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

Percentage Accurate: 69.0% → 99.8%

Time: 17.2s

Alternatives: 16

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

Alternative 2: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{t\_0}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y -8e+107)
     (/ (/ y t_0) (+ x y))
     (if (<= y 7.2e+100)
       (* (/ x (+ x y)) (/ y (* (+ x y) t_0)))
       (/ (/ x y) (* (+ x y) (/ t_0 y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_0) / (x + y);
	} else if (y <= 7.2e+100) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= (-8d+107)) then
        tmp = (y / t_0) / (x + y)
    else if (y <= 7.2d+100) then
        tmp = (x / (x + y)) * (y / ((x + y) * t_0))
    else
        tmp = (x / y) / ((x + y) * (t_0 / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_0) / (x + y);
	} else if (y <= 7.2e+100) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= -8e+107:
		tmp = (y / t_0) / (x + y)
	elif y <= 7.2e+100:
		tmp = (x / (x + y)) * (y / ((x + y) * t_0))
	else:
		tmp = (x / y) / ((x + y) * (t_0 / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -8e+107)
		tmp = Float64(Float64(y / t_0) / Float64(x + y));
	elseif (y <= 7.2e+100)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + y) * Float64(t_0 / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -8e+107)
		tmp = (y / t_0) / (x + y);
	elseif (y <= 7.2e+100)
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	else
		tmp = (x / y) / ((x + y) * (t_0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -8e+107], N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+100], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.9999999999999998e107

   1. Initial program 59.3%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 22.6%

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

   if -7.9999999999999998e107 < y < 7.2e100

   1. Initial program 75.6%

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

      1. *-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. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   if 7.2e100 < y

   1. Initial program 57.3%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 91.9

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

   9. Simplified 91.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{\left(x + y\right) + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{x + y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+176}:\\ \;\;\;\;\frac{y \cdot t\_0}{\left(x + y\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ x y) 1.0)))
   (if (<= y -8e+107)
     (/ (/ y t_1) (+ x y))
     (if (<= y 2.25e+176)
       (/ (* y t_0) (* (+ x y) t_1))
       (* t_0 (/ 1.0 (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 2.25e+176) {
		tmp = (y * t_0) / ((x + y) * t_1);
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = (x + y) + 1.0d0
    if (y <= (-8d+107)) then
        tmp = (y / t_1) / (x + y)
    else if (y <= 2.25d+176) then
        tmp = (y * t_0) / ((x + y) * t_1)
    else
        tmp = t_0 * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 2.25e+176) {
		tmp = (y * t_0) / ((x + y) * t_1);
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = (x + y) + 1.0
	tmp = 0
	if y <= -8e+107:
		tmp = (y / t_1) / (x + y)
	elif y <= 2.25e+176:
		tmp = (y * t_0) / ((x + y) * t_1)
	else:
		tmp = t_0 * (1.0 / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -8e+107)
		tmp = Float64(Float64(y / t_1) / Float64(x + y));
	elseif (y <= 2.25e+176)
		tmp = Float64(Float64(y * t_0) / Float64(Float64(x + y) * t_1));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -8e+107)
		tmp = (y / t_1) / (x + y);
	elseif (y <= 2.25e+176)
		tmp = (y * t_0) / ((x + y) * t_1);
	else
		tmp = t_0 * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -8e+107], N[(N[(y / t$95$1), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+176], N[(N[(y * t$95$0), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.9999999999999998e107

   1. Initial program 59.3%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 22.6%

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

   if -7.9999999999999998e107 < y < 2.25000000000000002e176

   1. Initial program 73.3%

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

      1. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 98.2

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

   6. Applied egg-rr 98.2%

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

   if 2.25000000000000002e176 < y

   1. Initial program 61.8%

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

      1. times-frac N/A

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

      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. Taylor expanded in y around inf

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

   7. Simplified 91.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+176}:\\ \;\;\;\;\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{x + y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+176}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ x y) 1.0)))
   (if (<= y -8e+107)
     (/ (/ y t_1) (+ x y))
     (if (<= y 2.1e+176)
       (* t_0 (/ y (* (+ x y) t_1)))
       (* t_0 (/ 1.0 (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 2.1e+176) {
		tmp = t_0 * (y / ((x + y) * t_1));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = (x + y) + 1.0d0
    if (y <= (-8d+107)) then
        tmp = (y / t_1) / (x + y)
    else if (y <= 2.1d+176) then
        tmp = t_0 * (y / ((x + y) * t_1))
    else
        tmp = t_0 * (1.0d0 / (x + y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -8e+107) {
		tmp = (y / t_1) / (x + y);
	} else if (y <= 2.1e+176) {
		tmp = t_0 * (y / ((x + y) * t_1));
	} else {
		tmp = t_0 * (1.0 / (x + y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	t_1 = (x + y) + 1.0
	tmp = 0
	if y <= -8e+107:
		tmp = (y / t_1) / (x + y)
	elif y <= 2.1e+176:
		tmp = t_0 * (y / ((x + y) * t_1))
	else:
		tmp = t_0 * (1.0 / (x + y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -8e+107)
		tmp = Float64(Float64(y / t_1) / Float64(x + y));
	elseif (y <= 2.1e+176)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * t_1)));
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(x + y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -8e+107)
		tmp = (y / t_1) / (x + y);
	elseif (y <= 2.1e+176)
		tmp = t_0 * (y / ((x + y) * t_1));
	else
		tmp = t_0 * (1.0 / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -8e+107], N[(N[(y / t$95$1), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+176], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.9999999999999998e107

   1. Initial program 59.3%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 22.6%

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

   if -7.9999999999999998e107 < y < 2.0999999999999999e176

   1. Initial program 73.3%

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

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

      4. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 98.2

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

   4. Applied egg-rr 98.2%

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

   if 2.0999999999999999e176 < y

   1. Initial program 61.8%

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

      1. times-frac N/A

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

      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. Taylor expanded in y around inf

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

   7. Simplified 91.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+102)
   (/ (/ y (+ (+ x y) 1.0)) (+ x y))
   (if (<= x -3.8e-162)
     (* y (/ x (* (+ x y) (* (+ x y) (+ x (+ y 1.0))))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	} else if (x <= -3.8e-162) {
		tmp = y * (x / ((x + y) * ((x + y) * (x + (y + 1.0)))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d+102)) then
        tmp = (y / ((x + y) + 1.0d0)) / (x + y)
    else if (x <= (-3.8d-162)) then
        tmp = y * (x / ((x + y) * ((x + y) * (x + (y + 1.0d0)))))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	} else if (x <= -3.8e-162) {
		tmp = y * (x / ((x + y) * ((x + y) * (x + (y + 1.0)))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.5e+102:
		tmp = (y / ((x + y) + 1.0)) / (x + y)
	elif x <= -3.8e-162:
		tmp = y * (x / ((x + y) * ((x + y) * (x + (y + 1.0)))))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+102)
		tmp = Float64(Float64(y / Float64(Float64(x + y) + 1.0)) / Float64(x + y));
	elseif (x <= -3.8e-162)
		tmp = Float64(y * Float64(x / Float64(Float64(x + y) * Float64(Float64(x + y) * Float64(x + Float64(y + 1.0))))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+102)
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	elseif (x <= -3.8e-162)
		tmp = y * (x / ((x + y) * ((x + y) * (x + (y + 1.0)))));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.5e+102], N[(N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-162], N[(y * N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.49999999999999981e102

   1. Initial program 47.2%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 85.9%

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

   if -5.49999999999999981e102 < x < -3.80000000000000005e-162

   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. 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. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.4

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

   6. Applied egg-rr 99.4%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. times-frac N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

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

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

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

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

      17. associate-+l+ N/A

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

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

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

      19. +-lowering-+.f64 96.5

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

   8. Applied egg-rr 96.5%

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

   if -3.80000000000000005e-162 < x

   1. Initial program 70.8%

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

      1. times-frac N/A

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

      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. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 54.5

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

   9. Simplified 54.5%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= x -5.5e+102)
     (/ (/ y t_0) (+ x y))
     (if (<= x -3.7e-164)
       (* y (/ x (* t_0 (* (+ x y) (+ x y)))))
       (/ (/ x (+ x y)) (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (y / t_0) / (x + y);
	} else if (x <= -3.7e-164) {
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (x <= (-5.5d+102)) then
        tmp = (y / t_0) / (x + y)
    else if (x <= (-3.7d-164)) then
        tmp = y * (x / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -5.5e+102) {
		tmp = (y / t_0) / (x + y);
	} else if (x <= -3.7e-164) {
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if x <= -5.5e+102:
		tmp = (y / t_0) / (x + y)
	elif x <= -3.7e-164:
		tmp = y * (x / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (x <= -5.5e+102)
		tmp = Float64(Float64(y / t_0) / Float64(x + y));
	elseif (x <= -3.7e-164)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (x <= -5.5e+102)
		tmp = (y / t_0) / (x + y);
	elseif (x <= -3.7e-164)
		tmp = y * (x / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -5.5e+102], N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.7e-164], N[(y * N[(x / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.49999999999999981e102

   1. Initial program 47.2%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 85.9%

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

   if -5.49999999999999981e102 < x < -3.6999999999999999e-164

   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 96.5

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

   4. Applied egg-rr 96.5%

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

   if -3.6999999999999999e-164 < x

   1. Initial program 70.8%

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

      1. times-frac N/A

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

      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. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 54.5

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

   9. Simplified 54.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-164}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+29)
   (/ (/ y (+ (+ x y) 1.0)) (+ x y))
   (if (<= x -3.5e-159)
     (* x (/ y (* (+ y 1.0) (* (+ x y) (+ x y)))))
     (/ (/ x (+ x y)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+29) {
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	} else if (x <= -3.5e-159) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d+29)) then
        tmp = (y / ((x + y) + 1.0d0)) / (x + y)
    else if (x <= (-3.5d-159)) then
        tmp = x * (y / ((y + 1.0d0) * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+29) {
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	} else if (x <= -3.5e-159) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e+29:
		tmp = (y / ((x + y) + 1.0)) / (x + y)
	elif x <= -3.5e-159:
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+29)
		tmp = Float64(Float64(y / Float64(Float64(x + y) + 1.0)) / Float64(x + y));
	elseif (x <= -3.5e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+29)
		tmp = (y / ((x + y) + 1.0)) / (x + y);
	elseif (x <= -3.5e-159)
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e+29], N[(N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-159], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999991e29

   1. Initial program 51.5%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 79.3%

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

   if -1.69999999999999991e29 < x < -3.50000000000000002e-159

   1. Initial program 88.5%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 93.3%

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

   if -3.50000000000000002e-159 < x

   1. Initial program 70.8%

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

      1. times-frac N/A

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

      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. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 54.5

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

   9. Simplified 54.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+29}:\\ \;\;\;\;t\_0 \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -6.8e+29)
     (* t_0 (/ y (* x x)))
     (if (<= x -3.5e-159)
       (* x (/ y (* (+ y 1.0) (* (+ x y) (+ x y)))))
       (/ t_0 (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -6.8e+29) {
		tmp = t_0 * (y / (x * x));
	} else if (x <= -3.5e-159) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (x <= (-6.8d+29)) then
        tmp = t_0 * (y / (x * x))
    else if (x <= (-3.5d-159)) then
        tmp = x * (y / ((y + 1.0d0) * ((x + y) * (x + y))))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -6.8e+29) {
		tmp = t_0 * (y / (x * x));
	} else if (x <= -3.5e-159) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -6.8e+29:
		tmp = t_0 * (y / (x * x))
	elif x <= -3.5e-159:
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -6.8e+29)
		tmp = Float64(t_0 * Float64(y / Float64(x * x)));
	elseif (x <= -3.5e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -6.8e+29)
		tmp = t_0 * (y / (x * x));
	elseif (x <= -3.5e-159)
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e+29], N[(t$95$0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-159], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999963e29

   1. Initial program 51.5%

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.8

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

   7. Simplified 75.8%

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

   if -6.79999999999999963e29 < x < -3.50000000000000002e-159

   1. Initial program 88.5%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 93.3%

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

   if -3.50000000000000002e-159 < x

   1. Initial program 70.8%

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

      1. times-frac N/A

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

      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. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 54.5

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

   9. Simplified 54.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (fma y y y))))
   (if (<= x -6e+29)
     (/ y (* x x))
     (if (<= x -6e-37) t_0 (if (<= x -8e-115) (/ y (fma x x x)) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / fma(y, y, y);
	double tmp;
	if (x <= -6e+29) {
		tmp = y / (x * x);
	} else if (x <= -6e-37) {
		tmp = t_0;
	} else if (x <= -8e-115) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / fma(y, y, y))
	tmp = 0.0
	if (x <= -6e+29)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -6e-37)
		tmp = t_0;
	elseif (x <= -8e-115)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = t_0;
	end
	return 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 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+29], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-37], t$95$0, If[LessEqual[x, -8e-115], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.9999999999999998e29

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 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 75.6

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

   5. Simplified 75.6%

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

   if -5.9999999999999998e29 < x < -6e-37 or -8.0000000000000004e-115 < x

   1. Initial program 74.1%

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

   3. Taylor expanded in x around 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 57.5

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

   5. Simplified 57.5%

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

   if -6e-37 < x < -8.0000000000000004e-115

   1. Initial program 79.6%

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

   3. Taylor expanded in y around 0

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

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

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

      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 56.0

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

   5. Simplified 56.0%

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

Alternative 10: 66.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2e+36)
     (/ y (* x x))
     (if (<= x -1.1e-202) t_0 (if (<= x 1.45e-40) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -2e+36) {
		tmp = y / (x * x);
	} else if (x <= -1.1e-202) {
		tmp = t_0;
	} else if (x <= 1.45e-40) {
		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 <= (-2d+36)) then
        tmp = y / (x * x)
    else if (x <= (-1.1d-202)) then
        tmp = t_0
    else if (x <= 1.45d-40) 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 <= -2e+36) {
		tmp = y / (x * x);
	} else if (x <= -1.1e-202) {
		tmp = t_0;
	} else if (x <= 1.45e-40) {
		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 <= -2e+36:
		tmp = y / (x * x)
	elif x <= -1.1e-202:
		tmp = t_0
	elif x <= 1.45e-40:
		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 <= -2e+36)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.1e-202)
		tmp = t_0;
	elseif (x <= 1.45e-40)
		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 <= -2e+36)
		tmp = y / (x * x);
	elseif (x <= -1.1e-202)
		tmp = t_0;
	elseif (x <= 1.45e-40)
		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, -2e+36], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-202], t$95$0, If[LessEqual[x, 1.45e-40], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000008e36

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 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 75.6

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

   5. Simplified 75.6%

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

   if -2.00000000000000008e36 < x < -1.10000000000000004e-202 or 1.4499999999999999e-40 < x

   1. Initial program 75.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 40.7

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

   5. Simplified 40.7%

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

   if -1.10000000000000004e-202 < x < 1.4499999999999999e-40

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 66.5

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

   5. Simplified 66.5%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 63.1

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

   8. Simplified 63.1%

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

Alternative 11: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 7e-48) (/ y (fma x x x)) (/ (/ x (+ x y)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 7e-48) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 7e-48)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 7e-48], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999982e-48

   1. Initial program 70.5%

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

   3. Taylor expanded in y around 0

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

      1. /-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 59.2

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

   5. Simplified 59.2%

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

   if 6.99999999999999982e-48 < y

   1. Initial program 66.7%

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

      1. times-frac N/A

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

      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 99.7

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

   4. Applied egg-rr 99.7%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. +-lowering-+.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 73.4

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

   9. Simplified 73.4%

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

Alternative 12: 48.4% accurate, 1.3× speedup

Math

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

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 (<= y -1.0) t_0 (if (<= y 1.0) (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 (y <= (-1.0d0)) then
        tmp = t_0
    else if (y <= 1.0d0) 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 (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		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 y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		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 (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		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 (y <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		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[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 63.6%

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

   3. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.3

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

   5. Simplified 75.3%

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

   if -1 < y < 1

   1. Initial program 75.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 10.7

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

   5. Simplified 10.7%

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

   6. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 18.3

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

   8. Simplified 18.3%

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

Alternative 13: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e+31) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e+31) {
		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 <= -3.1e+31)
		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, -3.1e+31], 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 < -3.1000000000000002e31

   1. Initial program 51.5%

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

   3. Taylor expanded in x around 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 75.6

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

   5. Simplified 75.6%

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

   if -3.1000000000000002e31 < x

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0

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

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

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

      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 56.3

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

   5. Simplified 56.3%

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

Alternative 14: 26.7% 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 69.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{{y}^{2} \cdot \left(1 + y\right)}} \]
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 36.3

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

5. Simplified 36.3%

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

6. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 23.6

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

8. Simplified 23.6%

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

Alternative 15: 4.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. div-inv N/A

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

   2. clear-num N/A

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

   3. frac-times N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. +-lowering-+.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in y around inf

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

9. Simplified 41.1%

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

10. Taylor expanded in x around inf

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

12. Simplified 4.3%

    \[\leadsto \frac{\color{blue}{1}}{y} \]
13. Add Preprocessing

Alternative 16: 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 69.4%

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

   1. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   7. +-lowering-+.f64 69.4

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

4. Applied egg-rr 69.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. accelerator-lowering-fma.f64 37.5

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

7. Simplified 37.5%

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

8. Taylor expanded in y around 0

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

10. Simplified 3.7%

    \[\leadsto \color{blue}{1} \]
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 2024196 
(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))))