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

Percentage Accurate: 69.1% → 99.8%

Time: 15.4s

Alternatives: 28

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: 69.1% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{y + x}\\ t_1 := \frac{\frac{t\_0}{y + x}}{y + x}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ x (+ y x)))) (t_1 (/ (/ t_0 (+ y x)) (+ y x))))
   (if (<= y -7e+14)
     t_1
     (if (<= y 3e+30) (/ t_0 (* (+ y x) (+ y (+ x 1.0)))) t_1))))

C

double code(double x, double y) {
	double t_0 = y * (x / (y + x));
	double t_1 = (t_0 / (y + x)) / (y + x);
	double tmp;
	if (y <= -7e+14) {
		tmp = t_1;
	} else if (y <= 3e+30) {
		tmp = t_0 / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * (x / (y + x))
    t_1 = (t_0 / (y + x)) / (y + x)
    if (y <= (-7d+14)) then
        tmp = t_1
    else if (y <= 3d+30) then
        tmp = t_0 / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x / (y + x));
	double t_1 = (t_0 / (y + x)) / (y + x);
	double tmp;
	if (y <= -7e+14) {
		tmp = t_1;
	} else if (y <= 3e+30) {
		tmp = t_0 / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x / (y + x))
	t_1 = (t_0 / (y + x)) / (y + x)
	tmp = 0
	if y <= -7e+14:
		tmp = t_1
	elif y <= 3e+30:
		tmp = t_0 / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x / Float64(y + x)))
	t_1 = Float64(Float64(t_0 / Float64(y + x)) / Float64(y + x))
	tmp = 0.0
	if (y <= -7e+14)
		tmp = t_1;
	elseif (y <= 3e+30)
		tmp = Float64(t_0 / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x / (y + x));
	t_1 = (t_0 / (y + x)) / (y + x);
	tmp = 0.0;
	if (y <= -7e+14)
		tmp = t_1;
	elseif (y <= 3e+30)
		tmp = t_0 / ((y + x) * (y + (x + 1.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+14], t$95$1, If[LessEqual[y, 3e+30], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e14 or 2.99999999999999978e30 < y

   1. Initial program 59.3%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

   9. Simplified 99.8%

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

   if -7e14 < y < 2.99999999999999978e30

   1. Initial program 77.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

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

   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. 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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{y + x}}{y + x}}{y + x}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{y + x}}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7.8e+117)
     (/ (/ y t_0) (+ y x))
     (if (<= x -7e-17)
       (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
       (if (<= x 1.02e-100)
         (/ (* y (/ x (+ y x))) (* (+ y x) (+ y 1.0)))
         (/ (/ x t_0) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.8e+117) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -7e-17) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / t_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 <= (-7.8d+117)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-7d-17)) then
        tmp = y * (x / (((y + x) + 1.0d0) * ((y + x) * (y + x))))
    else if (x <= 1.02d-100) then
        tmp = (y * (x / (y + x))) / ((y + x) * (y + 1.0d0))
    else
        tmp = (x / t_0) / (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 <= -7.8e+117) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -7e-17) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7.8e+117:
		tmp = (y / t_0) / (y + x)
	elif x <= -7e-17:
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))))
	elif x <= 1.02e-100:
		tmp = (y * (x / (y + x))) / ((y + x) * (y + 1.0))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.8e+117)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -7e-17)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (x <= 1.02e-100)
		tmp = Float64(Float64(y * Float64(x / Float64(y + x))) / Float64(Float64(y + x) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / t_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 <= -7.8e+117)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -7e-17)
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	elseif (x <= 1.02e-100)
		tmp = (y * (x / (y + x))) / ((y + x) * (y + 1.0));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+117], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-17], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-100], N[(N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.79999999999999981e117

   1. Initial program 47.2%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 90.4%

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

   if -7.79999999999999981e117 < x < -7.0000000000000003e-17

   1. Initial program 82.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   if -7.0000000000000003e-17 < x < 1.02e-100

   1. Initial program 73.4%

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 99.8%

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

   if 1.02e-100 < x

   1. Initial program 65.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 45.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \left(y + 1\right)\right) \cdot \frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -4.4e+119)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1e-16)
       (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
       (if (<= x -6.2e-246)
         (/ y (* (* (+ y x) (+ y 1.0)) (/ (+ y x) x)))
         (/ (/ x t_0) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -4.4e+119) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1e-16) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else if (x <= -6.2e-246) {
		tmp = y / (((y + x) * (y + 1.0)) * ((y + x) / x));
	} else {
		tmp = (x / t_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 <= (-4.4d+119)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1d-16)) then
        tmp = y * (x / (((y + x) + 1.0d0) * ((y + x) * (y + x))))
    else if (x <= (-6.2d-246)) then
        tmp = y / (((y + x) * (y + 1.0d0)) * ((y + x) / x))
    else
        tmp = (x / t_0) / (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 <= -4.4e+119) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1e-16) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else if (x <= -6.2e-246) {
		tmp = y / (((y + x) * (y + 1.0)) * ((y + x) / x));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -4.4e+119:
		tmp = (y / t_0) / (y + x)
	elif x <= -1e-16:
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))))
	elif x <= -6.2e-246:
		tmp = y / (((y + x) * (y + 1.0)) * ((y + x) / x))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.4e+119)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1e-16)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (x <= -6.2e-246)
		tmp = Float64(y / Float64(Float64(Float64(y + x) * Float64(y + 1.0)) * Float64(Float64(y + x) / x)));
	else
		tmp = Float64(Float64(x / t_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 <= -4.4e+119)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1e-16)
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	elseif (x <= -6.2e-246)
		tmp = y / (((y + x) * (y + 1.0)) * ((y + x) / x));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e+119], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-16], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-246], N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.4000000000000003e119

   1. Initial program 47.2%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 90.4%

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

   if -4.4000000000000003e119 < x < -9.9999999999999998e-17

   1. Initial program 82.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   if -9.9999999999999998e-17 < x < -6.2000000000000001e-246

   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. 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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 99.7

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 99.7%

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

       1. associate-/l* N/A

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

       2. clear-num N/A

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

       3. un-div-inv N/A

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

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

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

       5. div-inv N/A

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

       6. clear-num N/A

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

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

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

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

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

       9. +-commutative N/A

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

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

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

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

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

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

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

       13. +-commutative N/A

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

       14. +-lowering-+.f64 95.4

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

   11. Applied egg-rr 95.4%

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

   if -6.2000000000000001e-246 < x

   1. Initial program 67.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 60.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-246}:\\ \;\;\;\;\frac{y}{\left(\left(y + x\right) \cdot \left(y + 1\right)\right) \cdot \frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1} \cdot \frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7.6e+154)
     (/ (/ y t_0) (+ y x))
     (if (<= x 1.02e-100)
       (/ (* y (/ x (+ y x))) (* (+ y x) t_0))
       (/ (* (/ y (+ (+ y x) 1.0)) (/ x y)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.6e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = ((y / ((y + x) + 1.0)) * (x / y)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-7.6d+154)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= 1.02d-100) then
        tmp = (y * (x / (y + x))) / ((y + x) * t_0)
    else
        tmp = ((y / ((y + x) + 1.0d0)) * (x / y)) / (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 <= -7.6e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = ((y / ((y + x) + 1.0)) * (x / y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7.6e+154:
		tmp = (y / t_0) / (y + x)
	elif x <= 1.02e-100:
		tmp = (y * (x / (y + x))) / ((y + x) * t_0)
	else:
		tmp = ((y / ((y + x) + 1.0)) * (x / y)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.6e+154)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= 1.02e-100)
		tmp = Float64(Float64(y * Float64(x / Float64(y + x))) / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(Float64(y / Float64(Float64(y + x) + 1.0)) * Float64(x / y)) / 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 <= -7.6e+154)
		tmp = (y / t_0) / (y + x);
	elseif (x <= 1.02e-100)
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	else
		tmp = ((y / ((y + x) + 1.0)) * (x / y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-100], N[(N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5999999999999996e154

   1. Initial program 44.8%

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 89.4%

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

   if -7.5999999999999996e154 < x < 1.02e-100

   1. Initial program 75.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

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

   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. 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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 98.4

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

   6. Applied egg-rr 98.4%

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

   if 1.02e-100 < x

   1. Initial program 65.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 44.7

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

   7. Simplified 44.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) + 1} \cdot \frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-142}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-241}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{x + \mathsf{fma}\left(x, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -4.2e+119)
     (/ (/ y t_0) (+ y x))
     (if (<= x -2.4e-142)
       (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
       (if (<= x -3e-241)
         (/ (* y (/ x (+ y x))) (+ x (fma x y y)))
         (/ (/ x t_0) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -4.2e+119) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -2.4e-142) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else if (x <= -3e-241) {
		tmp = (y * (x / (y + x))) / (x + fma(x, y, y));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -4.2e+119)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -2.4e-142)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (x <= -3e-241)
		tmp = Float64(Float64(y * Float64(x / Float64(y + x))) / Float64(x + fma(x, y, y)));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e+119], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.4e-142], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3e-241], N[(N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(x * y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.19999999999999966e119

   1. Initial program 47.2%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 90.4%

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

   if -4.19999999999999966e119 < x < -2.39999999999999988e-142

   1. Initial program 86.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 89.0

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

   4. Applied egg-rr 89.0%

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

   if -2.39999999999999988e-142 < x < -2.9999999999999999e-241

   1. Initial program 63.1%

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around 0

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

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

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-lft-identity N/A

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

       5. accelerator-lowering-fma.f64 89.7

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

   12. Simplified 89.7%

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

   if -2.9999999999999999e-241 < x

   1. Initial program 67.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 60.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-142}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-241}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{x + \mathsf{fma}\left(x, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -3.7e+154)
     (/ (/ y t_0) (+ y x))
     (if (<= x 1.02e-100)
       (/ (* y (/ x (+ y x))) (* (+ y x) t_0))
       (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -3.7e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = (x / t_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 <= (-3.7d+154)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= 1.02d-100) then
        tmp = (y * (x / (y + x))) / ((y + x) * t_0)
    else
        tmp = (x / t_0) / (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 <= -3.7e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -3.7e+154:
		tmp = (y / t_0) / (y + x)
	elif x <= 1.02e-100:
		tmp = (y * (x / (y + x))) / ((y + x) * t_0)
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -3.7e+154)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= 1.02e-100)
		tmp = Float64(Float64(y * Float64(x / Float64(y + x))) / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / t_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 <= -3.7e+154)
		tmp = (y / t_0) / (y + x);
	elseif (x <= 1.02e-100)
		tmp = (y * (x / (y + x))) / ((y + x) * t_0);
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-100], N[(N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.69999999999999994e154

   1. Initial program 44.8%

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 89.4%

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

   if -3.69999999999999994e154 < x < 1.02e-100

   1. Initial program 75.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

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

   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. 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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 98.4

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

   6. Applied egg-rr 98.4%

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

   if 1.02e-100 < x

   1. Initial program 65.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 45.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -2.35e+126)
     (/ (/ y t_0) (+ y x))
     (if (<= x 1.02e-100)
       (* (/ y (+ y x)) (/ x (* (+ y x) (+ (+ y x) 1.0))))
       (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2.35e+126) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0)));
	} else {
		tmp = (x / t_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 <= (-2.35d+126)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= 1.02d-100) then
        tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0d0)))
    else
        tmp = (x / t_0) / (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 <= -2.35e+126) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 1.02e-100) {
		tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0)));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -2.35e+126:
		tmp = (y / t_0) / (y + x)
	elif x <= 1.02e-100:
		tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0)))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -2.35e+126)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= 1.02e-100)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0))));
	else
		tmp = Float64(Float64(x / t_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 <= -2.35e+126)
		tmp = (y / t_0) / (y + x);
	elseif (x <= 1.02e-100)
		tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0)));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e+126], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.02e-100], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3499999999999999e126

   1. Initial program 47.2%

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

      1. times-frac N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 90.4%

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

   if -2.3499999999999999e126 < x < 1.02e-100

   1. Initial program 75.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 98.3

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

   4. Applied egg-rr 98.3%

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

   if 1.02e-100 < x

   1. Initial program 65.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 45.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -8e+154)
     (/ (/ y t_0) (+ y x))
     (if (<= x -3.1e-9)
       (/ y (* (+ y x) t_0))
       (if (<= x -2e-162)
         (* y (/ x (* (* (+ y x) (+ y x)) (+ y 1.0))))
         (/ (/ x t_0) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -8e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.1e-9) {
		tmp = y / ((y + x) * t_0);
	} else if (x <= -2e-162) {
		tmp = y * (x / (((y + x) * (y + x)) * (y + 1.0)));
	} else {
		tmp = (x / t_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 <= (-8d+154)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-3.1d-9)) then
        tmp = y / ((y + x) * t_0)
    else if (x <= (-2d-162)) then
        tmp = y * (x / (((y + x) * (y + x)) * (y + 1.0d0)))
    else
        tmp = (x / t_0) / (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 <= -8e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.1e-9) {
		tmp = y / ((y + x) * t_0);
	} else if (x <= -2e-162) {
		tmp = y * (x / (((y + x) * (y + x)) * (y + 1.0)));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -8e+154:
		tmp = (y / t_0) / (y + x)
	elif x <= -3.1e-9:
		tmp = y / ((y + x) * t_0)
	elif x <= -2e-162:
		tmp = y * (x / (((y + x) * (y + x)) * (y + 1.0)))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -8e+154)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -3.1e-9)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	elseif (x <= -2e-162)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / t_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 <= -8e+154)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -3.1e-9)
		tmp = y / ((y + x) * t_0);
	elseif (x <= -2e-162)
		tmp = y * (x / (((y + x) * (y + x)) * (y + 1.0)));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.1e-9], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-162], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.0000000000000003e154

   1. Initial program 44.8%

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 89.4%

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

   if -8.0000000000000003e154 < x < -3.10000000000000005e-9

   1. Initial program 81.0%

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

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

   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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 95.3

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

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 77.8%

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

   if -3.10000000000000005e-9 < x < -1.99999999999999991e-162

   1. Initial program 89.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 94.3

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

   4. Applied egg-rr 94.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 94.3

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

   7. Simplified 94.3%

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

   if -1.99999999999999991e-162 < x

   1. Initial program 66.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. 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. /-lowering-/.f64 N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 61.8%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-162}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-178)
   (/ (/ y (+ y (+ x 1.0))) (+ y x))
   (if (<= y 7.5e+100)
     (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (* (/ x (+ y x)) (/ 1.0 (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-178) {
		tmp = (y / (y + (x + 1.0))) / (y + x);
	} else if (y <= 7.5e+100) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * (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 <= 2.1d-178) then
        tmp = (y / (y + (x + 1.0d0))) / (y + x)
    else if (y <= 7.5d+100) then
        tmp = y * (x / (((y + x) + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + x)) * (1.0d0 / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-178) {
		tmp = (y / (y + (x + 1.0))) / (y + x);
	} else if (y <= 7.5e+100) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * (1.0 / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-178:
		tmp = (y / (y + (x + 1.0))) / (y + x)
	elif y <= 7.5e+100:
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + x)) * (1.0 / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-178)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 7.5e+100)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(1.0 / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-178)
		tmp = (y / (y + (x + 1.0))) / (y + x);
	elseif (y <= 7.5e+100)
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + x)) * (1.0 / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-178], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+100], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1e-178

   1. Initial program 65.1%

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

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

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

      10. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 55.4%

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

   if 2.1e-178 < y < 7.49999999999999983e100

   1. Initial program 82.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 93.4

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

   4. Applied egg-rr 93.4%

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

   if 7.49999999999999983e100 < y

   1. Initial program 59.8%

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

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

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

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

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

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

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

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

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

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

      9. +-commutative N/A

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

      10. associate-+l+ N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 89.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{1}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 68.5%

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

   1. 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 11: 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 68.5%

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

   1. 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 12: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 55000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e+117)
   (/ (/ y x) (+ y x))
   (if (<= x -4.4e-77)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (if (<= x 55000000.0) (/ x (fma y y y)) (/ (/ x y) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+117) {
		tmp = (y / x) / (y + x);
	} else if (x <= -4.4e-77) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else if (x <= 55000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.6e+117)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -4.4e-77)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 55000000.0)
		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[x, -7.6e+117], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.4e-77], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 55000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.6000000000000003e117

   1. Initial program 45.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 87.3

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

   7. Simplified 87.3%

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

   if -7.6000000000000003e117 < x < -4.40000000000000014e-77

   1. Initial program 87.1%

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 95.8

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 77.7%

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

   if -4.40000000000000014e-77 < x < 5.5e7

   1. Initial program 73.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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 81.9

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

   5. Simplified 81.9%

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

   if 5.5e7 < x

   1. Initial program 59.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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 y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 32.6

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Simplified 32.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 55000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 54000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+154)
   (/ (/ y x) x)
   (if (<= x -4e-78)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (if (<= x 54000000.0) (/ x (fma y y y)) (/ (/ x y) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / x) / x;
	} else if (x <= -4e-78) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else if (x <= 54000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -4e-78)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 54000000.0)
		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[x, -1.32e+154], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4e-78], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 54000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 44.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot {x}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      5. *-lowering-*.f64 44.8

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   5. Simplified 44.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{y}{x \cdot x}} \]

      2. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \frac{y}{x \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      6. /-lowering-/.f64 88.7

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -1.31999999999999998e154 < x < -4e-78

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 96.1

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if -4e-78 < x < 5.4e7

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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 81.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 5.4e7 < x

   1. Initial program 59.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.7

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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 y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 32.6

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Simplified 32.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 54000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.32e+154)
     (/ (/ y t_0) (+ y x))
     (if (<= x -3.5e-77) (/ y (* (+ y x) t_0)) (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.5e-77) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / t_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 <= (-1.32d+154)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-3.5d-77)) then
        tmp = y / ((y + x) * t_0)
    else
        tmp = (x / t_0) / (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 <= -1.32e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -3.5e-77) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.32e+154:
		tmp = (y / t_0) / (y + x)
	elif x <= -3.5e-77:
		tmp = y / ((y + x) * t_0)
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -3.5e-77)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / t_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 <= -1.32e+154)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -3.5e-77)
		tmp = y / ((y + x) * t_0);
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-77], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 44.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) + 1}}}{x + y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      10. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + \left(x + 1\right)}}}{x + y} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{\color{blue}{y}}{y + \left(x + 1\right)}}{x + y} \]
   8. Step-by-step derivation

   9. Simplified 89.4%

      \[\leadsto \frac{\frac{\color{blue}{y}}{y + \left(x + 1\right)}}{x + y} \]

   if -1.31999999999999998e154 < x < -3.50000000000000013e-77

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 96.1

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if -3.50000000000000013e-77 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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. 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. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) + 1}}}{x + y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      10. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + \left(x + 1\right)}}}{x + y} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{\color{blue}{x}}{y + \left(x + 1\right)}}{x + y} \]
   8. Step-by-step derivation

   9. Simplified 62.8%

      \[\leadsto \frac{\frac{\color{blue}{x}}{y + \left(x + 1\right)}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t\_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7.6e+117)
     (/ (/ y x) (+ y x))
     (if (<= x -5e-79) (/ y (* (+ y x) t_0)) (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7.6e+117) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e-79) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / t_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 <= (-7.6d+117)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-5d-79)) then
        tmp = y / ((y + x) * t_0)
    else
        tmp = (x / t_0) / (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 <= -7.6e+117) {
		tmp = (y / x) / (y + x);
	} else if (x <= -5e-79) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -7.6e+117:
		tmp = (y / x) / (y + x)
	elif x <= -5e-79:
		tmp = y / ((y + x) * t_0)
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7.6e+117)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -5e-79)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / t_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 <= -7.6e+117)
		tmp = (y / x) / (y + x);
	elseif (x <= -5e-79)
		tmp = y / ((y + x) * t_0);
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e+117], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-79], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.6000000000000003e117

   1. Initial program 45.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Simplified 87.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   if -7.6000000000000003e117 < x < -4.99999999999999999e-79

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 95.8

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if -4.99999999999999999e-79 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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. 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. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) + 1}}}{x + y} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) + 1}}{x + y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) + 1}}{x + y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} + 1}}{x + y} \]

      8. associate-+l+ N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + \left(x + 1\right)}}}{x + y} \]

      10. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{y + \color{blue}{\left(x + 1\right)}}}{x + y} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + \left(x + 1\right)}}}{x + y} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\frac{\color{blue}{x}}{y + \left(x + 1\right)}}{x + y} \]
   8. Step-by-step derivation

   9. Simplified 62.8%

      \[\leadsto \frac{\frac{\color{blue}{x}}{y + \left(x + 1\right)}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+154)
   (/ (/ y x) x)
   (if (<= x -1.46e-77)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (if (<= x 3.9e+18) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / x) / x;
	} else if (x <= -1.46e-77) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else if (x <= 3.9e+18) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.46e-77)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	elseif (x <= 3.9e+18)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+154], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.46e-77], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e+18], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 44.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot {x}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      5. *-lowering-*.f64 44.8

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   5. Simplified 44.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{y}{x \cdot x}} \]

      2. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \frac{y}{x \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      6. /-lowering-/.f64 88.7

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -1.31999999999999998e154 < x < -1.45999999999999996e-77

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 96.1

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 77.5%

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if -1.45999999999999996e-77 < x < 3.9e18

   1. Initial program 73.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 80.6

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 3.9e18 < x

   1. Initial program 58.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 21.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 31.7

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 72000000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e+154)
   (/ (/ y x) x)
   (if (<= x -1.46e-77)
     (/ y (* (+ y x) (+ x 1.0)))
     (if (<= x 72000000000000.0) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+154) {
		tmp = (y / x) / x;
	} else if (x <= -1.46e-77) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (x <= 72000000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e+154)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.46e-77)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	elseif (x <= 72000000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e+154], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.46e-77], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 72000000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.5500000000000001e154

   1. Initial program 44.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot {x}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      5. *-lowering-*.f64 44.8

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   5. Simplified 44.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]
   6. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{x} \cdot \frac{y}{x \cdot x}} \]

      2. *-inverses N/A

         \[\leadsto \color{blue}{1} \cdot \frac{y}{x \cdot x} \]

      3. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      6. /-lowering-/.f64 88.7

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 88.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -1.5500000000000001e154 < x < -1.45999999999999996e-77

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 59.9

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 59.9%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   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. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      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. +-lowering-+.f64 63.8

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   9. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   if -1.45999999999999996e-77 < x < 7.2e13

   1. Initial program 72.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   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 81.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 7.2e13 < x

   1. Initial program 59.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-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 21.3

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 21.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 31.3

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 31.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.6e+117)
   (/ (/ y x) (+ y x))
   (if (<= x -4e-77)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.6e+117) {
		tmp = (y / x) / (y + x);
	} else if (x <= -4e-77) {
		tmp = y / ((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 <= (-7.6d+117)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-4d-77)) then
        tmp = y / ((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 <= -7.6e+117) {
		tmp = (y / x) / (y + x);
	} else if (x <= -4e-77) {
		tmp = y / ((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 <= -7.6e+117:
		tmp = (y / x) / (y + x)
	elif x <= -4e-77:
		tmp = y / ((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 <= -7.6e+117)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -4e-77)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(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 <= -7.6e+117)
		tmp = (y / x) / (y + x);
	elseif (x <= -4e-77)
		tmp = y / ((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, -7.6e+117], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-77], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 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 < -7.6000000000000003e117

   1. Initial program 45.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.3

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Simplified 87.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   if -7.6000000000000003e117 < x < -3.9999999999999997e-77

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 95.8

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 77.7%

      \[\leadsto \frac{\color{blue}{y}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if -3.9999999999999997e-77 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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 62.1

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   7. Simplified 62.1%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+194}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-160)
   (/ y (* (+ y x) (+ x 1.0)))
   (if (<= y 1.7e+194) (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-160) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (y <= 1.7e+194) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} 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 <= 3d-160) then
        tmp = y / ((y + x) * (x + 1.0d0))
    else if (y <= 1.7d+194) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-160) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (y <= 1.7e+194) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-160:
		tmp = y / ((y + x) * (x + 1.0))
	elif y <= 1.7e+194:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-160)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	elseif (y <= 1.7e+194)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-160)
		tmp = y / ((y + x) * (x + 1.0));
	elseif (y <= 1.7e+194)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-160], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+194], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.99999999999999997e-160

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   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 54.9

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   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. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      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. +-lowering-+.f64 54.9

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   9. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   if 2.99999999999999997e-160 < y < 1.7000000000000001e194

   1. Initial program 76.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. 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. 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

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{x + y}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{x + y}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{y + x}}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      14. +-lowering-+.f64 95.9

          \[\leadsto \frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \color{blue}{\left(x + 1\right)}\right)} \]

   6. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]
   8. Step-by-step derivation

   9. Simplified 72.1%

      \[\leadsto \frac{\color{blue}{x}}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \]

   if 1.7000000000000001e194 < y

   1. Initial program 61.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 77.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 77.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 94.2

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 94.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -4 \cdot 10^{-227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{-214}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 5000000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -4e-227)
     t_0
     (if (<= y 3.45e-214)
       (/ y x)
       (if (<= y 5000000000000.0) t_0 (/ x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -4e-227) {
		tmp = t_0;
	} else if (y <= 3.45e-214) {
		tmp = y / x;
	} else if (y <= 5000000000000.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-4d-227)) then
        tmp = t_0
    else if (y <= 3.45d-214) then
        tmp = y / x
    else if (y <= 5000000000000.0d0) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -4e-227) {
		tmp = t_0;
	} else if (y <= 3.45e-214) {
		tmp = y / x;
	} else if (y <= 5000000000000.0) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -4e-227:
		tmp = t_0
	elif y <= 3.45e-214:
		tmp = y / x
	elif y <= 5000000000000.0:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -4e-227)
		tmp = t_0;
	elseif (y <= 3.45e-214)
		tmp = Float64(y / x);
	elseif (y <= 5000000000000.0)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -4e-227)
		tmp = t_0;
	elseif (y <= 3.45e-214)
		tmp = y / x;
	elseif (y <= 5000000000000.0)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e-227], t$95$0, If[LessEqual[y, 3.45e-214], N[(y / x), $MachinePrecision], If[LessEqual[y, 5000000000000.0], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999978e-227 or 3.44999999999999996e-214 < y < 5e12

   1. Initial program 72.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 37.7

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -3.99999999999999978e-227 < y < 3.44999999999999996e-214

   1. Initial program 52.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto y \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      10. +-lowering-+.f64 69.7

          \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   4. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   5. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto y \cdot \frac{1}{\color{blue}{1 \cdot x + x \cdot x}} \]

      3. *-lft-identity N/A

         \[\leadsto y \cdot \frac{1}{\color{blue}{x} + x \cdot x} \]

      4. unpow2 N/A

         \[\leadsto y \cdot \frac{1}{x + \color{blue}{{x}^{2}}} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto y \cdot \frac{1}{\color{blue}{x + {x}^{2}}} \]

      6. unpow2 N/A

         \[\leadsto y \cdot \frac{1}{x + \color{blue}{x \cdot x}} \]

      7. *-lowering-*.f64 88.8

         \[\leadsto y \cdot \frac{1}{x + \color{blue}{x \cdot x}} \]

   7. Simplified 88.8%

      \[\leadsto y \cdot \color{blue}{\frac{1}{x + x \cdot x}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 85.8

         \[\leadsto \color{blue}{\frac{y}{x}} \]

   10. Simplified 85.8%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 5e12 < y

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-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 63.6

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-77)
   (/ y (* (+ y x) (+ x 1.0)))
   (if (<= x 7.8e+18) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-77) {
		tmp = y / ((y + x) * (x + 1.0));
	} else if (x <= 7.8e+18) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-77)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + 1.0)));
	elseif (x <= 7.8e+18)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.5e-77], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+18], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5000000000000001e-77

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 70.1

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 70.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   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. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      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. +-lowering-+.f64 69.6

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   9. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   if -4.5000000000000001e-77 < x < 7.8e18

   1. Initial program 73.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 80.6

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 7.8e18 < x

   1. Initial program 58.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 21.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   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 31.7

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -4.5e-77) (/ y (+ y x)) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.5e-77) {
		tmp = y / (y + x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -4.5e-77)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-77], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 65.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 68.8

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1 < x < -4.5000000000000001e-77

   1. Initial program 99.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 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 49.7

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 49.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
   9. Step-by-step derivation

   10. Simplified 39.3%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if -4.5000000000000001e-77 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 57.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-77) (/ y (* (+ y x) (+ x 1.0))) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-77) {
		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 (x <= -4.5e-77)
		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[x, -4.5e-77], 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 x < -4.5000000000000001e-77

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 70.1

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 70.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]
   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. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + 1\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

      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. +-lowering-+.f64 69.6

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \color{blue}{\left(x + 1\right)}} \]

   9. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}} \]

   if -4.5000000000000001e-77 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 57.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-77) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-77) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-77)
		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[x, -4.5e-77], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000001e-77

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. accelerator-lowering-fma.f64 66.6

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 66.6%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -4.5000000000000001e-77 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 57.9

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 43.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.46e-28) (/ y (+ y x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.46e-28) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.46d-28) then
        tmp = y / (y + x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.46e-28) {
		tmp = y / (y + x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.46e-28:
		tmp = y / (y + x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.46e-28)
		tmp = Float64(y / Float64(y + x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.46e-28)
		tmp = y / (y + x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.46e-28], N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.46e-28

   1. Initial program 68.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.4

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Simplified 55.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{y}}{x + y} \]
   9. Step-by-step derivation

   10. Simplified 28.6%

       \[\leadsto \frac{\color{blue}{y}}{x + y} \]

   if 1.46e-28 < y

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 26.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{y}{y + x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ y (+ y x)))

C

double code(double x, double y) {
	return y / (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / (y + x)
end function

Java

public static double code(double x, double y) {
	return y / (y + x);
}

Python

def code(x, y):
	return y / (y + x)

Julia

function code(x, y)
	return Float64(y / Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = y / (y + x);
end

Wolfram

code[x_, y_] := N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. 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 49.1

      \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

7. Simplified 49.1%

   \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{y}}{x + y} \]
9. Step-by-step derivation

10. Simplified 21.7%

    \[\leadsto \frac{\color{blue}{y}}{x + y} \]

11. Final simplification 21.7%

    \[\leadsto \frac{y}{y + x} \]
12. Add Preprocessing

Alternative 27: 26.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ y x))

C

double code(double x, double y) {
	return y / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / x
end function

Java

public static double code(double x, double y) {
	return y / x;
}

Python

def code(x, y):
	return y / x

Julia

function code(x, y)
	return Float64(y / x)
end

MATLAB

function tmp = code(x, y)
	tmp = y / x;
end

Wolfram

code[x_, y_] := N[(y / x), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   7. +-lowering-+.f64 N/A

      \[\leadsto y \cdot \frac{x}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   10. +-lowering-+.f64 81.4

       \[\leadsto y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

4. Applied egg-rr 81.4%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

5. Taylor expanded in y around 0

   \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto y \cdot \color{blue}{\frac{1}{x \cdot \left(1 + x\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto y \cdot \frac{1}{\color{blue}{1 \cdot x + x \cdot x}} \]

   3. *-lft-identity N/A

      \[\leadsto y \cdot \frac{1}{\color{blue}{x} + x \cdot x} \]

   4. unpow2 N/A

      \[\leadsto y \cdot \frac{1}{x + \color{blue}{{x}^{2}}} \]

   5. +-lowering-+.f64 N/A

      \[\leadsto y \cdot \frac{1}{\color{blue}{x + {x}^{2}}} \]

   6. unpow2 N/A

      \[\leadsto y \cdot \frac{1}{x + \color{blue}{x \cdot x}} \]

   7. *-lowering-*.f64 47.5

      \[\leadsto y \cdot \frac{1}{x + \color{blue}{x \cdot x}} \]

7. Simplified 47.5%

   \[\leadsto y \cdot \color{blue}{\frac{1}{x + x \cdot x}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 21.2

      \[\leadsto \color{blue}{\frac{y}{x}} \]

10. Simplified 21.2%

    \[\leadsto \color{blue}{\frac{y}{x}} \]
11. Add Preprocessing

Alternative 28: 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 68.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. 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 49.1

      \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

7. Simplified 49.1%

   \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 3.5%

    \[\leadsto \color{blue}{1} \]
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 2024198 
(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))))