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

Percentage Accurate: 68.5% → 99.8%

Time: 15.9s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. +-commutative 72.9%

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

   2. +-commutative 72.9%

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

   3. +-commutative 72.9%

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

   4. *-commutative 72.9%

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

   5. distribute-rgt1-in 61.5%

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

   6. fma-define 72.9%

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

   7. +-commutative 72.9%

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

   8. +-commutative 72.9%

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

   9. cube-unmult 72.9%

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

   10. +-commutative 72.9%

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

3. Simplified 72.9%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 72.9%

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

   2. fma-define 61.5%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 61.5%

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

   4. distribute-rgt1-in 72.9%

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

   5. *-commutative 72.9%

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

   6. associate-*l* 72.9%

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

   7. times-frac 93.7%

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

   8. associate-+r+ 93.7%

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

6. Applied egg-rr 93.7%

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

   1. clear-num 93.7%

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

   2. associate-/r* 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-+l+ 99.7%

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

   5. frac-times 99.6%

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

   6. +-commutative 99.6%

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

   7. +-commutative 99.6%

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

   8. +-commutative 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-lft-identity 99.6%

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

10. Simplified 99.6%

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

    1. *-un-lft-identity 99.6%

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

    2. +-commutative 99.6%

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

    3. times-frac 99.7%

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

    4. +-commutative 99.7%

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

    5. clear-num 99.8%

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

    6. +-commutative 99.8%

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

    7. associate-+l+ 99.8%

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

    8. +-commutative 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{y + x} \cdot t\_0\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ x (+ y 1.0))))
   (if (<= y 2e-289)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 5e-167)
       (* (/ y (+ y x)) t_0)
       (if (<= y 6.4e-13)
         (* x (/ y (* (+ x 1.0) (* (+ y x) (+ y x)))))
         (if (<= y 8e+116) (/ x (* (+ y x) t_1)) (/ t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= 2e-289) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 5e-167) {
		tmp = (y / (y + x)) * t_0;
	} else if (y <= 6.4e-13) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 / 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 = x / (y + x)
    t_1 = x + (y + 1.0d0)
    if (y <= 2d-289) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 5d-167) then
        tmp = (y / (y + x)) * t_0
    else if (y <= 6.4d-13) then
        tmp = x * (y / ((x + 1.0d0) * ((y + x) * (y + x))))
    else if (y <= 8d+116) then
        tmp = x / ((y + x) * t_1)
    else
        tmp = t_0 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= 2e-289) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 5e-167) {
		tmp = (y / (y + x)) * t_0;
	} else if (y <= 6.4e-13) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	t_1 = x + (y + 1.0)
	tmp = 0
	if y <= 2e-289:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 5e-167:
		tmp = (y / (y + x)) * t_0
	elif y <= 6.4e-13:
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))))
	elif y <= 8e+116:
		tmp = x / ((y + x) * t_1)
	else:
		tmp = t_0 / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 2e-289)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 5e-167)
		tmp = Float64(Float64(y / Float64(y + x)) * t_0);
	elseif (y <= 6.4e-13)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 8e+116)
		tmp = Float64(x / Float64(Float64(y + x) * t_1));
	else
		tmp = Float64(t_0 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 2e-289)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 5e-167)
		tmp = (y / (y + x)) * t_0;
	elseif (y <= 6.4e-13)
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	elseif (y <= 8e+116)
		tmp = x / ((y + x) * t_1);
	else
		tmp = t_0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2e-289], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-167], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 6.4e-13], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+116], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 2e-289

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. +-commutative 66.0%

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

      4. *-commutative 66.0%

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

      5. distribute-rgt1-in 50.4%

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

      6. fma-define 66.0%

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

      7. +-commutative 66.0%

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

      8. +-commutative 66.0%

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

      9. cube-unmult 66.0%

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

      10. +-commutative 66.0%

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

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.0%

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

      2. fma-define 50.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 50.4%

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

      4. distribute-rgt1-in 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

      7. times-frac 92.0%

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

      8. associate-+r+ 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around 0 58.8%

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

      1. +-commutative 58.8%

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

   9. Simplified 58.8%

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

       1. associate-*l/ 58.8%

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

       2. un-div-inv 58.8%

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

   11. Applied egg-rr 58.8%

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

   if 2e-289 < y < 5.0000000000000002e-167

   1. Initial program 79.1%

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

      1. +-commutative 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. *-commutative 79.1%

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

      5. distribute-rgt1-in 66.6%

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

      6. fma-define 79.1%

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

      7. +-commutative 79.1%

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

      8. +-commutative 79.1%

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

      9. cube-unmult 79.1%

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

      10. +-commutative 79.1%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 79.1%

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

      2. fma-define 66.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 66.6%

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

      4. distribute-rgt1-in 79.1%

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

      5. *-commutative 79.1%

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

      6. associate-*l* 79.1%

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

      7. times-frac 99.9%

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

      8. associate-+r+ 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 92.3%

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

      1. +-commutative 92.3%

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

   9. Simplified 92.3%

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

   10. Taylor expanded in y around 0 92.3%

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

   if 5.0000000000000002e-167 < y < 6.39999999999999999e-13

   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. Step-by-step derivation

      1. associate-/l* 92.6%

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

      2. associate-+l+ 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in y around 0 92.6%

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

      1. +-commutative 99.6%

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

   7. Simplified 92.6%

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

   if 6.39999999999999999e-13 < y < 8.00000000000000012e116

   1. Initial program 83.6%

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

      1. +-commutative 83.6%

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

      2. +-commutative 83.6%

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

      3. +-commutative 83.6%

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

      4. *-commutative 83.6%

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

      5. distribute-rgt1-in 77.0%

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

      6. fma-define 83.7%

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

      7. +-commutative 83.7%

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

      8. +-commutative 83.7%

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

      9. cube-unmult 83.6%

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

      10. +-commutative 83.6%

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

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 83.6%

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

      2. fma-define 76.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 77.0%

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

      4. distribute-rgt1-in 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l* 83.7%

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

      7. times-frac 96.7%

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

      8. associate-+r+ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in y around inf 78.0%

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

   if 8.00000000000000012e116 < y

   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. Step-by-step derivation

      1. +-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. +-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt1-in 70.5%

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

      6. fma-define 70.5%

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

      7. +-commutative 70.5%

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

      8. +-commutative 70.5%

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

      9. cube-unmult 70.5%

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

      10. +-commutative 70.5%

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

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. fma-define 70.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.5%

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

      4. distribute-rgt1-in 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 89.0%

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

      8. associate-+r+ 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

       1. *-un-lft-identity 99.9%

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

       2. +-commutative 99.9%

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

       3. times-frac 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+l+ 99.8%

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

       8. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around inf 93.6%

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{y + x}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-146)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 2.5e+138)
     (* (/ y (+ y x)) (/ x (* (+ y x) (+ x (+ y 1.0)))))
     (/ (/ x y) (* (/ (+ y x) y) (+ y (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.5e+138) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d-146)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 2.5d+138) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0d0))))
    else
        tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.5e+138) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-146:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 2.5e+138:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))))
	else:
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-146)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 2.5e+138)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(y + x) / y) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-146)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 2.5e+138)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + (y + 1.0))));
	else
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-146], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+138], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e-146

   1. Initial program 64.3%

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

      1. +-commutative 64.3%

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

      2. +-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. distribute-rgt1-in 48.6%

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

      6. fma-define 64.3%

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

      7. +-commutative 64.3%

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

      8. +-commutative 64.3%

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

      9. cube-unmult 64.4%

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

      10. +-commutative 64.4%

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

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 64.4%

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

      2. fma-define 48.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 48.6%

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

      4. distribute-rgt1-in 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 86.6%

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

      8. associate-+r+ 86.6%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in y around 0 38.1%

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

      1. +-commutative 38.1%

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

   9. Simplified 38.1%

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

       1. associate-*l/ 38.1%

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

       2. un-div-inv 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -2.2e-146 < y < 2.50000000000000008e138

   1. Initial program 78.3%

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

      1. +-commutative 78.3%

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

      2. +-commutative 78.3%

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

      3. +-commutative 78.3%

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

      4. *-commutative 78.3%

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

      5. distribute-rgt1-in 66.9%

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

      6. fma-define 78.3%

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

      7. +-commutative 78.3%

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

      8. +-commutative 78.3%

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

      9. cube-unmult 78.3%

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

      10. +-commutative 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 78.3%

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

      2. fma-define 66.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 66.9%

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

      4. distribute-rgt1-in 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-*l* 78.3%

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

      7. times-frac 99.2%

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

      8. associate-+r+ 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 2.50000000000000008e138 < y

   1. Initial program 67.9%

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

      1. +-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. +-commutative 67.9%

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

      4. *-commutative 67.9%

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

      5. distribute-rgt1-in 67.9%

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

      6. fma-define 67.9%

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

      7. +-commutative 67.9%

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

      8. +-commutative 67.9%

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

      9. cube-unmult 67.9%

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

      10. +-commutative 67.9%

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

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.9%

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

      2. fma-define 67.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 67.9%

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

      4. distribute-rgt1-in 67.9%

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

      5. *-commutative 67.9%

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

      6. associate-*l* 67.9%

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

      7. times-frac 84.7%

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

      8. associate-+r+ 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. clear-num 84.7%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 91.2%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y -2.2e-146)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 5.6e-10)
       (* (/ y (+ y x)) (/ x (* (+ y x) (+ x 1.0))))
       (if (<= y 8e+116) (/ x (* (+ y x) t_0)) (/ (/ x (+ y x)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 5.6e-10) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	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 = x + (y + 1.0d0)
    if (y <= (-2.2d-146)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 5.6d-10) then
        tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0d0)))
    else if (y <= 8d+116) then
        tmp = x / ((y + x) * t_0)
    else
        tmp = (x / (y + x)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 5.6e-10) {
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= -2.2e-146:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 5.6e-10:
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)))
	elif y <= 8e+116:
		tmp = x / ((y + x) * t_0)
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= -2.2e-146)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 5.6e-10)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(x + 1.0))));
	elseif (y <= 8e+116)
		tmp = Float64(x / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= -2.2e-146)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 5.6e-10)
		tmp = (y / (y + x)) * (x / ((y + x) * (x + 1.0)));
	elseif (y <= 8e+116)
		tmp = x / ((y + x) * t_0);
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e-146], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e-10], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+116], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2e-146

   1. Initial program 64.3%

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

      1. +-commutative 64.3%

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

      2. +-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. distribute-rgt1-in 48.6%

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

      6. fma-define 64.3%

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

      7. +-commutative 64.3%

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

      8. +-commutative 64.3%

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

      9. cube-unmult 64.4%

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

      10. +-commutative 64.4%

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

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 64.4%

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

      2. fma-define 48.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 48.6%

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

      4. distribute-rgt1-in 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 86.6%

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

      8. associate-+r+ 86.6%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in y around 0 38.1%

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

      1. +-commutative 38.1%

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

   9. Simplified 38.1%

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

       1. associate-*l/ 38.1%

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

       2. un-div-inv 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -2.2e-146 < y < 5.60000000000000031e-10

   1. Initial program 77.1%

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

      1. +-commutative 77.1%

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

      2. +-commutative 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. distribute-rgt1-in 63.2%

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

      6. fma-define 77.1%

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

      7. +-commutative 77.1%

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

      8. +-commutative 77.1%

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

      9. cube-unmult 77.2%

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

      10. +-commutative 77.2%

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

   3. Simplified 77.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 77.2%

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

      2. fma-define 63.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.2%

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

      4. distribute-rgt1-in 77.1%

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

      5. *-commutative 77.1%

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

      6. associate-*l* 77.1%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

   9. Simplified 99.8%

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

   if 5.60000000000000031e-10 < y < 8.00000000000000012e116

   1. Initial program 83.1%

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

      1. +-commutative 83.1%

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

      2. +-commutative 83.1%

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

      3. +-commutative 83.1%

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

      4. *-commutative 83.1%

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

      5. distribute-rgt1-in 76.2%

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

      6. fma-define 83.2%

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

      7. +-commutative 83.2%

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

      8. +-commutative 83.2%

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

      9. cube-unmult 83.1%

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

      10. +-commutative 83.1%

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

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 83.1%

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

      2. fma-define 76.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 76.2%

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

      4. distribute-rgt1-in 83.1%

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

      5. *-commutative 83.1%

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

      6. associate-*l* 83.2%

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

      7. times-frac 96.7%

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

      8. associate-+r+ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in y around inf 80.4%

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

   if 8.00000000000000012e116 < y

   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. Step-by-step derivation

      1. +-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. +-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt1-in 70.5%

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

      6. fma-define 70.5%

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

      7. +-commutative 70.5%

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

      8. +-commutative 70.5%

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

      9. cube-unmult 70.5%

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

      10. +-commutative 70.5%

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

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. fma-define 70.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.5%

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

      4. distribute-rgt1-in 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 89.0%

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

      8. associate-+r+ 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

       1. *-un-lft-identity 99.9%

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

       2. +-commutative 99.9%

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

       3. times-frac 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+l+ 99.8%

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

       8. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around inf 93.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 5 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + x} \cdot t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ x (+ y 1.0))))
   (if (<= y 5e-288)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 6.8e-86)
       (* (/ y (+ y x)) t_0)
       (if (<= y 8e+116) (/ x (* (+ y x) t_1)) (/ t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= 5e-288) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 6.8e-86) {
		tmp = (y / (y + x)) * t_0;
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 / 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 = x / (y + x)
    t_1 = x + (y + 1.0d0)
    if (y <= 5d-288) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 6.8d-86) then
        tmp = (y / (y + x)) * t_0
    else if (y <= 8d+116) then
        tmp = x / ((y + x) * t_1)
    else
        tmp = t_0 / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = x + (y + 1.0);
	double tmp;
	if (y <= 5e-288) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 6.8e-86) {
		tmp = (y / (y + x)) * t_0;
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_1);
	} else {
		tmp = t_0 / t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	t_1 = x + (y + 1.0)
	tmp = 0
	if y <= 5e-288:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 6.8e-86:
		tmp = (y / (y + x)) * t_0
	elif y <= 8e+116:
		tmp = x / ((y + x) * t_1)
	else:
		tmp = t_0 / t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 5e-288)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 6.8e-86)
		tmp = Float64(Float64(y / Float64(y + x)) * t_0);
	elseif (y <= 8e+116)
		tmp = Float64(x / Float64(Float64(y + x) * t_1));
	else
		tmp = Float64(t_0 / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 5e-288)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 6.8e-86)
		tmp = (y / (y + x)) * t_0;
	elseif (y <= 8e+116)
		tmp = x / ((y + x) * t_1);
	else
		tmp = t_0 / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5e-288], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e-86], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 8e+116], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 5.00000000000000011e-288

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. +-commutative 66.0%

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

      4. *-commutative 66.0%

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

      5. distribute-rgt1-in 50.4%

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

      6. fma-define 66.0%

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

      7. +-commutative 66.0%

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

      8. +-commutative 66.0%

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

      9. cube-unmult 66.0%

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

      10. +-commutative 66.0%

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

   3. Simplified 66.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 66.0%

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

      2. fma-define 50.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 50.4%

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

      4. distribute-rgt1-in 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

      7. times-frac 92.0%

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

      8. associate-+r+ 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around 0 58.8%

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

      1. +-commutative 58.8%

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

   9. Simplified 58.8%

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

       1. associate-*l/ 58.8%

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

       2. un-div-inv 58.8%

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

   11. Applied egg-rr 58.8%

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

   if 5.00000000000000011e-288 < y < 6.8000000000000001e-86

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

      2. +-commutative 81.8%

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

      3. +-commutative 81.8%

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

      4. *-commutative 81.8%

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

      5. distribute-rgt1-in 69.3%

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

      6. fma-define 81.8%

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

      7. +-commutative 81.8%

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

      8. +-commutative 81.8%

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

      9. cube-unmult 81.9%

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

      10. +-commutative 81.9%

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

   3. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 81.9%

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

      2. fma-define 69.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 69.3%

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

      4. distribute-rgt1-in 81.8%

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

      5. *-commutative 81.8%

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

      6. associate-*l* 81.8%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 83.0%

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

      1. +-commutative 83.0%

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

   9. Simplified 83.0%

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

   10. Taylor expanded in y around 0 83.0%

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

   if 6.8000000000000001e-86 < y < 8.00000000000000012e116

   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. Step-by-step derivation

      1. +-commutative 86.9%

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

      2. +-commutative 86.9%

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

      3. +-commutative 86.9%

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

      4. *-commutative 86.9%

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

      5. distribute-rgt1-in 78.0%

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

      6. fma-define 86.9%

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

      7. +-commutative 86.9%

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

      8. +-commutative 86.9%

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

      9. cube-unmult 86.9%

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

      10. +-commutative 86.9%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 86.9%

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

      2. fma-define 78.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 78.0%

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

      4. distribute-rgt1-in 86.9%

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

      5. *-commutative 86.9%

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

      6. associate-*l* 87.0%

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

      7. times-frac 97.7%

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

      8. associate-+r+ 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in y around inf 70.6%

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

   if 8.00000000000000012e116 < y

   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. Step-by-step derivation

      1. +-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. +-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt1-in 70.5%

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

      6. fma-define 70.5%

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

      7. +-commutative 70.5%

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

      8. +-commutative 70.5%

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

      9. cube-unmult 70.5%

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

      10. +-commutative 70.5%

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

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. fma-define 70.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.5%

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

      4. distribute-rgt1-in 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 89.0%

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

      8. associate-+r+ 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

       1. *-un-lft-identity 99.9%

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

       2. +-commutative 99.9%

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

       3. times-frac 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+l+ 99.8%

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

       8. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around inf 93.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{y + x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-146)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 8e-9)
     (/ (* y (/ x (+ y x))) (* (+ y x) (+ x 1.0)))
     (/ (/ x y) (* (/ (+ y x) y) (+ y (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 8e-9) {
		tmp = (y * (x / (y + x))) / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.2d-146)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 8d-9) then
        tmp = (y * (x / (y + x))) / ((y + x) * (x + 1.0d0))
    else
        tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-146) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 8e-9) {
		tmp = (y * (x / (y + x))) / ((y + x) * (x + 1.0));
	} else {
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.2e-146:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 8e-9:
		tmp = (y * (x / (y + x))) / ((y + x) * (x + 1.0))
	else:
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-146)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 8e-9)
		tmp = Float64(Float64(y * Float64(x / Float64(y + x))) / Float64(Float64(y + x) * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(Float64(y + x) / y) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.2e-146)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 8e-9)
		tmp = (y * (x / (y + x))) / ((y + x) * (x + 1.0));
	else
		tmp = (x / y) / (((y + x) / y) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-146], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-9], N[(N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e-146

   1. Initial program 64.3%

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

      1. +-commutative 64.3%

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

      2. +-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. *-commutative 64.3%

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

      5. distribute-rgt1-in 48.6%

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

      6. fma-define 64.3%

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

      7. +-commutative 64.3%

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

      8. +-commutative 64.3%

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

      9. cube-unmult 64.4%

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

      10. +-commutative 64.4%

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

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 64.4%

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

      2. fma-define 48.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 48.6%

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

      4. distribute-rgt1-in 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l* 64.3%

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

      7. times-frac 86.6%

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

      8. associate-+r+ 86.6%

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

   6. Applied egg-rr 86.6%

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

   7. Taylor expanded in y around 0 38.1%

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

      1. +-commutative 38.1%

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

   9. Simplified 38.1%

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

       1. associate-*l/ 38.1%

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

       2. un-div-inv 38.1%

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

   11. Applied egg-rr 38.1%

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

   if -2.2e-146 < y < 8.0000000000000005e-9

   1. Initial program 77.3%

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

      1. +-commutative 77.3%

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

      2. +-commutative 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. distribute-rgt1-in 63.6%

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

      6. fma-define 77.3%

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

      7. +-commutative 77.3%

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

      8. +-commutative 77.3%

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

      9. cube-unmult 77.4%

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

      10. +-commutative 77.4%

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

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 77.4%

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

      2. fma-define 63.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 77.3%

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

      5. *-commutative 77.3%

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

      6. associate-*l* 77.4%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

   9. Simplified 99.8%

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

       1. *-commutative 99.8%

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

       2. associate-/r* 99.9%

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

       3. +-commutative 99.9%

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

       4. frac-times 99.8%

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

       5. +-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   if 8.0000000000000005e-9 < y

   1. Initial program 75.3%

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

      1. +-commutative 75.3%

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

      2. +-commutative 75.3%

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

      3. +-commutative 75.3%

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

      4. *-commutative 75.3%

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

      5. distribute-rgt1-in 72.5%

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

      6. fma-define 75.3%

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

      7. +-commutative 75.3%

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

      8. +-commutative 75.3%

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

      9. cube-unmult 75.3%

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

      10. +-commutative 75.3%

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

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 75.3%

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

      2. fma-define 72.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 72.5%

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

      4. distribute-rgt1-in 75.3%

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

      5. *-commutative 75.3%

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

      6. associate-*l* 75.3%

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

      7. times-frac 92.0%

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

      8. associate-+r+ 92.0%

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

   6. Applied egg-rr 92.0%

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

      1. clear-num 92.0%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. frac-times 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 89.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-9}:\\ \;\;\;\;\frac{y \cdot \frac{x}{y + x}}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 1.12 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1.12e-172)
     (/ (/ y x) (+ x 1.0))
     (if (<= y 8e+116) (/ x (* (+ y x) t_0)) (/ (/ x (+ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.12e-172) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	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 = x + (y + 1.0d0)
    if (y <= 1.12d-172) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 8d+116) then
        tmp = x / ((y + x) * t_0)
    else
        tmp = (x / (y + x)) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.12e-172) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 8e+116) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + x)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 1.12e-172:
		tmp = (y / x) / (x + 1.0)
	elif y <= 8e+116:
		tmp = x / ((y + x) * t_0)
	else:
		tmp = (x / (y + x)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.12e-172)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 8e+116)
		tmp = Float64(x / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 1.12e-172)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 8e+116)
		tmp = x / ((y + x) * t_0);
	else
		tmp = (x / (y + x)) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.12e-172], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+116], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.11999999999999996e-172

   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. Step-by-step derivation

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. +-commutative 68.5%

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

      4. *-commutative 68.5%

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

      5. distribute-rgt1-in 53.3%

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

      6. fma-define 68.5%

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

      7. +-commutative 68.5%

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

      8. +-commutative 68.5%

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

      9. cube-unmult 68.5%

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

      10. +-commutative 68.5%

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

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.5%

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

      2. fma-define 53.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.3%

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

      4. distribute-rgt1-in 68.5%

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

      5. *-commutative 68.5%

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

      6. associate-*l* 68.5%

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

      7. times-frac 93.2%

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

      8. associate-+r+ 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. clear-num 93.1%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

      5. frac-times 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in y around 0 61.8%

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

       1. associate-/r* 63.5%

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

       2. +-commutative 63.5%

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

   13. Simplified 63.5%

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

   if 1.11999999999999996e-172 < y < 8.00000000000000012e116

   1. Initial program 85.2%

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

      1. +-commutative 85.2%

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

      2. +-commutative 85.2%

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

      3. +-commutative 85.2%

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

      4. *-commutative 85.2%

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

      5. distribute-rgt1-in 75.5%

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

      6. fma-define 85.2%

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

      7. +-commutative 85.2%

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

      8. +-commutative 85.2%

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

      9. cube-unmult 85.2%

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

      10. +-commutative 85.2%

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

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 85.2%

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

      2. fma-define 75.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 75.5%

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

      4. distribute-rgt1-in 85.2%

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

      5. *-commutative 85.2%

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

      6. associate-*l* 85.3%

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

      7. times-frac 98.2%

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

      8. associate-+r+ 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in y around inf 66.6%

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

   if 8.00000000000000012e116 < y

   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. Step-by-step derivation

      1. +-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. +-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt1-in 70.5%

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

      6. fma-define 70.5%

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

      7. +-commutative 70.5%

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

      8. +-commutative 70.5%

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

      9. cube-unmult 70.5%

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

      10. +-commutative 70.5%

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

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.5%

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

      2. fma-define 70.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 70.5%

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

      4. distribute-rgt1-in 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 89.0%

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

      8. associate-+r+ 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

       1. *-un-lft-identity 99.9%

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

       2. +-commutative 99.9%

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

       3. times-frac 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.8%

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

       6. +-commutative 99.8%

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

       7. associate-+l+ 99.8%

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

       8. +-commutative 99.8%

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

   12. Applied egg-rr 99.8%

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

   13. Taylor expanded in y around inf 93.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-173)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.5e+138) (/ x (* (+ y x) (+ x (+ y 1.0)))) (/ (/ x (+ y x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-173) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.5e+138) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-173) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.5d+138) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-173) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.5e+138) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-173:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.5e+138:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-173)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.5e+138)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-173)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.5e+138)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-173], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+138], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.10000000000000005e-173

   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. Step-by-step derivation

      1. +-commutative 68.5%

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

      2. +-commutative 68.5%

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

      3. +-commutative 68.5%

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

      4. *-commutative 68.5%

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

      5. distribute-rgt1-in 53.3%

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

      6. fma-define 68.5%

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

      7. +-commutative 68.5%

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

      8. +-commutative 68.5%

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

      9. cube-unmult 68.5%

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

      10. +-commutative 68.5%

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

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.5%

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

      2. fma-define 53.3%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.3%

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

      4. distribute-rgt1-in 68.5%

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

      5. *-commutative 68.5%

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

      6. associate-*l* 68.5%

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

      7. times-frac 93.2%

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

      8. associate-+r+ 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. clear-num 93.1%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-+l+ 99.7%

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

      5. frac-times 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in y around 0 61.8%

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

       1. associate-/r* 63.5%

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

       2. +-commutative 63.5%

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

   13. Simplified 63.5%

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

   if 3.10000000000000005e-173 < y < 2.50000000000000008e138

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. +-commutative 83.9%

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

      3. +-commutative 83.9%

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

      4. *-commutative 83.9%

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

      5. distribute-rgt1-in 75.8%

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

      6. fma-define 83.9%

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

      7. +-commutative 83.9%

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

      8. +-commutative 83.9%

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

      9. cube-unmult 83.9%

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

      10. +-commutative 83.9%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 83.9%

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

      2. fma-define 75.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 75.8%

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

      4. distribute-rgt1-in 83.9%

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

      5. *-commutative 83.9%

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

      6. associate-*l* 83.9%

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

      7. times-frac 98.5%

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

      8. associate-+r+ 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in y around inf 72.0%

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

   if 2.50000000000000008e138 < y

   1. Initial program 67.9%

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

      1. +-commutative 67.9%

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

      2. +-commutative 67.9%

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

      3. +-commutative 67.9%

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

      4. *-commutative 67.9%

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

      5. distribute-rgt1-in 67.9%

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

      6. fma-define 67.9%

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

      7. +-commutative 67.9%

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

      8. +-commutative 67.9%

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

      9. cube-unmult 67.9%

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

      10. +-commutative 67.9%

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

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.9%

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

      2. fma-define 67.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 67.9%

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

      4. distribute-rgt1-in 67.9%

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

      5. *-commutative 67.9%

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

      6. associate-*l* 67.9%

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

      7. times-frac 84.7%

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

      8. associate-+r+ 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. clear-num 84.7%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. frac-times 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in y around inf 91.1%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+138}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -940000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -940000.0)
   (/ (/ y (+ x 1.0)) (+ y x))
   (* (/ y (+ y x)) (/ x (* (+ y x) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -940000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (y / (y + x)) * (x / ((y + x) * (y + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-940000.0d0)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (y / (y + x)) * (x / ((y + x) * (y + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -940000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (y / (y + x)) * (x / ((y + x) * (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -940000.0:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (y / (y + x)) * (x / ((y + x) * (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -940000.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(y + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -940000.0)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (y / (y + x)) * (x / ((y + x) * (y + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -940000.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4e5

   1. Initial program 67.3%

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

      1. +-commutative 67.3%

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

      2. +-commutative 67.3%

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

      3. +-commutative 67.3%

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

      4. *-commutative 67.3%

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

      5. distribute-rgt1-in 32.7%

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

      6. fma-define 67.3%

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

      7. +-commutative 67.3%

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

      8. +-commutative 67.3%

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

      9. cube-unmult 67.4%

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

      10. +-commutative 67.4%

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

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.4%

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

      2. fma-define 32.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 32.7%

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

      4. distribute-rgt1-in 67.3%

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

      5. *-commutative 67.3%

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

      6. associate-*l* 67.3%

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

      7. times-frac 88.8%

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

      8. associate-+r+ 88.8%

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

   6. Applied egg-rr 88.8%

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

   7. Taylor expanded in y around 0 83.0%

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

      1. +-commutative 83.0%

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

   9. Simplified 83.0%

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

       1. associate-*l/ 82.9%

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

       2. un-div-inv 83.0%

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

   11. Applied egg-rr 83.0%

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

   if -9.4e5 < x

   1. Initial program 74.5%

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

      1. +-commutative 74.5%

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

      2. +-commutative 74.5%

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

      3. +-commutative 74.5%

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

      4. *-commutative 74.5%

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

      5. distribute-rgt1-in 69.9%

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

      6. fma-define 74.5%

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

      7. +-commutative 74.5%

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

      8. +-commutative 74.5%

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

      9. cube-unmult 74.5%

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

      10. +-commutative 74.5%

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

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 74.5%

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

      2. fma-define 69.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 69.9%

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

      4. distribute-rgt1-in 74.5%

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

      5. *-commutative 74.5%

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

      6. associate-*l* 74.5%

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

      7. times-frac 95.2%

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

      8. associate-+r+ 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in x around 0 83.7%

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

      1. +-commutative 83.7%

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

   9. Simplified 83.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -940000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-52)
   (/ y (* x (+ x 1.0)))
   (if (<= y 2e+144) (/ x (* y (+ y 1.0))) (/ 1.0 (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-52) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2e+144) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-52) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 2d+144) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-52) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2e+144) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-52:
		tmp = y / (x * (x + 1.0))
	elif y <= 2e+144:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-52)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 2e+144)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-52)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 2e+144)
		tmp = x / (y * (y + 1.0));
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-52], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+144], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2999999999999999e-52

   1. Initial program 70.6%

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

      1. associate-/l* 78.7%

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

      2. associate-+l+ 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in y around 0 64.0%

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

      1. +-commutative 64.0%

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

   7. Simplified 64.0%

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

   if 1.2999999999999999e-52 < y < 2.00000000000000005e144

   1. Initial program 83.2%

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

      1. associate-/l* 87.2%

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

      2. associate-+l+ 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in x around 0 69.0%

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

   if 2.00000000000000005e144 < y

   1. Initial program 69.2%

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

      1. associate-/l* 84.2%

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

      2. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in x around 0 84.2%

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

      1. clear-num 84.2%

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

      2. inv-pow 84.2%

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

      3. distribute-rgt-in 84.2%

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

      4. *-un-lft-identity 84.2%

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

      5. pow2 84.2%

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

   7. Applied egg-rr 84.2%

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

      1. unpow-1 84.2%

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

      2. *-rgt-identity 84.2%

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

      3. unpow2 84.2%

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

      4. distribute-lft-in 84.2%

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

      5. associate-/l* 90.7%

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

      6. +-commutative 90.7%

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

   9. Simplified 90.7%

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

   10. Taylor expanded in y around inf 90.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-155)
   (/ y x)
   (if (<= y 2e+144) (/ x (* y (+ y 1.0))) (/ 1.0 (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-155) {
		tmp = y / x;
	} else if (y <= 2e+144) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.6d-155) then
        tmp = y / x
    else if (y <= 2d+144) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-155) {
		tmp = y / x;
	} else if (y <= 2e+144) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e-155:
		tmp = y / x
	elif y <= 2e+144:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-155)
		tmp = Float64(y / x);
	elseif (y <= 2e+144)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-155)
		tmp = y / x;
	elseif (y <= 2e+144)
		tmp = x / (y * (y + 1.0));
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-155], N[(y / x), $MachinePrecision], If[LessEqual[y, 2e+144], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.60000000000000006e-155

   1. Initial program 68.7%

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

      1. +-commutative 68.7%

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

      2. +-commutative 68.7%

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

      3. +-commutative 68.7%

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

      4. *-commutative 68.7%

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

      5. distribute-rgt1-in 52.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 78.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   10. Taylor expanded in y around 0 43.9%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.60000000000000006e-155 < y < 2.00000000000000005e144

   1. Initial program 83.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 87.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 87.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if 2.00000000000000005e144 < y

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.2%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 84.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. clear-num 84.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \left(1 + y\right)}{x}}} \]

      2. inv-pow 84.2%

         \[\leadsto \color{blue}{{\left(\frac{y \cdot \left(1 + y\right)}{x}\right)}^{-1}} \]

      3. distribute-rgt-in 84.2%

         \[\leadsto {\left(\frac{\color{blue}{1 \cdot y + y \cdot y}}{x}\right)}^{-1} \]

      4. *-un-lft-identity 84.2%

         \[\leadsto {\left(\frac{\color{blue}{y} + y \cdot y}{x}\right)}^{-1} \]

      5. pow2 84.2%

         \[\leadsto {\left(\frac{y + \color{blue}{{y}^{2}}}{x}\right)}^{-1} \]

   7. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{{\left(\frac{y + {y}^{2}}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 84.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + {y}^{2}}{x}}} \]

      2. *-rgt-identity 84.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot 1} + {y}^{2}}{x}} \]

      3. unpow2 84.2%

         \[\leadsto \frac{1}{\frac{y \cdot 1 + \color{blue}{y \cdot y}}{x}} \]

      4. distribute-lft-in 84.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(1 + y\right)}}{x}} \]

      5. associate-/l* 90.7%

         \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1 + y}{x}}} \]

      6. +-commutative 90.7%

         \[\leadsto \frac{1}{y \cdot \frac{\color{blue}{y + 1}}{x}} \]

   9. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \frac{y + 1}{x}}} \]

   10. Taylor expanded in y around inf 90.7%

       \[\leadsto \frac{1}{y \cdot \color{blue}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e-155) (/ y x) (if (<= y 1.0) (/ x y) (/ 1.0 (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-155) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d-155) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e-155) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-155:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-155)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-155)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-155], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.00000000000000001e-155

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 52.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 78.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   10. Taylor expanded in y around 0 43.9%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.00000000000000001e-155 < y < 1

   1. Initial program 89.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 92.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 24.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 23.0%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. clear-num 79.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot \left(1 + y\right)}{x}}} \]

      2. inv-pow 79.7%

         \[\leadsto \color{blue}{{\left(\frac{y \cdot \left(1 + y\right)}{x}\right)}^{-1}} \]

      3. distribute-rgt-in 79.7%

         \[\leadsto {\left(\frac{\color{blue}{1 \cdot y + y \cdot y}}{x}\right)}^{-1} \]

      4. *-un-lft-identity 79.7%

         \[\leadsto {\left(\frac{\color{blue}{y} + y \cdot y}{x}\right)}^{-1} \]

      5. pow2 79.7%

         \[\leadsto {\left(\frac{y + \color{blue}{{y}^{2}}}{x}\right)}^{-1} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{{\left(\frac{y + {y}^{2}}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 79.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + {y}^{2}}{x}}} \]

      2. *-rgt-identity 79.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot 1} + {y}^{2}}{x}} \]

      3. unpow2 79.7%

         \[\leadsto \frac{1}{\frac{y \cdot 1 + \color{blue}{y \cdot y}}{x}} \]

      4. distribute-lft-in 79.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(1 + y\right)}}{x}} \]

      5. associate-/l* 82.5%

         \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1 + y}{x}}} \]

      6. +-commutative 82.5%

         \[\leadsto \frac{1}{y \cdot \frac{\color{blue}{y + 1}}{x}} \]

   9. Simplified 82.5%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \frac{y + 1}{x}}} \]

   10. Taylor expanded in y around inf 82.5%

       \[\leadsto \frac{1}{y \cdot \color{blue}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-155) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-155) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-155) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-155) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-155:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-155)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-155)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-155], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8999999999999999e-155

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 52.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 78.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   10. Taylor expanded in y around 0 43.9%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 1.8999999999999999e-155 < y < 1

   1. Initial program 89.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 92.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 92.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 24.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 23.0%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 81.7%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-29) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-29) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d-29)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-29) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-29:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-29)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e-29)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-29], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999984e-29

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 39.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 70.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 70.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 39.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 39.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 90.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 90.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in y around 0 79.8%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

   9. Simplified 79.8%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]
   10. Step-by-step derivation

       1. associate-*l/ 79.7%

          \[\leadsto \color{blue}{\frac{y \cdot \frac{1}{x + 1}}{x + y}} \]

       2. un-div-inv 79.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   11. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + 1}}{x + y}} \]

   if -4.79999999999999984e-29 < x

   1. Initial program 73.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 57.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 57.8%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-52) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y x)) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-52) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-52) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-52) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-52:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-52)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-52)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-52], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e-52

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 55.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 70.6%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 55.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 55.8%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 94.1%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 94.1%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 94.1%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 94.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. +-commutative 99.6%

         \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. +-commutative 99.6%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. +-commutative 99.6%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.6%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)} \]

   10. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   11. Taylor expanded in y around 0 64.0%

       \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   12. Step-by-step derivation

       1. associate-/r* 65.5%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 65.5%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   13. Simplified 65.5%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 3e-52 < y

   1. Initial program 78.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 78.0%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 78.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 78.0%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 78.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 74.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 78.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 78.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 78.1%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 78.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 78.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 78.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 74.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 74.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 78.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 78.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 78.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 92.9%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 92.9%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 92.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. +-commutative 99.8%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   11. Taylor expanded in x around 0 77.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{1 + y}} \]
   12. Step-by-step derivation

       1. +-commutative 77.3%

          \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]

   13. Simplified 77.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.3e-30) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-30) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.3d-30)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.3e-30) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.3e-30:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.3e-30)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.3e-30)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.3e-30], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999984e-30

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 39.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 70.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 70.6%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 70.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 70.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 70.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 39.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 39.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 70.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 90.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 90.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. clear-num 90.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)} \]

      2. associate-/r* 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}} \]

      3. +-commutative 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{x + \color{blue}{\left(1 + y\right)}} \]

      4. associate-+l+ 99.8%

         \[\leadsto \frac{1}{\frac{x + y}{y}} \cdot \frac{\frac{x}{x + y}}{\color{blue}{\left(x + 1\right) + y}} \]

      5. frac-times 99.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)}} \]

      6. +-commutative 99.3%

         \[\leadsto \frac{1 \cdot \frac{x}{\color{blue}{y + x}}}{\frac{x + y}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      7. +-commutative 99.3%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{\color{blue}{y + x}}{y} \cdot \left(\left(x + 1\right) + y\right)} \]

      8. +-commutative 99.3%

         \[\leadsto \frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   8. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 99.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)} \]

   10. Simplified 99.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)}} \]

   11. Taylor expanded in y around 0 77.7%

       \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   12. Step-by-step derivation

       1. associate-/r* 79.3%

          \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

       2. +-commutative 79.3%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   13. Simplified 79.3%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -2.29999999999999984e-30 < x

   1. Initial program 73.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 56.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 57.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 57.8%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 57.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.8e-52) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-52) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.8d-52) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-52) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.8e-52:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.8e-52)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.8e-52)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.8e-52], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.8000000000000003e-52

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 78.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 78.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 78.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 5.8000000000000003e-52 < y

   1. Initial program 78.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 86.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 86.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 86.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 74.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 76.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 76.9%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 33.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.5e-156) (/ y x) (/ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-156) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-156) then
        tmp = y / x
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-156) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-156:
		tmp = y / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-156)
		tmp = Float64(y / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-156)
		tmp = y / x;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-156], N[(y / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999986e-156

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 52.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 78.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   10. Taylor expanded in y around 0 43.9%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 4.49999999999999986e-156 < y

   1. Initial program 79.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 86.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 86.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 31.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

      1. clear-num 32.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

      2. inv-pow 32.1%

         \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

   8. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 32.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   10. Simplified 32.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 33.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.2e-156) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-156) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-156) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-156) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-156:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-156)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-156)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-156], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999998e-156

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 52.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 68.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 68.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 68.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 68.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 93.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 93.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative 78.5%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 78.5%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   10. Taylor expanded in y around 0 43.9%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 9.1999999999999998e-156 < y

   1. Initial program 79.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 86.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 86.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 31.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 81.0%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 81.0%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 46.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 23.9%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× 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 2024170 
(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))))