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

Percentage Accurate: 69.7% → 99.8%

Time: 15.8s

Alternatives: 26

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: 69.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e+157)
   (* (/ (/ x (+ y x)) (+ y x)) (/ y x))
   (if (<= x 1.8e-44)
     (/ (* x (/ y (+ y x))) (* (+ y x) (+ (+ y x) 1.0)))
     (/ (- -1.0) (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+157) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= 1.8e-44) {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = -(-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 (x <= (-9.5d+157)) then
        tmp = ((x / (y + x)) / (y + x)) * (y / x)
    else if (x <= 1.8d-44) then
        tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0d0))
    else
        tmp = -(-1.0d0) / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+157) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= 1.8e-44) {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = -(-1.0) / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e+157:
		tmp = ((x / (y + x)) / (y + x)) * (y / x)
	elif x <= 1.8e-44:
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0))
	else:
		tmp = -(-1.0) / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e+157)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * Float64(y / x));
	elseif (x <= 1.8e-44)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(-(-1.0)) / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e+157)
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	elseif (x <= 1.8e-44)
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	else
		tmp = -(-1.0) / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e+157], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-44], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((--1.0) / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999996e157

   1. Initial program 70.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 91.3%

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. +-commutative N/A

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

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

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

      16. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   if -9.4999999999999996e157 < x < 1.7999999999999999e-44

   1. Initial program 76.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 97.4

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

   4. Applied egg-rr 97.4%

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

   if 1.7999999999999999e-44 < x

   1. Initial program 72.5%

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

   3. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 23.5

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

   5. Simplified 23.5%

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

      1. *-lft-identity N/A

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

      2. *-inverses N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. associate-/r* N/A

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

      9. *-inverses N/A

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

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

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

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

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

      12. /-lowering-/.f64 27.0

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

   7. Applied egg-rr 27.0%

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

      1. associate-/l/ N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

      8. distribute-neg-frac2 N/A

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

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

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

      10. neg-lowering-neg.f64 27.0

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

   9. Applied egg-rr 27.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{--1}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e+74)
   (* (/ (/ x (+ y x)) (+ y x)) (/ y x))
   (if (<= x -2.1e-298)
     (* x (/ (/ y (+ y x)) (* (+ y x) (+ (+ y x) 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+74) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= -2.1e-298) {
		tmp = x * ((y / (y + x)) / ((y + x) * ((y + x) + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (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 (x <= (-1.8d+74)) then
        tmp = ((x / (y + x)) / (y + x)) * (y / x)
    else if (x <= (-2.1d-298)) then
        tmp = x * ((y / (y + x)) / ((y + x) * ((y + x) + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e+74) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= -2.1e-298) {
		tmp = x * ((y / (y + x)) / ((y + x) * ((y + x) + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.8e+74:
		tmp = ((x / (y + x)) / (y + x)) * (y / x)
	elif x <= -2.1e-298:
		tmp = x * ((y / (y + x)) / ((y + x) * ((y + x) + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.8e+74)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * Float64(y / x));
	elseif (x <= -2.1e-298)
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e+74)
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	elseif (x <= -2.1e-298)
		tmp = x * ((y / (y + x)) / ((y + x) * ((y + x) + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.8e+74], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-298], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999994e74

   1. Initial program 67.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 81.3

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 81.3%

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. +-commutative N/A

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

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

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

      16. /-lowering-/.f64 91.2

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

   9. Applied egg-rr 91.2%

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

   if -1.79999999999999994e74 < x < -2.10000000000000005e-298

   1. Initial program 79.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 86.6

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

   4. Applied egg-rr 86.6%

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

      1. associate-*l* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. +-commutative N/A

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

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

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

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

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 97.7

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

   6. Applied egg-rr 97.7%

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

   if -2.10000000000000005e-298 < x

   1. Initial program 73.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 46.6

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

   7. Simplified 46.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-296}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e+157)
   (* (/ (/ x (+ y x)) (+ y x)) (/ y x))
   (if (<= x -4e-296)
     (* y (/ (/ x (* (+ y x) (+ (+ y x) 1.0))) (+ y x)))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+157) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= -4e-296) {
		tmp = y * ((x / ((y + x) * ((y + x) + 1.0))) / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (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 (x <= (-9.5d+157)) then
        tmp = ((x / (y + x)) / (y + x)) * (y / x)
    else if (x <= (-4d-296)) then
        tmp = y * ((x / ((y + x) * ((y + x) + 1.0d0))) / (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+157) {
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	} else if (x <= -4e-296) {
		tmp = y * ((x / ((y + x) * ((y + x) + 1.0))) / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.5e+157:
		tmp = ((x / (y + x)) / (y + x)) * (y / x)
	elif x <= -4e-296:
		tmp = y * ((x / ((y + x) * ((y + x) + 1.0))) / (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.5e+157)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(y + x)) * Float64(y / x));
	elseif (x <= -4e-296)
		tmp = Float64(y * Float64(Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0))) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e+157)
		tmp = ((x / (y + x)) / (y + x)) * (y / x);
	elseif (x <= -4e-296)
		tmp = y * ((x / ((y + x) * ((y + x) + 1.0))) / (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.5e+157], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-296], N[(y * N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999996e157

   1. Initial program 70.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 91.3%

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

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

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

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

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

      14. +-commutative N/A

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

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

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

      16. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   if -9.4999999999999996e157 < x < -4e-296

   1. Initial program 76.7%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.4

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

   6. Applied egg-rr 99.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 92.4

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

   8. Applied egg-rr 92.4%

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

   if -4e-296 < x

   1. Initial program 73.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 46.6

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

   7. Simplified 46.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-296}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-293}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -9.5e+157)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.1e-293)
       (* y (/ (/ x (* (+ y x) t_0)) (+ y x)))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -9.5e+157) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.1e-293) {
		tmp = y * ((x / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-9.5d+157)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.1d-293)) then
        tmp = y * ((x / ((y + x) * t_0)) / (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -9.5e+157) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.1e-293) {
		tmp = y * ((x / ((y + x) * t_0)) / (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -9.5e+157:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.1e-293:
		tmp = y * ((x / ((y + x) * t_0)) / (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -9.5e+157)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.1e-293)
		tmp = Float64(y * Float64(Float64(x / Float64(Float64(y + x) * t_0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -9.5e+157)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.1e-293)
		tmp = y * ((x / ((y + x) * t_0)) / (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -9.5e+157], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-293], N[(y * N[(N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.4999999999999996e157

   1. Initial program 70.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 100.0%

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

   if -9.4999999999999996e157 < x < -1.1e-293

   1. Initial program 76.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

      10. +-commutative N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.4

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

   6. Applied egg-rr 99.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-lowering-+.f64 92.3

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

   8. Applied egg-rr 92.3%

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

   if -1.1e-293 < x

   1. Initial program 73.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 47.0

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

   7. Simplified 47.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-293}:\\ \;\;\;\;y \cdot \frac{\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{+74}:\\ \;\;\;\;\frac{y \cdot \frac{1}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= x -1.85e+74)
     (/ (* y (/ 1.0 t_0)) (+ y x))
     (if (<= x -3.4e-159)
       (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
       (/ (/ x (+ y 1.0)) y)))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.85e+74) {
		tmp = (y * (1.0 / t_0)) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (x <= (-1.85d+74)) then
        tmp = (y * (1.0d0 / t_0)) / (y + x)
    else if (x <= (-3.4d-159)) then
        tmp = x * (y / (t_0 * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (x <= -1.85e+74) {
		tmp = (y * (1.0 / t_0)) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if x <= -1.85e+74:
		tmp = (y * (1.0 / t_0)) / (y + x)
	elif x <= -3.4e-159:
		tmp = x * (y / (t_0 * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (x <= -1.85e+74)
		tmp = Float64(Float64(y * Float64(1.0 / t_0)) / Float64(y + x));
	elseif (x <= -3.4e-159)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (x <= -1.85e+74)
		tmp = (y * (1.0 / t_0)) / (y + x);
	elseif (x <= -3.4e-159)
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.85e+74], N[(N[(y * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-159], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8500000000000001e74

   1. Initial program 67.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

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

   9. Simplified 91.1%

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

   if -1.8500000000000001e74 < x < -3.39999999999999984e-159

   1. Initial program 84.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 93.5

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

   4. Applied egg-rr 93.5%

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

   if -3.39999999999999984e-159 < x

   1. Initial program 72.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 77.0

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.5

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

   7. Simplified 52.5%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 54.3

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

   9. Applied egg-rr 54.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+74}:\\ \;\;\;\;\frac{y \cdot \frac{1}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. times-frac N/A

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

   2. associate-*l/ N/A

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

   3. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8500000000000:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8500000000000.0)
   (/ (/ y (+ (+ y x) 1.0)) (+ y x))
   (if (<= x -3.4e-159)
     (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8500000000000.0) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8500000000000.0d0)) then
        tmp = (y / ((y + x) + 1.0d0)) / (y + x)
    else if (x <= (-3.4d-159)) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8500000000000.0) {
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8500000000000.0:
		tmp = (y / ((y + x) + 1.0)) / (y + x)
	elif x <= -3.4e-159:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8500000000000.0)
		tmp = Float64(Float64(y / Float64(Float64(y + x) + 1.0)) / Float64(y + x));
	elseif (x <= -3.4e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8500000000000.0)
		tmp = (y / ((y + x) + 1.0)) / (y + x);
	elseif (x <= -3.4e-159)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8500000000000.0], N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-159], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e12

   1. Initial program 72.3%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 78.4%

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

   if -8.5e12 < x < -3.39999999999999984e-159

   1. Initial program 84.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 95.3

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

   4. Applied egg-rr 95.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 90.7

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

   7. Simplified 90.7%

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

   if -3.39999999999999984e-159 < x

   1. Initial program 72.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 77.0

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.5

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

   7. Simplified 52.5%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 54.3

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

   9. Applied egg-rr 54.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8500000000000:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{y \cdot \frac{1}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+16)
   (/ (* y (/ 1.0 (+ (+ y x) 1.0))) (+ y x))
   (if (<= x -3.4e-159)
     (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+16) {
		tmp = (y * (1.0 / ((y + x) + 1.0))) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d+16)) then
        tmp = (y * (1.0d0 / ((y + x) + 1.0d0))) / (y + x)
    else if (x <= (-3.4d-159)) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+16) {
		tmp = (y * (1.0 / ((y + x) + 1.0))) / (y + x);
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e+16:
		tmp = (y * (1.0 / ((y + x) + 1.0))) / (y + x)
	elif x <= -3.4e-159:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+16)
		tmp = Float64(Float64(y * Float64(1.0 / Float64(Float64(y + x) + 1.0))) / Float64(y + x));
	elseif (x <= -3.4e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e+16)
		tmp = (y * (1.0 / ((y + x) + 1.0))) / (y + x);
	elseif (x <= -3.4e-159)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e+16], N[(N[(y * N[(1.0 / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-159], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e16

   1. Initial program 72.3%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.7

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

   9. Simplified 78.4%

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

   if -1.05e16 < x < -3.39999999999999984e-159

   1. Initial program 84.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 95.3

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

   4. Applied egg-rr 95.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 90.7

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

   7. Simplified 90.7%

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

   if -3.39999999999999984e-159 < x

   1. Initial program 72.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 77.0

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

   4. Applied egg-rr 77.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 52.5

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

   7. Simplified 52.5%

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

      1. *-commutative N/A

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

      2. un-div-inv N/A

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

      3. distribute-rgt1-in N/A

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

      4. associate-/r* N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

      8. +-lowering-+.f64 54.3

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{y \cdot \frac{1}{\left(y + x\right) + 1}}{y + x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6500000000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6500000000000.0)
   (/ (/ y (+ y x)) x)
   (if (<= x -3.4e-159)
     (* x (/ y (* (+ y 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6500000000000.0) {
		tmp = (y / (y + x)) / x;
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6500000000000.0d0)) then
        tmp = (y / (y + x)) / x
    else if (x <= (-3.4d-159)) then
        tmp = x * (y / ((y + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6500000000000.0) {
		tmp = (y / (y + x)) / x;
	} else if (x <= -3.4e-159) {
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6500000000000.0:
		tmp = (y / (y + x)) / x
	elif x <= -3.4e-159:
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6500000000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) / x);
	elseif (x <= -3.4e-159)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6500000000000.0)
		tmp = (y / (y + x)) / x;
	elseif (x <= -3.4e-159)
		tmp = x * (y / ((y + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6500000000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.4e-159], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e12

   1. Initial program 72.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 78.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Simplified 78.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{x} \]

      6. +-lowering-+.f64 78.3

         \[\leadsto \frac{\frac{y}{\color{blue}{y + x}}}{x} \]

   9. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x}}{x}} \]

   if -6.5e12 < x < -3.39999999999999984e-159

   1. Initial program 84.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 95.3

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

      2. +-lowering-+.f64 90.7

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   7. Simplified 90.7%

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   if -3.39999999999999984e-159 < x

   1. Initial program 72.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 77.0

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 52.5

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 52.5%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{y + y \cdot y}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right) \cdot y}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

      8. +-lowering-+.f64 54.3

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 54.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6500000000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-159}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-71)
   (/ y (* (+ y x) (+ x 1.0)))
   (if (<= y 1.72e+37) (/ x (fma y y y)) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-71) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (y <= 1.72e+37) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-71)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	elseif (y <= 1.72e+37)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-71], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e+37], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.39999999999999995e-71

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

      7. +-lowering-+.f64 62.0

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(1 + x\right)}} \]

   if 4.39999999999999995e-71 < y < 1.72000000000000002e37

   1. Initial program 92.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 46.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 1.72000000000000002e37 < y

   1. Initial program 67.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 74.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Simplified 74.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e-144)
   (/ y (* x x))
   (if (<= y 2.2e-210) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-144) {
		tmp = y / (x * x);
	} else if (y <= 2.2e-210) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} 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 <= (-4.6d-144)) then
        tmp = y / (x * x)
    else if (y <= 2.2d-210) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e-144) {
		tmp = y / (x * x);
	} else if (y <= 2.2e-210) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e-144:
		tmp = y / (x * x)
	elif y <= 2.2e-210:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e-144)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.2e-210)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e-144)
		tmp = y / (x * x);
	elseif (y <= 2.2e-210)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e-144], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-210], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.6e-144

   1. Initial program 78.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 32.8

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -4.6e-144 < y < 2.19999999999999989e-210

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 88.5

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 88.5%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
   9. Step-by-step derivation

   10. Simplified 73.8%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 2.19999999999999989e-210 < y < 1

   1. Initial program 77.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 76.6

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 37.2

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 35.1

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 35.1%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 67.4

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.2e-72)
   (/ y (* (+ y x) (+ x 1.0)))
   (if (<= y 2e+37) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.2e-72) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (y <= 2e+37) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.2e-72)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	elseif (y <= 2e+37)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.2e-72], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+37], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.20000000000000007e-72

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.2

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

      7. +-lowering-+.f64 62.0

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   9. Applied egg-rr 62.0%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(1 + x\right)}} \]

   if 8.20000000000000007e-72 < y < 1.99999999999999991e37

   1. Initial program 92.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 46.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 1.99999999999999991e37 < y

   1. Initial program 67.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 66.1

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. /-lowering-/.f64 74.3

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-76) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-76) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (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 <= 4.4d-76) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-76) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e-76:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-76)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-76)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-76], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999999e-76

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 59.7

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 59.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   if 4.39999999999999999e-76 < y

   1. Initial program 76.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

      3. +-lowering-+.f64 64.5

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   7. Simplified 64.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-69) (/ (/ y (+ x 1.0)) x) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-69) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + 1.0)) / (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 <= 3.8d-69) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-69) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e-69:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-69)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-69)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e-69], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998e-69

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.6

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y}{x + 1}}{\color{blue}{x}} \]
   9. Step-by-step derivation

   10. Simplified 60.0%

       \[\leadsto \frac{\frac{y}{x + 1}}{\color{blue}{x}} \]

   if 3.7999999999999998e-69 < y

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

      3. +-lowering-+.f64 67.7

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   7. Simplified 67.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.8e-69) (/ (/ y (+ x 1.0)) x) (/ (/ x (+ y 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-69) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + 1.0)) / 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 <= 3.8d-69) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.8e-69) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.8e-69:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.8e-69)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.8e-69)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.8e-69], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998e-69

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.6

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y}{x + 1}}{\color{blue}{x}} \]
   9. Step-by-step derivation

   10. Simplified 60.0%

       \[\leadsto \frac{\frac{y}{x + 1}}{\color{blue}{x}} \]

   if 3.7999999999999998e-69 < y

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 79.9

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 61.4

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{y + y \cdot y}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right) \cdot y}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

      8. +-lowering-+.f64 67.1

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 62.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e-69) (/ y (* (+ y x) (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-69) {
		tmp = y / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / 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 <= 3.9d-69) then
        tmp = y / ((y + x) * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e-69) {
		tmp = y / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.9e-69:
		tmp = y / ((y + x) * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.9e-69)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e-69)
		tmp = y / ((y + x) * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.9e-69], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.89999999999999981e-69

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.6

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

      7. +-lowering-+.f64 62.5

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   9. Applied egg-rr 62.5%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(1 + x\right)}} \]

   if 3.89999999999999981e-69 < y

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 79.9

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 61.4

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{y + y \cdot y}} \]

      2. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + 1\right) \cdot y}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y + 1}}{y}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

      8. +-lowering-+.f64 67.1

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-210) (/ y (+ y x)) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-210) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} 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 <= 2.2d-210) then
        tmp = y / (y + x)
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-210) {
		tmp = y / (y + x);
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-210:
		tmp = y / (y + x)
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-210)
		tmp = Float64(y / Float64(y + x));
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-210)
		tmp = y / (y + x);
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-210], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.19999999999999989e-210

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 59.3

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 59.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
   9. Step-by-step derivation

   10. Simplified 35.5%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 2.19999999999999989e-210 < y < 1

   1. Initial program 77.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 76.6

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 76.6%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 37.2

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 37.2%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 35.1

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 35.1%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 67.4

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-69) (/ y (* (+ y x) (+ x 1.0))) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-69) {
		tmp = y / ((y + x) * (x + 1.0));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-69)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-69], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3000000000000001e-69

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 60.6

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 60.6%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + 1\right)}} \]

      3. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

      7. +-lowering-+.f64 62.5

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(1 + x\right)}} \]

   9. Applied egg-rr 62.5%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(1 + x\right)}} \]

   if 2.3000000000000001e-69 < y

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 61.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e-69) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e-69) {
		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 <= 3.7e-69)
		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, 3.7e-69], 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 < 3.7000000000000002e-69

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. accelerator-lowering-fma.f64 60.3

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 3.7000000000000002e-69 < y

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 61.4

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.2e+14) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.2e+14) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.2e+14)
		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, -8.2e+14], 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 < -8.2e14

   1. Initial program 72.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 73.7

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -8.2e14 < x

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 54.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 34.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -6.1e-134) (/ y (+ y x)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.1e-134) {
		tmp = y / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.1d-134)) then
        tmp = y / (y + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.1e-134) {
		tmp = y / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.1e-134:
		tmp = y / (y + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.1e-134)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.1e-134)
		tmp = y / (y + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.1e-134], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.0999999999999996e-134

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. +-lowering-+.f64 68.7

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
   9. Step-by-step derivation

   10. Simplified 34.5%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if -6.0999999999999996e-134 < x

   1. Initial program 74.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 78.6

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 78.6%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 55.6

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 35.9

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 35.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 27.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5.3e+14) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.3e+14) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.3d+14)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.3e+14) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.3e+14:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.3e+14)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.3e+14)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.3e+14], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.3e14

   1. Initial program 72.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 78.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Simplified 78.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 5.7

         \[\leadsto \color{blue}{\frac{1}{x}} \]

   10. Simplified 5.7%

       \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -5.3e14 < x

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 81.3

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot 1 + y \cdot y}} \cdot x \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{y} + y \cdot y} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{{y}^{2}}} \cdot x \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y + {y}^{2}}} \cdot x \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

      7. *-lowering-*.f64 54.2

         \[\leadsto \frac{1}{y + \color{blue}{y \cdot y}} \cdot x \]

   7. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{1}{y + y \cdot y}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 31.9

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 31.9%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 4.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.8

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 39.3

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

7. Simplified 39.3%

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.2

      \[\leadsto \color{blue}{\frac{1}{x}} \]

10. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Alternative 25: 4.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 y))

C

double code(double x, double y) {
	return 0.5 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / y
end function

Java

public static double code(double x, double y) {
	return 0.5 / y;
}

Python

def code(x, y):
	return 0.5 / y

Julia

function code(x, y)
	return Float64(0.5 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / y;
end

Wolfram

code[x_, y_] := N[(0.5 / y), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

   10. +-lowering-+.f64 81.7

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

4. Applied egg-rr 81.7%

   \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{y}{\color{blue}{\left(2 \cdot \left(x \cdot y\right) + {x}^{2}\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{y}{\color{blue}{\left({x}^{2} + 2 \cdot \left(x \cdot y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   2. unpow2 N/A

      \[\leadsto \frac{y}{\left(\color{blue}{x \cdot x} + 2 \cdot \left(x \cdot y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   3. *-commutative N/A

      \[\leadsto \frac{y}{\left(x \cdot x + 2 \cdot \color{blue}{\left(y \cdot x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   4. associate-*r* N/A

      \[\leadsto \frac{y}{\left(x \cdot x + \color{blue}{\left(2 \cdot y\right) \cdot x}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{y}{\color{blue}{\left(x \cdot \left(x + 2 \cdot y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{y}{\color{blue}{\left(x \cdot \left(x + 2 \cdot y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   7. +-commutative N/A

      \[\leadsto \frac{y}{\left(x \cdot \color{blue}{\left(2 \cdot y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   8. *-commutative N/A

      \[\leadsto \frac{y}{\left(x \cdot \left(\color{blue}{y \cdot 2} + x\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

   9. accelerator-lowering-fma.f64 56.4

      \[\leadsto \frac{y}{\left(x \cdot \color{blue}{\mathsf{fma}\left(y, 2, x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

7. Simplified 56.4%

   \[\leadsto \frac{y}{\color{blue}{\left(x \cdot \mathsf{fma}\left(y, 2, x\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.1

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

10. Simplified 4.1%

    \[\leadsto \color{blue}{\frac{0.5}{y}} \]
11. Add Preprocessing

Alternative 26: 3.5% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 74.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.8

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   3. +-lowering-+.f64 53.0

      \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

7. Simplified 53.0%

   \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 3.7%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024204 
(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))))