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

Percentage Accurate: 69.8% → 99.8%

Time: 8.1s

Alternatives: 17

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.6%

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

   1. 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. associate-/r* N/A

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

   19. +-commutative N/A

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

   20. lift-+.f64 N/A

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

   21. lift-/.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 65.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-129)
   (/ (/ y (+ 1.0 x)) (+ y x))
   (if (<= y 5e+133)
     (* 1.0 (/ x (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ x (+ y x)) (pow y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / (y + x)) * pow(y, -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-129) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (y <= 5d+133) then
        tmp = 1.0d0 * (x / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (x / (y + x)) * (y ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / (y + x)) * Math.pow(y, -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-129:
		tmp = (y / (1.0 + x)) / (y + x)
	elif y <= 5e+133:
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (x / (y + x)) * math.pow(y, -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-129)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (y <= 5e+133)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (y ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-129)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (y <= 5e+133)
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (x / (y + x)) * (y ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-129], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+133], N[(1.0 * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1e-129

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.6

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

   9. Applied rewrites 56.6%

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

   if 2.1e-129 < y < 4.99999999999999961e133

   1. Initial program 80.0%

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

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 91.8

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 71.9%

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

   if 4.99999999999999961e133 < y

   1. Initial program 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 y around inf

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

      1. lower-/.f64 86.5

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

   7. Applied rewrites 86.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y - y \cdot \frac{\mathsf{fma}\left(3, y, 1\right)}{x}}{x}}{x}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.9e+157)
   (/ (/ (- y (* y (/ (fma 3.0 y 1.0) x))) x) x)
   (if (<= x 8.8e+30)
     (* (/ y (+ y x)) (/ x (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ 1.0 (+ y x)) (/ x (+ (+ y x) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+157) {
		tmp = ((y - (y * (fma(3.0, y, 1.0) / x))) / x) / x;
	} else if (x <= 8.8e+30) {
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.9e+157)
		tmp = Float64(Float64(Float64(y - Float64(y * Float64(fma(3.0, y, 1.0) / x))) / x) / x);
	elseif (x <= 8.8e+30)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(x / Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.9e+157], N[(N[(N[(y - N[(y * N[(N[(3.0 * y + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.8e+30], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999971e157

   1. Initial program 39.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\frac{\frac{y - y \cdot \frac{\mathsf{fma}\left(3, y, 1\right)}{x}}{x}}{x}} \]

   if -3.89999999999999971e157 < x < 8.7999999999999999e30

   1. Initial program 75.6%

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

      1. 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. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 8.7999999999999999e30 < x

   1. Initial program 48.8%

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

      1. lift-/.f64 N/A

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

      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. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 80.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 80.5

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 80.5

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 35.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.8

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

   9. Applied rewrites 35.8%

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

Alternative 4: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.9e+157)
   (/ (/ y x) (+ y x))
   (if (<= x 8.8e+30)
     (* (/ y (+ y x)) (/ x (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ 1.0 (+ y x)) (/ x (+ (+ y x) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+157) {
		tmp = (y / x) / (y + x);
	} else if (x <= 8.8e+30) {
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.9d+157)) then
        tmp = (y / x) / (y + x)
    else if (x <= 8.8d+30) then
        tmp = (y / (y + x)) * (x / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (1.0d0 / (y + x)) * (x / ((y + x) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.9e+157) {
		tmp = (y / x) / (y + x);
	} else if (x <= 8.8e+30) {
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.9e+157:
		tmp = (y / x) / (y + x)
	elif x <= 8.8e+30:
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.9e+157)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= 8.8e+30)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(x / Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.9e+157)
		tmp = (y / x) / (y + x);
	elseif (x <= 8.8e+30)
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.9e+157], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e+30], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.89999999999999971e157

   1. Initial program 39.3%

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

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

      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 99.9

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

      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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 87.2

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

   9. Applied rewrites 87.2%

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

   if -3.89999999999999971e157 < x < 8.7999999999999999e30

   1. Initial program 75.6%

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

      1. 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. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 8.7999999999999999e30 < x

   1. Initial program 48.8%

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

      1. lift-/.f64 N/A

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

      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. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 80.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 80.5

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 80.5

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 80.5

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

   4. Applied rewrites 80.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 35.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 35.8

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

   9. Applied rewrites 35.8%

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

Alternative 5: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-132)
   (/ (/ y (+ 1.0 x)) (+ y x))
   (if (<= y 3.2e+101)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-132) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 3.2e+101) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.8d-132) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (y <= 3.2d+101) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-132) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 3.2e+101) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-132:
		tmp = (y / (1.0 + x)) / (y + x)
	elif y <= 3.2e+101:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-132)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (y <= 3.2e+101)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-132)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (y <= 3.2e+101)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-132], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+101], 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], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.80000000000000031e-132

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.6

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

   9. Applied rewrites 56.6%

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

   if 4.80000000000000031e-132 < y < 3.20000000000000005e101

   1. Initial program 91.0%

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

   if 3.20000000000000005e101 < y

   1. Initial program 41.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 82.0

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

   9. Applied rewrites 82.0%

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

Alternative 6: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.6%

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

   1. 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. Add Preprocessing

Alternative 7: 65.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{x}{\left(y + x\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-129)
   (/ (/ y (+ 1.0 x)) (+ y x))
   (if (<= y 5e+133)
     (* 1.0 (/ x (* (+ 1.0 (+ y x)) (+ y x))))
     (* (/ 1.0 (+ y x)) (/ x (+ (+ y x) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-129) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (y <= 5d+133) then
        tmp = 1.0d0 * (x / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (1.0d0 / (y + x)) * (x / ((y + x) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-129:
		tmp = (y / (1.0 + x)) / (y + x)
	elif y <= 5e+133:
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-129)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (y <= 5e+133)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(x / Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-129)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (y <= 5e+133)
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (1.0 / (y + x)) * (x / ((y + x) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-129], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+133], N[(1.0 * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1e-129

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.6

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

   9. Applied rewrites 56.6%

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

   if 2.1e-129 < y < 4.99999999999999961e133

   1. Initial program 80.0%

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

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 91.8

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 71.9%

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

   if 4.99999999999999961e133 < y

   1. Initial program 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 87.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 87.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 87.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 87.2

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 87.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 86.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 86.9

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

   9. Applied rewrites 86.9%

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

Alternative 8: 60.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 4600000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (fma x x x))))
   (if (<= y 4.8e-115)
     t_0
     (if (<= y 1.85e-51)
       (/ x (fma y y y))
       (if (<= y 4600000000.0) t_0 (/ (/ x y) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y / fma(x, x, x);
	double tmp;
	if (y <= 4.8e-115) {
		tmp = t_0;
	} else if (y <= 1.85e-51) {
		tmp = x / fma(y, y, y);
	} else if (y <= 4600000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / fma(x, x, x))
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = t_0;
	elseif (y <= 1.85e-51)
		tmp = Float64(x / fma(y, y, y));
	elseif (y <= 4600000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-115], t$95$0, If[LessEqual[y, 1.85e-51], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4600000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.80000000000000042e-115 or 1.84999999999999987e-51 < y < 4.6e9

   1. Initial program 66.3%

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

   3. Taylor expanded in 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 55.6

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

   5. Applied rewrites 55.6%

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

   if 4.80000000000000042e-115 < y < 1.84999999999999987e-51

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

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

      1. 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 53.5

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

   5. Applied rewrites 53.5%

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

   if 4.6e9 < y

   1. Initial program 54.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 99.7

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

      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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 80.3

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

   9. Applied rewrites 80.3%

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

Alternative 9: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;1 \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-129)
   (/ (/ y (+ 1.0 x)) (+ y x))
   (if (<= y 5e+133)
     (* 1.0 (/ x (* (+ 1.0 (+ y x)) (+ y x))))
     (/ (/ x (+ 1.0 y)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-129) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (y <= 5d+133) then
        tmp = 1.0d0 * (x / ((1.0d0 + (y + x)) * (y + x)))
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-129) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (y <= 5e+133) {
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-129:
		tmp = (y / (1.0 + x)) / (y + x)
	elif y <= 5e+133:
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)))
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-129)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (y <= 5e+133)
		tmp = Float64(1.0 * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-129)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (y <= 5e+133)
		tmp = 1.0 * (x / ((1.0 + (y + x)) * (y + x)));
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-129], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+133], N[(1.0 * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1e-129

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.6

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

   9. Applied rewrites 56.6%

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

   if 2.1e-129 < y < 4.99999999999999961e133

   1. Initial program 80.0%

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

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 91.8

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 91.8

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 91.8

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 91.8

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 71.9%

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

   if 4.99999999999999961e133 < y

   1. Initial program 46.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 86.5

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

   9. Applied rewrites 86.5%

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

Alternative 10: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{elif}\;y \leq 4600000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (fma x x x))))
   (if (<= y 4.8e-115)
     t_0
     (if (<= y 1.85e-51)
       (/ x (fma y y y))
       (if (<= y 4600000000.0) t_0 (/ (/ x y) y))))))

C

double code(double x, double y) {
	double t_0 = y / fma(x, x, x);
	double tmp;
	if (y <= 4.8e-115) {
		tmp = t_0;
	} else if (y <= 1.85e-51) {
		tmp = x / fma(y, y, y);
	} else if (y <= 4600000000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / fma(x, x, x))
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = t_0;
	elseif (y <= 1.85e-51)
		tmp = Float64(x / fma(y, y, y));
	elseif (y <= 4600000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-115], t$95$0, If[LessEqual[y, 1.85e-51], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4600000000.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.80000000000000042e-115 or 1.84999999999999987e-51 < y < 4.6e9

   1. Initial program 66.3%

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

   3. Taylor expanded in 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 55.6

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

   5. Applied rewrites 55.6%

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

   if 4.80000000000000042e-115 < y < 1.84999999999999987e-51

   1. Initial program 87.7%

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

   3. Taylor expanded in x around 0

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

      1. 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 53.5

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

   5. Applied rewrites 53.5%

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

   if 4.6e9 < y

   1. Initial program 54.4%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

Alternative 11: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-115) (/ (/ y (+ 1.0 x)) (+ y x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-115) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.8d-115) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-115) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-115:
		tmp = (y / (1.0 + x)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-115)
		tmp = (y / (1.0 + x)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-115], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000042e-115

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 56.6

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

   9. Applied rewrites 56.6%

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

   if 4.80000000000000042e-115 < y

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 99.7

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

      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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 68.5

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

   9. Applied rewrites 68.5%

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

Alternative 12: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-115) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-115) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-115], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000042e-115

   1. Initial program 64.2%

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

   3. Taylor expanded in y around 0

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

      1. 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 54.0

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

   5. Applied rewrites 54.0%

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

   if 4.80000000000000042e-115 < y

   1. Initial program 65.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 99.7

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

      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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/r* N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. lift-/.f64 N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 68.5

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

   9. Applied rewrites 68.5%

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

Alternative 13: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(1 + y\right) \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-115) (/ y (fma x x x)) (/ x (* (+ 1.0 y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-115) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / ((1.0 + y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / Float64(Float64(1.0 + y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-115], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(1.0 + y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000042e-115

   1. Initial program 64.2%

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

   3. Taylor expanded in y around 0

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

      1. 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 54.0

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

   5. Applied rewrites 54.0%

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

   if 4.80000000000000042e-115 < y

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 68.4%

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

Alternative 14: 59.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-115) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-115) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-115)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-115], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000042e-115

   1. Initial program 64.2%

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

   3. Taylor expanded in y around 0

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

      1. 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 54.0

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

   5. Applied rewrites 54.0%

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

   if 4.80000000000000042e-115 < y

   1. Initial program 65.4%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 68.4

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

   5. Applied rewrites 68.4%

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

Alternative 15: 61.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70000000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70000000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70000000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70000000000.0], 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 < -7e10

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 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 71.5

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

   7. Applied rewrites 71.5%

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

   if -7e10 < x

   1. Initial program 69.7%

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

   3. Taylor expanded in x around 0

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

      1. 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 58.9

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

   5. Applied rewrites 58.9%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4600000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4600000000.0) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4600000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4600000000.0d0) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4600000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4600000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4600000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4600000000.0)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4600000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.6e9

   1. Initial program 67.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. 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 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 36.4

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

   7. Applied rewrites 36.4%

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

   if 4.6e9 < y

   1. Initial program 54.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 99.7

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

      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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. 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 80.6

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

   7. Applied rewrites 80.6%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4600000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.6%

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

   1. 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 y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
6. 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 41.1

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

7. Applied rewrites 41.1%

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

8. Final simplification 41.1%

   \[\leadsto \frac{x}{y \cdot y} \]
9. 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 2024337 
(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))))