3.2e-236) {
		tmp = (y / (x + 1.0)) / (x * (1.0 + (y / x)));
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-70000000000.0d0)) then
        tmp = (x / (y + x)) * ((y / (y + x)) / (y + x))
    else if (x <= (-3.2d-236)) then
        tmp = (y / (x + 1.0d0)) / (x * (1.0d0 + (y / x)))
    else
        tmp = (x / (x + (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -70000000000.0) {
		tmp = (x / (y + x)) * ((y / (y + x)) / (y + x));
	} else if (x <= -3.2e-236) {
		tmp = (y / (x + 1.0)) / (x * (1.0 + (y / x)));
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -70000000000.0:
		tmp = (x / (y + x)) * ((y / (y + x)) / (y + x))
	elif x <= -3.2e-236:
		tmp = (y / (x + 1.0)) / (x * (1.0 + (y / x)))
	else:
		tmp = (x / (x + (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -70000000000.0)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / Float64(y + x)) / Float64(y + x)));
	elseif (x <= -3.2e-236)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x * Float64(1.0 + Float64(y / x))));
	else
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -70000000000.0)
		tmp = (x / (y + x)) * ((y / (y + x)) / (y + x));
	elseif (x <= -3.2e-236)
		tmp = (y / (x + 1.0)) / (x * (1.0 + (y / x)));
	else
		tmp = (x / (x + (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -70000000000.0], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.2e-236], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e10

   1. Initial program 52.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 99.8%

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

   if -7e10 < x < -3.2e-236

   1. Initial program 81.8%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in x around inf

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

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

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

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

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

      3. /-lowering-/.f64 54.9%

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

   10. Simplified 54.9%

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

   if -3.2e-236 < x

   1. Initial program 61.3%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 48.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000000:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x \cdot \left(1 + \frac{y}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.35e+69)
   (/ (/ y x) x)
   (if (<= y 3.2e-190)
     (/ y (* x (+ x 1.0)))
     (if (<= y 4.05e+15) (/ x (* y (+ y 1.0))) (/ (/ x y) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.35e+69) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4.05e+15) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.35d+69)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-190) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 4.05d+15) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -3.35e+69) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4.05e+15) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.35e+69:
		tmp = (y / x) / x
	elif y <= 3.2e-190:
		tmp = y / (x * (x + 1.0))
	elif y <= 4.05e+15:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.35e+69)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-190)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 4.05e+15)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.35e+69)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-190)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 4.05e+15)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -3.35e+69], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.05e+15], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.35000000000000005e69

   1. Initial program 37.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({x}^{3}\right)}\right) \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 1.4%

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

   5. Simplified 1.4%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 28.4%

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

   7. Applied egg-rr 28.4%

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

   if -3.35000000000000005e69 < y < 3.2000000000000001e-190

   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 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 79.8%

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

   5. Simplified 79.8%

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

   if 3.2000000000000001e-190 < y < 4.05e15

   1. Initial program 82.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 35.5%

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

   5. Simplified 35.5%

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

   if 4.05e15 < y

   1. Initial program 64.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 75.8%

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

   7. Simplified 75.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -3.5e+69)
   (/ (/ y x) x)
   (if (<= y 3.2e-190)
     (/ y (* x (+ x 1.0)))
     (if (<= y 3e+103) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -3.5e+69) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 3e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.5d+69)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-190) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 3d+103) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -3.5e+69) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 3e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -3.5e+69:
		tmp = (y / x) / x
	elif y <= 3.2e-190:
		tmp = y / (x * (x + 1.0))
	elif y <= 3e+103:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -3.5e+69)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-190)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 3e+103)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.5e+69)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-190)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 3e+103)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -3.5e+69], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+103], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.49999999999999987e69

   1. Initial program 37.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({x}^{3}\right)}\right) \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 1.4%

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

   5. Simplified 1.4%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 28.4%

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

   7. Applied egg-rr 28.4%

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

   if -3.49999999999999987e69 < y < 3.2000000000000001e-190

   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 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 79.8%

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

   5. Simplified 79.8%

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

   if 3.2000000000000001e-190 < y < 3e103

   1. Initial program 79.5%

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

   3. Taylor expanded in 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 39.4%

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

   5. Simplified 39.4%

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

   if 3e103 < y

   1. Initial program 59.8%

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

   3. Taylor expanded in 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 80.3%

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

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 91.7%

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

   7. Applied egg-rr 91.7%

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

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e-188)
   (/ (/ y x) x)
   (if (<= y 3.2e-190)
     (/ y (+ y x))
     (if (<= y 4e+101) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-188) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 4e+101) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d-188)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-190) then
        tmp = y / (y + x)
    else if (y <= 4d+101) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-188) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 4e+101) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -4.6e-188:
		tmp = (y / x) / x
	elif y <= 3.2e-190:
		tmp = y / (y + x)
	elif y <= 4e+101:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -4.6e-188)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-190)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 4e+101)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e-188)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-190)
		tmp = y / (y + x);
	elseif (y <= 4e+101)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -4.6e-188], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+101], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.6e-188

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({x}^{3}\right)}\right) \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 20.7%

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

   5. Simplified 20.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 39.2%

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

   7. Applied egg-rr 39.2%

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

   if -4.6e-188 < y < 3.2000000000000001e-190

   1. Initial program 55.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

   if 3.2000000000000001e-190 < y < 3.9999999999999999e101

   1. Initial program 79.5%

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

   3. Taylor expanded in 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 39.4%

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

   5. Simplified 39.4%

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

   if 3.9999999999999999e101 < y

   1. Initial program 59.8%

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

   3. Taylor expanded in 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 80.3%

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

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 91.7%

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

   7. Applied egg-rr 91.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -9e-188)
   (/ (/ y x) x)
   (if (<= y 3.2e-190) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -9e-188) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9d-188)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-190) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -9e-188) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -9e-188:
		tmp = (y / x) / x
	elif y <= 3.2e-190:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -9e-188)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-190)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9e-188)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-190)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -9e-188], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.99999999999999986e-188

   1. Initial program 57.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({x}^{3}\right)}\right) \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 20.7%

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

   5. Simplified 20.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 39.2%

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

   7. Applied egg-rr 39.2%

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

   if -8.99999999999999986e-188 < y < 3.2000000000000001e-190

   1. Initial program 55.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

   if 3.2000000000000001e-190 < y < 1

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 34.9%

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

   8. Simplified 34.9%

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

   if 1 < y

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 67.1%

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

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 74.1%

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

   7. Applied egg-rr 74.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e-187)
   (/ y (* x x))
   (if (<= y 2.7e-190) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-187) {
		tmp = y / (x * x);
	} else if (y <= 2.7e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d-187)) then
        tmp = y / (x * x)
    else if (y <= 2.7d-190) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-187) {
		tmp = y / (x * x);
	} else if (y <= 2.7e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.4e-187:
		tmp = y / (x * x)
	elif y <= 2.7e-190:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-187)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.7e-190)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e-187)
		tmp = y / (x * x);
	elseif (y <= 2.7e-190)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.4e-187], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e-190], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4e-187

   1. Initial program 57.0%

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

   3. Taylor expanded in 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 32.1%

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

   5. Simplified 32.1%

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

   if -1.4e-187 < y < 2.6999999999999999e-190

   1. Initial program 55.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

   if 2.6999999999999999e-190 < y < 1

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 34.9%

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

   8. Simplified 34.9%

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

   if 1 < y

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 67.1%

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

   5. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 74.1%

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

   7. Applied egg-rr 74.1%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -1.85e-187)
   (/ y (* x x))
   (if (<= y 3.2e-190) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -1.85e-187) {
		tmp = y / (x * x);
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.85d-187)) then
        tmp = y / (x * x)
    else if (y <= 3.2d-190) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85e-187) {
		tmp = y / (x * x);
	} else if (y <= 3.2e-190) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -1.85e-187:
		tmp = y / (x * x)
	elif y <= 3.2e-190:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -1.85e-187)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 3.2e-190)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.85e-187)
		tmp = y / (x * x);
	elseif (y <= 3.2e-190)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -1.85e-187], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.85000000000000005e-187

   1. Initial program 57.0%

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

   3. Taylor expanded in 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 32.1%

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

   5. Simplified 32.1%

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

   if -1.85000000000000005e-187 < y < 3.2000000000000001e-190

   1. Initial program 55.9%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 72.5%

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

   if 3.2000000000000001e-190 < y < 1

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 34.9%

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

   8. Simplified 34.9%

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

   if 1 < y

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 67.1%

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

   5. Simplified 67.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-187}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2e-187)
   (/ y (* x x))
   (if (<= y 3.2e-190) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2e-187) {
		tmp = y / (x * x);
	} else if (y <= 3.2e-190) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d-187)) then
        tmp = y / (x * x)
    else if (y <= 3.2d-190) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= -2e-187) {
		tmp = y / (x * x);
	} else if (y <= 3.2e-190) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= -2e-187:
		tmp = y / (x * x)
	elif y <= 3.2e-190:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2e-187)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 3.2e-190)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e-187)
		tmp = y / (x * x);
	elseif (y <= 3.2e-190)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2e-187], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e-190], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2e-187

   1. Initial program 57.0%

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

   3. Taylor expanded in 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 32.1%

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

   5. Simplified 32.1%

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

   if -2e-187 < y < 3.2000000000000001e-190

   1. Initial program 55.9%

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

   3. Taylor expanded in 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 86.6%

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

   5. Simplified 86.6%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 72.5%

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

   8. Simplified 72.5%

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

   if 3.2000000000000001e-190 < y < 1

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 34.9%

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

   8. Simplified 34.9%

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

   if 1 < y

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 67.1%

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

   5. Simplified 67.1%

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

Alternative 10: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 55.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.4%

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

   if -1 < x

   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. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

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

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

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

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

      9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. +-lowering-+.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 86.6%

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

   7. Simplified 86.6%

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

4. Final simplification 89.2%

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

Alternative 11: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+65}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -7e+65)
   (/ (/ y x) x)
   (if (<= y 3.2e-190) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -7e+65) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		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 (y <= (-7d+65)) then
        tmp = (y / x) / x
    else if (y <= 3.2d-190) 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 (y <= -7e+65) {
		tmp = (y / x) / x;
	} else if (y <= 3.2e-190) {
		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 y <= -7e+65:
		tmp = (y / x) / x
	elif y <= 3.2e-190:
		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 (y <= -7e+65)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.2e-190)
		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 (y <= -7e+65)
		tmp = (y / x) / x;
	elseif (y <= 3.2e-190)
		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[y, -7e+65], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.2e-190], 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 y < -7.0000000000000002e65

   1. Initial program 38.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({x}^{3}\right)}\right) \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 3.2%

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

   5. Simplified 3.2%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 29.7%

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

   7. Applied egg-rr 29.7%

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

   if -7.0000000000000002e65 < y < 3.2000000000000001e-190

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 79.6%

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

   5. Simplified 79.6%

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

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 53.5%

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

   5. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   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 57.5%

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

   7. Applied egg-rr 57.5%

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

Alternative 12: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. associate-/r* N/A

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

   2. associate-/r* N/A

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

   3. associate-/r* N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

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

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

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

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

   9. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

   15. +-lowering-+.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 13: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190)
   (/ (/ y (+ y x)) (+ x 1.0))
   (/ (/ x (+ x (+ y 1.0))) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-190) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (x + (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (x + (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-190:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (x + (y + 1.0))) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-190)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-190)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (x + (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-190], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.4%

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

   7. Simplified 61.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-lowering-+.f64 61.4%

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

   9. Applied egg-rr 61.4%

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

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 58.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-190) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-190:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-190)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-190)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-190], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 36.3%

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

   8. Simplified 36.3%

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

   if 3.2000000000000001e-190 < y < 1

   1. Initial program 84.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 34.9%

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

   8. Simplified 34.9%

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

   if 1 < y

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 67.1%

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

   5. Simplified 67.1%

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

Alternative 15: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-190) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-190:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-190)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-190)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-190], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.4%

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

   7. Simplified 61.4%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

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

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

      6. +-lowering-+.f64 61.4%

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

   9. Applied egg-rr 61.4%

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

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 57.9%

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

   7. Simplified 57.9%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-190) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-190:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-190)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-190)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-190], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.4%

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

   7. Simplified 61.4%

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

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

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

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

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

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

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

      13. associate-+r+ N/A

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

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

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

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

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

      16. +-lowering-+.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 57.9%

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

   7. Simplified 57.9%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.2d-190) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 3.2e-190:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-190)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-190)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 3.2e-190], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

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

   3. Taylor expanded in 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 56.2%

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

   5. Simplified 56.2%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. +-lowering-+.f64 60.8%

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

   7. Applied egg-rr 60.8%

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

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{x}{x + y}\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \left(x + y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

      13. associate-+r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(x + y\right)\right) \]

      16. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{x}{1 + y}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(1 + y\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

      3. +-lowering-+.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   7. Simplified 57.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-190}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-190) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-190) {
		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 (y <= 3.2d-190) 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 (y <= 3.2e-190) {
		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 y <= 3.2e-190:
		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 (y <= 3.2e-190)
		tmp = Float64(Float64(y / 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 (y <= 3.2e-190)
		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[y, 3.2e-190], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2000000000000001e-190

   1. Initial program 56.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 56.2%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y}{x}\right), \color{blue}{\left(x + 1\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \left(\color{blue}{x} + 1\right)\right) \]

      4. +-lowering-+.f64 60.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right) \]

   7. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3.2000000000000001e-190 < y

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 53.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   5. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
   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 57.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right), y\right) \]

   7. Applied egg-rr 57.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 42.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.2e-236) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-236) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d-236)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-236) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.2e-236:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-236)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e-236)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.2e-236], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e-236

   1. Initial program 66.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 54.1%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   5. Simplified 54.1%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 34.9%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{x}\right) \]

   8. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -3.2e-236 < x

   1. Initial program 61.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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 45.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 30.3%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

   8. Simplified 30.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 63.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in 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 44.3%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(y, \color{blue}{1}\right)\right)\right) \]

5. Simplified 44.3%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 23.8%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{y}\right) \]

8. Simplified 23.8%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 21: 4.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 y))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / 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 = 1.0d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 63.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{\left({y}^{2}\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(x, y\right), 1\right)\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(y \cdot y\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, y\right)}, 1\right)\right)\right) \]

   2. *-lowering-*.f64 29.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{+.f64}\left(x, y\right)}, 1\right)\right)\right) \]

5. Simplified 29.6%

   \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 4.2%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{y}\right) \]

8. Simplified 4.2%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
9. Add Preprocessing

Alternative 22: 3.4% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 63.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{\left(x + y\right) + 1}} \]

   2. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{x + y}}{\color{blue}{\left(x + y\right)} + 1} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{x \cdot y}{x + y}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{x \cdot y}{x + y}}{\left(\left(x + y\right) + 1\right) \cdot \color{blue}{\left(x + y\right)}} \]

   5. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{x \cdot y}{x + y}}{\left(x + y\right) + 1}}{\color{blue}{x + y}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot y}{x + y}}{\left(x + y\right) + 1}\right), \color{blue}{\left(x + y\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot y}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), \left(\color{blue}{x} + y\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{y \cdot x}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(y \cdot \frac{x}{x + y}\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{x}{x + y}\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \left(x + y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(\left(x + y\right) + 1\right)\right), \left(x + y\right)\right) \]

   13. associate-+r+ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \left(x + \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \left(y + 1\right)\right)\right), \left(x + y\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \left(x + y\right)\right) \]

   16. +-lowering-+.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(x, y\right)\right)\right), \mathsf{+.f64}\left(x, \mathsf{+.f64}\left(y, 1\right)\right)\right), \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y \cdot \frac{x}{x + y}}{x + \left(y + 1\right)}}{x + y}} \]

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{y}{1 + x}\right)}, \mathsf{+.f64}\left(x, y\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(1 + x\right)\right), \mathsf{+.f64}\left(\color{blue}{x}, y\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \left(x + 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

   3. +-lowering-+.f64 54.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, \mathsf{+.f64}\left(x, 1\right)\right), \mathsf{+.f64}\left(x, y\right)\right) \]

7. Simplified 54.0%

   \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 3.6%

    \[\leadsto \color{blue}{1} \]
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 2024152 
(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))))