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

Percentage Accurate: 68.9% → 99.8%

Time: 10.8s

Alternatives: 12

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: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+162)
   (/ (/ y x) x)
   (if (<= x -1.7e-11)
     (* 1.0 (/ y (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ x (* (+ 1.0 y) (+ y x))) (/ y (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -1.7e-11) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / ((1.0 + y) * (y + x))) * (y / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+162)) then
        tmp = (y / x) / x
    else if (x <= (-1.7d-11)) then
        tmp = 1.0d0 * (y / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (x / ((1.0d0 + y) * (y + x))) * (y / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -1.7e-11) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / ((1.0 + y) * (y + x))) * (y / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+162:
		tmp = (y / x) / x
	elif x <= -1.7e-11:
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (x / ((1.0 + y) * (y + x))) * (y / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+162)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.7e-11)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(Float64(1.0 + y) * Float64(y + x))) * Float64(y / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+162)
		tmp = (y / x) / x;
	elseif (x <= -1.7e-11)
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (x / ((1.0 + y) * (y + x))) * (y / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+162], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.7e-11], N[(1.0 * N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45000000000000017e162

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 58.6

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 96.2%

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

   if -2.45000000000000017e162 < x < -1.6999999999999999e-11

   1. Initial program 60.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 95.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 81.7%

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

   if -1.6999999999999999e-11 < x

   1. Initial program 76.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 86.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+162)
   (/ (/ y x) x)
   (* (/ y (* (+ 1.0 (+ y x)) (+ y x))) (/ x (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * (x / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+162)) then
        tmp = (y / x) / x
    else
        tmp = (y / ((1.0d0 + (y + x)) * (y + x))) * (x / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else {
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * (x / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+162:
		tmp = (y / x) / x
	else:
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * (x / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+162)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))) * Float64(x / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+162)
		tmp = (y / x) / x;
	else
		tmp = (y / ((1.0 + (y + x)) * (y + x))) * (x / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+162], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.45000000000000017e162

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 58.6

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 96.2%

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

   if -2.45000000000000017e162 < x

   1. Initial program 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.2%

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

Alternative 4: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-192}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot \frac{x}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+162)
   (/ (/ y x) x)
   (if (<= x -6e-192)
     (* 1.0 (/ y (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ 1.0 (+ 1.0 y)) (/ x (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -6e-192) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (y + x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+162)) then
        tmp = (y / x) / x
    else if (x <= (-6d-192)) then
        tmp = 1.0d0 * (y / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (1.0d0 / (1.0d0 + y)) * (x / (y + x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -6e-192) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (1.0 + y)) * (x / (y + x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+162:
		tmp = (y / x) / x
	elif x <= -6e-192:
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (1.0 / (1.0 + y)) * (x / (y + x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+162)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -6e-192)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + y)) * Float64(x / Float64(y + x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+162)
		tmp = (y / x) / x;
	elseif (x <= -6e-192)
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (1.0 / (1.0 + y)) * (x / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+162], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -6e-192], N[(1.0 * N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45000000000000017e162

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 58.6

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 96.2%

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

   if -2.45000000000000017e162 < x < -5.9999999999999998e-192

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.9%

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

   if -5.9999999999999998e-192 < x

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 61.5

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

   7. Applied rewrites 61.5%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-192}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + y} \cdot \frac{x}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-192}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.45e+162)
   (/ (/ y x) x)
   (if (<= x -6e-192)
     (* 1.0 (/ y (* (+ 1.0 (+ y x)) (+ y x))))
     (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -6e-192) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.45d+162)) then
        tmp = (y / x) / x
    else if (x <= (-6d-192)) then
        tmp = 1.0d0 * (y / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45e+162) {
		tmp = (y / x) / x;
	} else if (x <= -6e-192) {
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.45e+162:
		tmp = (y / x) / x
	elif x <= -6e-192:
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.45e+162)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -6e-192)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45e+162)
		tmp = (y / x) / x;
	elseif (x <= -6e-192)
		tmp = 1.0 * (y / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.45e+162], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -6e-192], N[(1.0 * N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45000000000000017e162

   1. Initial program 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 58.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 58.6

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   9. Applied rewrites 96.2%

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

   if -2.45000000000000017e162 < x < -5.9999999999999998e-192

   1. Initial program 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.9%

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

   if -5.9999999999999998e-192 < x

   1. Initial program 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 59.9

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

   7. Applied rewrites 59.9%

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

   9. Applied rewrites 61.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-192}:\\ \;\;\;\;1 \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+70)
   (/ (/ y x) x)
   (if (<= x -5.8e-97) (/ y (fma x x x)) (/ (/ x y) (+ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+70) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-97) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.5e+70)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-97)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.5e+70], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-97], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999988e70

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 54.3

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 81.3%

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

   if -1.49999999999999988e70 < x < -5.7999999999999999e-97

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 46.0

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

   5. Applied rewrites 46.0%

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

   if -5.7999999999999999e-97 < x

   1. Initial program 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 74.8

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.0

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

   7. Applied rewrites 61.0%

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

   9. Applied rewrites 62.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.75 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -2600.0)
     (/ y (* x x))
     (if (<= x -4.75e-132) t_0 (if (<= x 6e-204) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -2600.0) {
		tmp = y / (x * x);
	} else if (x <= -4.75e-132) {
		tmp = t_0;
	} else if (x <= 6e-204) {
		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 <= (-2600.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-4.75d-132)) then
        tmp = t_0
    else if (x <= 6d-204) 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 <= -2600.0) {
		tmp = y / (x * x);
	} else if (x <= -4.75e-132) {
		tmp = t_0;
	} else if (x <= 6e-204) {
		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 <= -2600.0:
		tmp = y / (x * x)
	elif x <= -4.75e-132:
		tmp = t_0
	elif x <= 6e-204:
		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 <= -2600.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -4.75e-132)
		tmp = t_0;
	elseif (x <= 6e-204)
		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 <= -2600.0)
		tmp = y / (x * x);
	elseif (x <= -4.75e-132)
		tmp = t_0;
	elseif (x <= 6e-204)
		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, -2600.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.75e-132], t$95$0, If[LessEqual[x, 6e-204], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2600

   1. Initial program 57.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -2600 < x < -4.74999999999999993e-132 or 5.9999999999999997e-204 < x

   1. Initial program 80.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

   if -4.74999999999999993e-132 < x < 5.9999999999999997e-204

   1. Initial program 68.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 68.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 68.5

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 77.2%

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

Alternative 8: 80.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+70)
   (/ (/ y x) x)
   (if (<= x -5.8e-97) (/ y (fma x x x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+70) {
		tmp = (y / x) / x;
	} else if (x <= -5.8e-97) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.5e+70)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.8e-97)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.5e+70], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.8e-97], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.49999999999999988e70

   1. Initial program 54.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 54.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 54.3

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

   4. Applied rewrites 54.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.2

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

   7. Applied rewrites 75.2%

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

   9. Applied rewrites 81.3%

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

   if -1.49999999999999988e70 < x < -5.7999999999999999e-97

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 46.0

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

   5. Applied rewrites 46.0%

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

   if -5.7999999999999999e-97 < x

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.0

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

   5. Applied rewrites 61.0%

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

Alternative 9: 78.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-97) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-97) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.8e-97)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.8e-97], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999999e-97

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if -5.7999999999999999e-97 < x

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.0

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

   5. Applied rewrites 61.0%

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

Alternative 10: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2600:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2600.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2600.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2600.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2600.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2600

   1. Initial program 57.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -2600 < x

   1. Initial program 77.1%

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

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 61.2

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

   5. Applied rewrites 61.2%

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

Alternative 11: 47.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 74.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 44.2

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 24.1%

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

   if 1 < y

   1. Initial program 70.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

Alternative 12: 26.8% 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 73.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 73.0

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 73.0

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

4. Applied rewrites 73.0%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 53.8

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

7. Applied rewrites 53.8%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 27.2%

    \[\leadsto \frac{x}{\color{blue}{y}} \]
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 2024255 
(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))))