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

Percentage Accurate: 69.0% → 99.8%

Time: 14.6s

Alternatives: 20

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{t\_0}{y}}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y -1.4e-64)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 9.8e+151)
       (/ (* x (/ y (+ y x))) (* (+ y x) t_0))
       (/ (/ x (* (+ y x) (/ t_0 y))) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= -1.4e-64) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 9.8e+151) {
		tmp = (x * (y / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = (x / ((y + x) * (t_0 / y))) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= (-1.4d-64)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 9.8d+151) then
        tmp = (x * (y / (y + x))) / ((y + x) * t_0)
    else
        tmp = (x / ((y + x) * (t_0 / y))) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= -1.4e-64) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 9.8e+151) {
		tmp = (x * (y / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = (x / ((y + x) * (t_0 / y))) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= -1.4e-64:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 9.8e+151:
		tmp = (x * (y / (y + x))) / ((y + x) * t_0)
	else:
		tmp = (x / ((y + x) * (t_0 / y))) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= -1.4e-64)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 9.8e+151)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(t_0 / y))) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= -1.4e-64)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 9.8e+151)
		tmp = (x * (y / (y + x))) / ((y + x) * t_0);
	else
		tmp = (x / ((y + x) * (t_0 / y))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -1.4e-64], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e+151], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.40000000000000002e-64

   1. Initial program 67.3%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 27.9

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

   7. Simplified 27.9%

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

   if -1.40000000000000002e-64 < y < 9.7999999999999998e151

   1. Initial program 77.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 98.6

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

   4. Applied egg-rr 98.6%

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

   if 9.7999999999999998e151 < y

   1. Initial program 58.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

      1. *-rgt-identity N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. +-lowering-+.f64 99.9

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 90.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\left(y + x\right) \cdot \frac{\left(y + x\right) + 1}{y}}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -4.6e+156)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.6e-162)
       (/ (* y (/ x (* (+ y x) (+ y x)))) t_0)
       (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -4.6e+156) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.6e-162) {
		tmp = (y * (x / ((y + x) * (y + x)))) / t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-4.6d+156)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.6d-162)) then
        tmp = (y * (x / ((y + x) * (y + x)))) / t_0
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -4.6e+156) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.6e-162) {
		tmp = (y * (x / ((y + x) * (y + x)))) / t_0;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -4.6e+156:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.6e-162:
		tmp = (y * (x / ((y + x) * (y + x)))) / t_0
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -4.6e+156)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.6e-162)
		tmp = Float64(Float64(y * Float64(x / Float64(Float64(y + x) * Float64(y + x)))) / t_0);
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -4.6e+156)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.6e-162)
		tmp = (y * (x / ((y + x) * (y + x)))) / t_0;
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -4.6e+156], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-162], N[(N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999998e156

   1. Initial program 60.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 92.4%

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

   if -4.5999999999999998e156 < x < -1.59999999999999988e-162

   1. Initial program 73.9%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 94.6

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

   4. Applied egg-rr 94.6%

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

   if -1.59999999999999988e-162 < x

   1. Initial program 73.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 59.0

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

   7. Simplified 59.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 58.5%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.2e+54)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.6e-162)
       (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
       (/ (/ x (+ y 1.0)) y)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+54) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.6e-162) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.2d+54)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.6d-162)) then
        tmp = x * (y / (t_0 * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.2e+54) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.6e-162) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.2e+54:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.6e-162:
		tmp = x * (y / (t_0 * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.2e+54)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.6e-162)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.2e+54)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.6e-162)
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.2e+54], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-162], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999999e54

   1. Initial program 56.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 77.5%

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

   if -1.19999999999999999e54 < x < -1.59999999999999988e-162

   1. Initial program 88.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 94.0

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

   4. Applied egg-rr 94.0%

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

   if -1.59999999999999988e-162 < x

   1. Initial program 73.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 59.0

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

   7. Simplified 59.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 58.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e+81)
   (/ (/ y x) (+ y x))
   (if (<= x -2.3e-137)
     (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+81) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-137) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d+81)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-2.3d-137)) then
        tmp = y * (x / (((y + x) + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e+81) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-137) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.4e+81:
		tmp = (y / x) / (y + x)
	elif x <= -2.3e-137:
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.4e+81)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -2.3e-137)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4e+81)
		tmp = (y / x) / (y + x);
	elseif (x <= -2.3e-137)
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.4e+81], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-137], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.39999999999999997e81

   1. Initial program 56.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 80.7

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

   7. Simplified 80.7%

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

   if -1.39999999999999997e81 < x < -2.30000000000000008e-137

   1. Initial program 84.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 91.8

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

   4. Applied egg-rr 91.8%

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

   if -2.30000000000000008e-137 < x

   1. Initial program 73.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 60.0

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

   7. Simplified 60.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 59.6%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot t\_0}\\ \end{array} \end{array} \]

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 -2.2e+160)
     (/ (/ y t_0) (+ y x))
     (/ (* x (/ y (+ y x))) (* (+ y x) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -2.2e+160) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * 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 <= (-2.2d+160)) then
        tmp = (y / t_0) / (y + x)
    else
        tmp = (x * (y / (y + x))) / ((y + x) * 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 <= -2.2e+160) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -2.2e+160:
		tmp = (y / t_0) / (y + x)
	else:
		tmp = (x * (y / (y + x))) / ((y + x) * t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -2.2e+160)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * 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 <= -2.2e+160)
		tmp = (y / t_0) / (y + x);
	else
		tmp = (x * (y / (y + x))) / ((y + x) * 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -2.2e+160], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999992e160

   1. Initial program 62.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 95.4%

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

   if -2.19999999999999992e160 < x

   1. Initial program 73.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 94.7

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

   4. Applied egg-rr 94.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. times-frac N/A

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

   2. associate-*l/ N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.95e+15)
   (/ (/ y x) (+ y x))
   (if (<= x -5.8e-92) (/ y (fma x x x)) (/ (/ x (+ y 1.0)) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.95e+15) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5.8e-92) {
		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 <= -2.95e+15)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5.8e-92)
		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, -2.95e+15], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.8e-92], 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 < -2.95e15

   1. Initial program 58.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 73.1

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

   7. Simplified 73.1%

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

   if -2.95e15 < x < -5.79999999999999969e-92

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 48.4

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

   5. Simplified 48.4%

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

   if -5.79999999999999969e-92 < x

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.7

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 61.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+131)
   (/ (/ y x) x)
   (if (<= x -5.8e-92) (/ 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+131) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-92) {
		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+131)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-92)
		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+131], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-92], 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 < -4.99999999999999995e131

   1. Initial program 56.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 78.5

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

   4. Applied egg-rr 78.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. *-lowering-*.f64 80.6

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

   7. Simplified 80.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 83.7

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

   9. Applied egg-rr 83.7%

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

   if -4.99999999999999995e131 < x < -5.79999999999999969e-92

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0

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

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

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

      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. accelerator-lowering-fma.f64 51.1

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

   5. Simplified 51.1%

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

   if -5.79999999999999969e-92 < x

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.7

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 61.3%

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

Alternative 10: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-92) (/ (/ y (+ (+ y x) 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-92) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.5d-92)) then
        tmp = (y / ((y + x) + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-92) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.5e-92:
		tmp = (y / ((y + x) + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-92)
		tmp = Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-92)
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.5e-92], N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e-92

   1. Initial program 66.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 67.6%

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

   if -3.5e-92 < x

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.7

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 61.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -0.00045:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -0.00045)
     (/ y (* x x))
     (if (<= x -9.6e-218) t_0 (if (<= x 9.8e-228) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -0.00045) {
		tmp = y / (x * x);
	} else if (x <= -9.6e-218) {
		tmp = t_0;
	} else if (x <= 9.8e-228) {
		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 <= (-0.00045d0)) then
        tmp = y / (x * x)
    else if (x <= (-9.6d-218)) then
        tmp = t_0
    else if (x <= 9.8d-228) 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 <= -0.00045) {
		tmp = y / (x * x);
	} else if (x <= -9.6e-218) {
		tmp = t_0;
	} else if (x <= 9.8e-228) {
		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 <= -0.00045:
		tmp = y / (x * x)
	elif x <= -9.6e-218:
		tmp = t_0
	elif x <= 9.8e-228:
		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 <= -0.00045)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -9.6e-218)
		tmp = t_0;
	elseif (x <= 9.8e-228)
		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 <= -0.00045)
		tmp = y / (x * x);
	elseif (x <= -9.6e-218)
		tmp = t_0;
	elseif (x <= 9.8e-228)
		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, -0.00045], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.6e-218], t$95$0, If[LessEqual[x, 9.8e-228], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4999999999999999e-4

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 69.2

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

   5. Simplified 69.2%

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

   if -4.4999999999999999e-4 < x < -9.6000000000000003e-218 or 9.79999999999999976e-228 < x

   1. Initial program 79.9%

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

   3. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 43.0

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

   5. Simplified 43.0%

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

   if -9.6000000000000003e-218 < x < 9.79999999999999976e-228

   1. Initial program 58.2%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. accelerator-lowering-fma.f64 58.3

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

   5. Simplified 58.3%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 81.2

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

   8. Simplified 81.2%

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

Alternative 12: 83.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-92) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-92) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.8d-92)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-92) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.8e-92:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.8e-92)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e-92)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.8e-92], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999969e-92

   1. Initial program 66.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 67.0

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

   7. Simplified 67.0%

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

   if -5.79999999999999969e-92 < x

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 61.7

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 61.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+131)
   (/ (/ y x) x)
   (if (<= x -5.8e-92) (/ y (fma x x x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+131) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-92) {
		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 (x <= -5e+131)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-92)
		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[x, -5e+131], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-92], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999995e131

   1. Initial program 56.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 78.5

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

   4. Applied egg-rr 78.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   6. Step-by-step derivation

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

         \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. *-lowering-*.f64 80.6

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

   7. Simplified 80.6%

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

      1. un-div-inv N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 83.7

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

   9. Applied egg-rr 83.7%

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

   if -4.99999999999999995e131 < x < -5.79999999999999969e-92

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0

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

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

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

      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. accelerator-lowering-fma.f64 51.1

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

   5. Simplified 51.1%

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

   if -5.79999999999999969e-92 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 61.5

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

   5. Simplified 61.5%

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

Alternative 14: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.3e-93) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.3e-93) {
		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 (x <= -7.3e-93)
		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[x, -7.3e-93], 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 x < -7.29999999999999976e-93

   1. Initial program 66.2%

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

   3. Taylor expanded in y around 0

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

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

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

      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. accelerator-lowering-fma.f64 65.1

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

   5. Simplified 65.1%

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

   if -7.29999999999999976e-93 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 61.5

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

   5. Simplified 61.5%

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

Alternative 15: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2200:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2200.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2200.0) {
		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 <= -2200.0)
		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, -2200.0], 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 < -2200

   1. Initial program 61.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 69.2

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

   5. Simplified 69.2%

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

   if -2200 < x

   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. Taylor expanded in x around 0

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

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

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

      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. accelerator-lowering-fma.f64 61.3

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

   5. Simplified 61.3%

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

Alternative 16: 46.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2400000:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2400000.0) (/ x (+ y x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2400000.0) {
		tmp = x / (y + x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2400000.0d0) then
        tmp = x / (y + x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2400000.0) {
		tmp = x / (y + x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2400000.0:
		tmp = x / (y + x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2400000.0)
		tmp = Float64(x / Float64(y + x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2400000.0)
		tmp = x / (y + x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 2400000.0], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e6

   1. Initial program 74.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 45.2

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

   7. Simplified 45.2%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 24.3%

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

   if 2.4e6 < y

   1. Initial program 65.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 73.3

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

   5. Simplified 73.3%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2400000:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.9%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.8

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   3. +-lowering-+.f64 52.3

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

7. Simplified 52.3%

   \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]

8. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{x}}{x + y} \]
9. Step-by-step derivation

10. Simplified 23.7%

    \[\leadsto \frac{\color{blue}{x}}{x + y} \]

11. Final simplification 23.7%

    \[\leadsto \frac{x}{y + x} \]
12. Add Preprocessing

Alternative 18: 26.0% 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 71.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{{y}^{2} \cdot \left(1 + y\right)}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + y\right)} \]

   2. associate-*l* N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{y \cdot \left(y \cdot \left(1 + y\right)\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{y \cdot \left(y \cdot \left(1 + y\right)\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{x \cdot y}{y \cdot \left(y \cdot \color{blue}{\left(y + 1\right)}\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto \frac{x \cdot y}{y \cdot \color{blue}{\left(y \cdot y + y \cdot 1\right)}} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{x \cdot y}{y \cdot \left(y \cdot y + \color{blue}{y}\right)} \]

   7. accelerator-lowering-fma.f64 35.9

      \[\leadsto \frac{x \cdot y}{y \cdot \color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

5. Simplified 35.9%

   \[\leadsto \frac{x \cdot y}{\color{blue}{y \cdot \mathsf{fma}\left(y, y, y\right)}} \]

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 23.3

      \[\leadsto \color{blue}{\frac{x}{y}} \]

8. Simplified 23.3%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 19: 4.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{0.5}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 0.5 y))

C

assert(x < y);
double code(double x, double y) {
	return 0.5 / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 0.5 / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 0.5 / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{\left(2 \cdot \left(x \cdot y\right) + {x}^{2}\right)} \cdot \left(\left(x + y\right) + 1\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(2 \cdot \color{blue}{\left(y \cdot x\right)} + {x}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(2 \cdot y\right) \cdot x} + {x}^{2}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(2 \cdot y\right) \cdot x + \color{blue}{x \cdot x}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot \left(2 \cdot y + x\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(x \cdot \color{blue}{\left(x + 2 \cdot y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot \left(x + 2 \cdot y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(x \cdot \color{blue}{\left(2 \cdot y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{x \cdot y}{\left(x \cdot \left(\color{blue}{y \cdot 2} + x\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. accelerator-lowering-fma.f64 45.9

      \[\leadsto \frac{x \cdot y}{\left(x \cdot \color{blue}{\mathsf{fma}\left(y, 2, x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

5. Simplified 45.9%

   \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot \mathsf{fma}\left(y, 2, x\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 4.2

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

8. Simplified 4.2%

   \[\leadsto \color{blue}{\frac{0.5}{y}} \]
9. Add Preprocessing

Alternative 20: 3.5% accurate, 39.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 71.9%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.8

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   3. +-lowering-+.f64 52.3

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

7. Simplified 52.3%

   \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 3.3%

    \[\leadsto \color{blue}{1} \]
11. 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 2024205 
(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))))