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

Percentage Accurate: 69.0% → 99.8%

Time: 12.5s

Alternatives: 15

Speedup: 1.6×

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: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.7e+85)
     (/ (/ y x) t_0)
     (if (<= x -1e-157)
       (* y (/ x (* t_0 (* (+ x y) (+ x y)))))
       (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.7e+85) {
		tmp = (y / x) / t_0;
	} else if (x <= -1e-157) {
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.7d+85)) then
        tmp = (y / x) / t_0
    else if (x <= (-1d-157)) then
        tmp = y * (x / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.7e+85) {
		tmp = (y / x) / t_0;
	} else if (x <= -1e-157) {
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.7e+85:
		tmp = (y / x) / t_0
	elif x <= -1e-157:
		tmp = y * (x / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.7e+85)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -1e-157)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.7e+85)
		tmp = (y / x) / t_0;
	elseif (x <= -1e-157)
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+85], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1e-157], N[(y * N[(x / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000002e85

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.3

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 71.3

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

   4. Applied rewrites 71.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if -1.7000000000000002e85 < x < -9.99999999999999943e-158

   1. Initial program 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 92.1

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

   if -9.99999999999999943e-158 < x

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.5

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 62.1

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

   7. Applied rewrites 62.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + \left(x + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -5.1e+135)
     (/ (/ y x) t_0)
     (/ (* y (/ x (+ x y))) (* (+ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (y * (x / (x + y))) / ((x + y) * t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-5.1d+135)) then
        tmp = (y / x) / t_0
    else
        tmp = (y * (x / (x + y))) / ((x + y) * t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (y * (x / (x + y))) / ((x + y) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -5.1e+135:
		tmp = (y / x) / t_0
	else:
		tmp = (y * (x / (x + y))) / ((x + y) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5.1e+135)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(y * Float64(x / Float64(x + y))) / Float64(Float64(x + y) * t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -5.1e+135)
		tmp = (y / x) / t_0;
	else
		tmp = (y * (x / (x + y))) / ((x + y) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.1e+135], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(y * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.09999999999999982e135

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -5.09999999999999982e135 < x

   1. Initial program 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 95.5

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -5.1e+135)
     (/ (/ y x) t_0)
     (* (/ x (+ x y)) (/ y (* (+ x y) t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-5.1d+135)) then
        tmp = (y / x) / t_0
    else
        tmp = (x / (x + y)) * (y / ((x + y) * t_0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -5.1e+135:
		tmp = (y / x) / t_0
	else:
		tmp = (x / (x + y)) * (y / ((x + y) * t_0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5.1e+135)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * t_0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -5.1e+135)
		tmp = (y / x) / t_0;
	else
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.1e+135], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.09999999999999982e135

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -5.09999999999999982e135 < x

   1. Initial program 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -5.1e+135)
     (/ (/ y x) t_0)
     (if (<= x -3.1e-87) (/ (* y 1.0) (* (+ x y) t_0)) (/ (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else if (x <= -3.1e-87) {
		tmp = (y * 1.0) / ((x + y) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-5.1d+135)) then
        tmp = (y / x) / t_0
    else if (x <= (-3.1d-87)) then
        tmp = (y * 1.0d0) / ((x + y) * t_0)
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -5.1e+135) {
		tmp = (y / x) / t_0;
	} else if (x <= -3.1e-87) {
		tmp = (y * 1.0) / ((x + y) * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -5.1e+135:
		tmp = (y / x) / t_0
	elif x <= -3.1e-87:
		tmp = (y * 1.0) / ((x + y) * t_0)
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -5.1e+135)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -3.1e-87)
		tmp = Float64(Float64(y * 1.0) / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -5.1e+135)
		tmp = (y / x) / t_0;
	elseif (x <= -3.1e-87)
		tmp = (y * 1.0) / ((x + y) * t_0);
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.1e+135], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -3.1e-87], N[(N[(y * 1.0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.09999999999999982e135

   1. Initial program 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -5.09999999999999982e135 < x < -3.09999999999999998e-87

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 95.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 95.4

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 71.1%

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

   if -3.09999999999999998e-87 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{y \cdot 1}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+31)
   (/ (/ y x) x)
   (if (<= x -1.3e-86) (/ y (fma x x x)) (/ (/ x y) (+ y (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+31) {
		tmp = (y / x) / x;
	} else if (x <= -1.3e-86) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / (y + (x + 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+31)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.3e-86)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + Float64(x + 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+31], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.3e-86], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000027e31

   1. Initial program 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   9. Applied rewrites 85.3%

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

   if -5.00000000000000027e31 < x < -1.3000000000000001e-86

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -1.3000000000000001e-86 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+31)
   (/ (/ y x) x)
   (if (<= x -1.3e-86) (/ y (fma x x x)) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+31) {
		tmp = (y / x) / x;
	} else if (x <= -1.3e-86) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+31)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.3e-86)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+31], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.3e-86], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000027e31

   1. Initial program 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   9. Applied rewrites 85.3%

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

   if -5.00000000000000027e31 < x < -1.3000000000000001e-86

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -1.3000000000000001e-86 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.2

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

   9. Applied rewrites 61.2%

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

   11. Applied rewrites 63.5%

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

Alternative 8: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -0.001:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -0.001)
     (/ y (* x x))
     (if (<= x -7.5e-178) t_0 (if (<= x 5.5e-129) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -0.001) {
		tmp = y / (x * x);
	} else if (x <= -7.5e-178) {
		tmp = t_0;
	} else if (x <= 5.5e-129) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-0.001d0)) then
        tmp = y / (x * x)
    else if (x <= (-7.5d-178)) then
        tmp = t_0
    else if (x <= 5.5d-129) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -0.001) {
		tmp = y / (x * x);
	} else if (x <= -7.5e-178) {
		tmp = t_0;
	} else if (x <= 5.5e-129) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -0.001:
		tmp = y / (x * x)
	elif x <= -7.5e-178:
		tmp = t_0
	elif x <= 5.5e-129:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -0.001)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -7.5e-178)
		tmp = t_0;
	elseif (x <= 5.5e-129)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -0.001)
		tmp = y / (x * x);
	elseif (x <= -7.5e-178)
		tmp = t_0;
	elseif (x <= 5.5e-129)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.001], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-178], t$95$0, If[LessEqual[x, 5.5e-129], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e-3

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -1e-3 < x < -7.50000000000000019e-178 or 5.50000000000000023e-129 < x

   1. Initial program 74.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 40.7

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

   5. Applied rewrites 40.7%

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

   if -7.50000000000000019e-178 < x < 5.50000000000000023e-129

   1. Initial program 65.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 65.6

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 65.6

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

   4. Applied rewrites 65.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 87.0

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

   9. Applied rewrites 87.0%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 79.1%

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

Alternative 9: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.3e-86) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.3e-86) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.3d-86)) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.3e-86) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.3e-86:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.3e-86)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.3e-86)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-86], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3000000000000001e-86

   1. Initial program 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.7

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 80.7

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if -1.3000000000000001e-86 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+31)
   (/ (/ y x) x)
   (if (<= x -1.3e-86) (/ y (fma x x x)) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+31) {
		tmp = (y / x) / x;
	} else if (x <= -1.3e-86) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+31)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.3e-86)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+31], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.3e-86], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000027e31

   1. Initial program 71.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 75.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   9. Applied rewrites 85.3%

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

   if -5.00000000000000027e31 < x < -1.3000000000000001e-86

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -1.3000000000000001e-86 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.2

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

   9. Applied rewrites 61.2%

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

   11. Applied rewrites 61.2%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-73) (/ y (fma x x x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-73) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-73)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.3e-73], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2999999999999999e-73

   1. Initial program 68.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if 4.2999999999999999e-73 < y

   1. Initial program 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 81.1

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 81.1

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

   4. Applied rewrites 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 70.9

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

   9. Applied rewrites 70.9%

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

   11. Applied rewrites 70.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-73) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-73) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-73)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.3e-73], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2999999999999999e-73

   1. Initial program 68.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if 4.2999999999999999e-73 < y

   1. Initial program 77.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 70.9

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

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7500:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7500.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7500.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7500.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7500.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7500

   1. Initial program 72.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -7500 < x

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.1

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

   5. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 48.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.0

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 42.1

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

   9. Applied rewrites 42.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 28.5%

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

   if 1 < y

   1. Initial program 74.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

Alternative 15: 27.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 74.8

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

   6. lift-+.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. associate-+l+ N/A

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

   11. lower-+.f64 N/A

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

   12. lower-+.f64 74.8

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

4. Applied rewrites 74.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 50.5

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

9. Applied rewrites 50.5%

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

10. Taylor expanded in y around 0

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

12. Applied rewrites 28.6%

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

Developer Target 1: 99.8% accurate, 0.6× 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 2024233 
(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))))