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

Percentage Accurate: 69.2% → 99.8%

Time: 15.6s

Alternatives: 22

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: 69.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. *-commutative 69.0%

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

   2. associate-*l* 69.0%

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

   3. times-frac 91.9%

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

   4. +-commutative 91.9%

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

   5. +-commutative 91.9%

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

   6. associate-+r+ 91.9%

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

   7. +-commutative 91.9%

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

   8. associate-+l+ 91.9%

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

4. Applied egg-rr 91.9%

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

   1. *-commutative 91.9%

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

   2. associate-/r* 99.8%

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

   3. clear-num 99.8%

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

   4. frac-times 99.5%

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

   5. +-commutative 99.5%

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

   6. +-commutative 99.5%

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

   7. +-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-rgt-identity 99.5%

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

   2. *-commutative 99.5%

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

   3. times-frac 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. associate-+l+ 99.7%

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

   7. *-rgt-identity 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in x around 0 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{x}{x}}{\frac{x + y}{y}}}{t\_0}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y -1.45e-69)
     (/ (/ (/ x x) (/ (+ x y) y)) t_0)
     (if (<= y 1.2e-48)
       (/ y (* (+ x y) (+ (+ x 1.0) (* y (/ 1.0 x)))))
       (if (<= y 1.6e+99)
         (* x (/ y (* t_0 (* (+ x y) (+ x y)))))
         (/ (/ x (+ x y)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= -1.45e-69) {
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	} else if (y <= 1.2e-48) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 1.6e+99) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / 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 <= (-1.45d-69)) then
        tmp = ((x / x) / ((x + y) / y)) / t_0
    else if (y <= 1.2d-48) then
        tmp = y / ((x + y) * ((x + 1.0d0) + (y * (1.0d0 / x))))
    else if (y <= 1.6d+99) then
        tmp = x * (y / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / 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 <= -1.45e-69) {
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	} else if (y <= 1.2e-48) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 1.6e+99) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= -1.45e-69:
		tmp = ((x / x) / ((x + y) / y)) / t_0
	elif y <= 1.2e-48:
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))))
	elif y <= 1.6e+99:
		tmp = x * (y / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1.45e-69)
		tmp = Float64(Float64(Float64(x / x) / Float64(Float64(x + y) / y)) / t_0);
	elseif (y <= 1.2e-48)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(Float64(x + 1.0) + Float64(y * Float64(1.0 / x)))));
	elseif (y <= 1.6e+99)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= -1.45e-69)
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	elseif (y <= 1.2e-48)
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	elseif (y <= 1.6e+99)
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / 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, -1.45e-69], N[(N[(N[(x / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.2e-48], N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+99], N[(x * N[(y / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4499999999999999e-69

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. times-frac 87.1%

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

      4. +-commutative 87.1%

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

      5. +-commutative 87.1%

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

      6. associate-+r+ 87.1%

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

      7. +-commutative 87.1%

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

      8. associate-+l+ 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 40.1%

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

   if -1.4499999999999999e-69 < y < 1.2e-48

   1. Initial program 66.6%

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

      1. *-commutative 66.6%

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

      2. associate-*l* 66.5%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.7%

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

      4. *-un-lft-identity 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0 99.7%

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

   if 1.2e-48 < y < 1.6e99

   1. Initial program 90.7%

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

      1. associate-/l* 90.7%

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

      2. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   if 1.6e99 < y

   1. Initial program 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l* 56.2%

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

      3. times-frac 77.8%

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

      4. +-commutative 77.8%

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

      5. +-commutative 77.8%

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

      6. associate-+r+ 77.8%

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

      7. +-commutative 77.8%

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

      8. associate-+l+ 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.7%

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

      5. +-commutative 98.7%

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

      6. +-commutative 98.7%

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

      7. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. *-rgt-identity 98.7%

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

      2. *-commutative 98.7%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 84.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{x}{x}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot 2\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.15e+181)
     (/ (/ y x) t_0)
     (if (<= x -2.35e-7)
       (/ y (* (+ x y) (+ (+ x 1.0) (* y 2.0))))
       (if (<= x 5.4e+71)
         (* (/ y (+ x y)) (/ x (* (+ x y) (+ y 1.0))))
         (/ (/ x (+ x y)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.15e+181) {
		tmp = (y / x) / t_0;
	} else if (x <= -2.35e-7) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	} else if (x <= 5.4e+71) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / 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 (x <= (-1.15d+181)) then
        tmp = (y / x) / t_0
    else if (x <= (-2.35d-7)) then
        tmp = y / ((x + y) * ((x + 1.0d0) + (y * 2.0d0)))
    else if (x <= 5.4d+71) then
        tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0d0)))
    else
        tmp = (x / (x + y)) / 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 (x <= -1.15e+181) {
		tmp = (y / x) / t_0;
	} else if (x <= -2.35e-7) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	} else if (x <= 5.4e+71) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	} else {
		tmp = (x / (x + y)) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.15e+181:
		tmp = (y / x) / t_0
	elif x <= -2.35e-7:
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)))
	elif x <= 5.4e+71:
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)))
	else:
		tmp = (x / (x + y)) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.15e+181)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -2.35e-7)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(Float64(x + 1.0) + Float64(y * 2.0))));
	elseif (x <= 5.4e+71)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.15e+181)
		tmp = (y / x) / t_0;
	elseif (x <= -2.35e-7)
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	elseif (x <= 5.4e+71)
		tmp = (y / (x + y)) * (x / ((x + y) * (y + 1.0)));
	else
		tmp = (x / (x + y)) / 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[x, -1.15e+181], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -2.35e-7], N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+71], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.1499999999999999e181

   1. Initial program 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*l* 48.1%

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

      3. times-frac 73.8%

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

      4. +-commutative 73.8%

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

      5. +-commutative 73.8%

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

      6. associate-+r+ 73.8%

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

      7. +-commutative 73.8%

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

      8. associate-+l+ 73.8%

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

   4. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-rgt-identity 98.0%

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

      2. *-commutative 98.0%

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

      3. times-frac 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. *-rgt-identity 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 86.3%

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

   if -1.1499999999999999e181 < x < -2.35e-7

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l* 77.2%

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

      3. times-frac 94.9%

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

      4. +-commutative 94.9%

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

      5. +-commutative 94.9%

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

      6. associate-+r+ 94.9%

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

      7. +-commutative 94.9%

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

      8. associate-+l+ 94.9%

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

   4. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. clear-num 94.9%

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

      3. frac-times 95.1%

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

      4. *-un-lft-identity 95.1%

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

      5. +-commutative 95.1%

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

      6. +-commutative 95.1%

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

      7. +-commutative 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. +-commutative 84.7%

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

   9. Simplified 84.7%

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

   10. Taylor expanded in x around inf 84.5%

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

       1. *-commutative 84.5%

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

   12. Simplified 84.5%

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

   if -2.35e-7 < x < 5.39999999999999993e71

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*l* 77.1%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 96.5%

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

      1. +-commutative 96.5%

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

   7. Simplified 96.5%

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

   if 5.39999999999999993e71 < x

   1. Initial program 52.1%

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

      1. *-commutative 52.1%

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

      2. associate-*l* 52.1%

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

      3. times-frac 77.3%

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

      4. +-commutative 77.3%

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

      5. +-commutative 77.3%

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

      6. associate-+r+ 77.3%

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

      7. +-commutative 77.3%

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

      8. associate-+l+ 77.3%

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

   4. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-rgt-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 26.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot 2\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+71}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot 2\right)}\\ \mathbf{elif}\;x \leq -1.58 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.5e+181)
     (/ (/ y x) t_0)
     (if (<= x -1e-7)
       (/ y (* (+ x y) (+ (+ x 1.0) (* y 2.0))))
       (if (<= x -1.58e-162)
         (* x (/ y (* (+ y 1.0) (* (+ x y) (+ x y)))))
         (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.5e+181) {
		tmp = (y / x) / t_0;
	} else if (x <= -1e-7) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	} else if (x <= -1.58e-162) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / 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 (x <= (-1.5d+181)) then
        tmp = (y / x) / t_0
    else if (x <= (-1d-7)) then
        tmp = y / ((x + y) * ((x + 1.0d0) + (y * 2.0d0)))
    else if (x <= (-1.58d-162)) then
        tmp = x * (y / ((y + 1.0d0) * ((x + y) * (x + y))))
    else
        tmp = (x / y) / 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 (x <= -1.5e+181) {
		tmp = (y / x) / t_0;
	} else if (x <= -1e-7) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	} else if (x <= -1.58e-162) {
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.5e+181:
		tmp = (y / x) / t_0
	elif x <= -1e-7:
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)))
	elif x <= -1.58e-162:
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.5e+181)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (x <= -1e-7)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(Float64(x + 1.0) + Float64(y * 2.0))));
	elseif (x <= -1.58e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(y + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.5e+181)
		tmp = (y / x) / t_0;
	elseif (x <= -1e-7)
		tmp = y / ((x + y) * ((x + 1.0) + (y * 2.0)));
	elseif (x <= -1.58e-162)
		tmp = x * (y / ((y + 1.0) * ((x + y) * (x + y))));
	else
		tmp = (x / y) / 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[x, -1.5e+181], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -1e-7], N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.58e-162], N[(x * N[(y / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.50000000000000006e181

   1. Initial program 48.1%

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

      1. *-commutative 48.1%

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

      2. associate-*l* 48.1%

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

      3. times-frac 73.8%

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

      4. +-commutative 73.8%

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

      5. +-commutative 73.8%

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

      6. associate-+r+ 73.8%

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

      7. +-commutative 73.8%

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

      8. associate-+l+ 73.8%

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

   4. Applied egg-rr 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-rgt-identity 98.0%

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

      2. *-commutative 98.0%

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

      3. times-frac 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. *-rgt-identity 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 86.3%

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

   if -1.50000000000000006e181 < x < -9.9999999999999995e-8

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l* 77.2%

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

      3. times-frac 94.9%

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

      4. +-commutative 94.9%

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

      5. +-commutative 94.9%

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

      6. associate-+r+ 94.9%

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

      7. +-commutative 94.9%

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

      8. associate-+l+ 94.9%

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

   4. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. clear-num 94.9%

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

      3. frac-times 95.1%

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

      4. *-un-lft-identity 95.1%

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

      5. +-commutative 95.1%

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

      6. +-commutative 95.1%

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

      7. +-commutative 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around 0 84.7%

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

      1. associate-+r+ 84.7%

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

      2. +-commutative 84.7%

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

   9. Simplified 84.7%

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

   10. Taylor expanded in x around inf 84.5%

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

       1. *-commutative 84.5%

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

   12. Simplified 84.5%

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

   if -9.9999999999999995e-8 < x < -1.5800000000000001e-162

   1. Initial program 90.3%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 97.1%

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

      1. +-commutative 97.1%

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

   7. Simplified 97.1%

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

   if -1.5800000000000001e-162 < x

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 67.3%

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

      2. associate-*l* 67.3%

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

      3. times-frac 93.2%

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

      4. +-commutative 93.2%

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

      5. +-commutative 93.2%

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

      6. associate-+r+ 93.2%

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

      7. +-commutative 93.2%

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

      8. associate-+l+ 93.2%

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

   4. Applied egg-rr 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-rgt-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 61.6%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot 2\right)}\\ \mathbf{elif}\;x \leq -1.58 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7e+75)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 3.8e+154)
     (/ (* y (/ x (* (+ x y) (+ y (+ x 1.0))))) (+ x y))
     (/ (/ x y) (+ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7e+75) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3.8e+154) {
		tmp = (y * (x / ((x + y) * (y + (x + 1.0))))) / (x + y);
	} 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 <= (-1.7d+75)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 3.8d+154) then
        tmp = (y * (x / ((x + y) * (y + (x + 1.0d0))))) / (x + y)
    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 <= -1.7e+75) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 3.8e+154) {
		tmp = (y * (x / ((x + y) * (y + (x + 1.0))))) / (x + y);
	} else {
		tmp = (x / y) / (x + (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7e+75:
		tmp = (y / x) / (x + 1.0)
	elif y <= 3.8e+154:
		tmp = (y * (x / ((x + y) * (y + (x + 1.0))))) / (x + y)
	else:
		tmp = (x / y) / (x + (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7e+75)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 3.8e+154)
		tmp = Float64(Float64(y * Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))))) / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.7e+75)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 3.8e+154)
		tmp = (y * (x / ((x + y) * (y + (x + 1.0))))) / (x + y);
	else
		tmp = (x / y) / (x + (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7e+75], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+154], N[(N[(y * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.70000000000000006e75

   1. Initial program 61.2%

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

      1. associate-/l* 79.7%

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

      2. associate-+l+ 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y around 0 19.8%

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

      1. associate-/r* 30.7%

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

      2. +-commutative 30.7%

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

   7. Simplified 30.7%

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

   if -1.70000000000000006e75 < y < 3.7999999999999998e154

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l* 74.9%

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

      3. times-frac 98.5%

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

      4. +-commutative 98.5%

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

      5. +-commutative 98.5%

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

      6. associate-+r+ 98.5%

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

      7. +-commutative 98.5%

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

      8. associate-+l+ 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.6%

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

      2. +-commutative 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

   6. Applied egg-rr 98.6%

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

   if 3.7999999999999998e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

Alternative 6: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+105)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 6.8e+154)
     (* (/ y (+ x y)) (/ x (* (+ x y) (+ y (+ x 1.0)))))
     (/ (/ x y) (+ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6.8e+154) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (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 <= (-4.1d+105)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 6.8d+154) then
        tmp = (y / (x + y)) * (x / ((x + y) * (y + (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 <= -4.1e+105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6.8e+154) {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (x / y) / (x + (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+105:
		tmp = (y / x) / (x + 1.0)
	elif y <= 6.8e+154:
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = (x / y) / (x + (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+105)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 6.8e+154)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+105)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 6.8e+154)
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))));
	else
		tmp = (x / y) / (x + (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+105], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+154], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1000000000000002e105

   1. Initial program 55.8%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around 0 18.9%

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

      1. associate-/r* 32.0%

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

      2. +-commutative 32.0%

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

   7. Simplified 32.0%

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

   if -4.1000000000000002e105 < y < 6.79999999999999948e154

   1. Initial program 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l* 75.5%

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

      3. times-frac 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

      6. associate-+r+ 98.6%

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

      7. +-commutative 98.6%

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

      8. associate-+l+ 98.6%

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

   4. Applied egg-rr 98.6%

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

   if 6.79999999999999948e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.1e+105)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.5e+154)
     (* (/ x (+ x y)) (/ y (* (+ x y) (+ y (+ x 1.0)))))
     (/ (/ x y) (+ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.1e+105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.5e+154) {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (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 <= (-4.1d+105)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.5d+154) then
        tmp = (x / (x + y)) * (y / ((x + y) * (y + (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 <= -4.1e+105) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.5e+154) {
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = (x / y) / (x + (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.1e+105:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.5e+154:
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = (x / y) / (x + (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.1e+105)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.5e+154)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.1e+105)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.5e+154)
		tmp = (x / (x + y)) * (y / ((x + y) * (y + (x + 1.0))));
	else
		tmp = (x / y) / (x + (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.1e+105], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+154], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1000000000000002e105

   1. Initial program 55.8%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around 0 18.9%

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

      1. associate-/r* 32.0%

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

      2. +-commutative 32.0%

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

   7. Simplified 32.0%

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

   if -4.1000000000000002e105 < y < 1.50000000000000013e154

   1. Initial program 75.6%

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

      1. associate-*l* 75.5%

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

      2. times-frac 98.6%

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

      3. +-commutative 98.6%

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

      4. +-commutative 98.6%

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

      5. associate-+r+ 98.6%

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

      6. +-commutative 98.6%

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

      7. associate-+l+ 98.6%

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

   4. Applied egg-rr 98.6%

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

   if 1.50000000000000013e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{x}{x}}{\frac{x + y}{y}}}{t\_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y -1.45e-69)
     (/ (/ (/ x x) (/ (+ x y) y)) t_0)
     (if (<= y 5.8e-15)
       (/ y (* (+ x y) (+ (+ x 1.0) (* y (/ 1.0 x)))))
       (if (<= y 2.7e+154)
         (/ x (* (+ x y) (+ y (+ x 1.0))))
         (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= -1.45e-69) {
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 2.7e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / 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 <= (-1.45d-69)) then
        tmp = ((x / x) / ((x + y) / y)) / t_0
    else if (y <= 5.8d-15) then
        tmp = y / ((x + y) * ((x + 1.0d0) + (y * (1.0d0 / x))))
    else if (y <= 2.7d+154) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / 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 <= -1.45e-69) {
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 2.7e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= -1.45e-69:
		tmp = ((x / x) / ((x + y) / y)) / t_0
	elif y <= 5.8e-15:
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))))
	elif y <= 2.7e+154:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1.45e-69)
		tmp = Float64(Float64(Float64(x / x) / Float64(Float64(x + y) / y)) / t_0);
	elseif (y <= 5.8e-15)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(Float64(x + 1.0) + Float64(y * Float64(1.0 / x)))));
	elseif (y <= 2.7e+154)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= -1.45e-69)
		tmp = ((x / x) / ((x + y) / y)) / t_0;
	elseif (y <= 5.8e-15)
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	elseif (y <= 2.7e+154)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / 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, -1.45e-69], N[(N[(N[(x / x), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e-15], N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+154], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4499999999999999e-69

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. times-frac 87.1%

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

      4. +-commutative 87.1%

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

      5. +-commutative 87.1%

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

      6. associate-+r+ 87.1%

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

      7. +-commutative 87.1%

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

      8. associate-+l+ 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 40.1%

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

   if -1.4499999999999999e-69 < y < 5.80000000000000037e-15

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around 0 99.6%

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

   if 5.80000000000000037e-15 < y < 2.70000000000000006e154

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l* 85.5%

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

      3. times-frac 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-+r+ 96.5%

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

      7. +-commutative 96.5%

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

      8. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf 77.4%

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

   if 2.70000000000000006e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\frac{x}{x}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y -1.45e-69)
     (/ (/ y x) t_0)
     (if (<= y 5.8e-15)
       (/ y (* (+ x y) (+ (+ x 1.0) (* y (/ 1.0 x)))))
       (if (<= y 2.5e+154)
         (/ x (* (+ x y) (+ y (+ x 1.0))))
         (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= -1.45e-69) {
		tmp = (y / x) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 2.5e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / 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 <= (-1.45d-69)) then
        tmp = (y / x) / t_0
    else if (y <= 5.8d-15) then
        tmp = y / ((x + y) * ((x + 1.0d0) + (y * (1.0d0 / x))))
    else if (y <= 2.5d+154) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / 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 <= -1.45e-69) {
		tmp = (y / x) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	} else if (y <= 2.5e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= -1.45e-69:
		tmp = (y / x) / t_0
	elif y <= 5.8e-15:
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))))
	elif y <= 2.5e+154:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= -1.45e-69)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 5.8e-15)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(Float64(x + 1.0) + Float64(y * Float64(1.0 / x)))));
	elseif (y <= 2.5e+154)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= -1.45e-69)
		tmp = (y / x) / t_0;
	elseif (y <= 5.8e-15)
		tmp = y / ((x + y) * ((x + 1.0) + (y * (1.0 / x))));
	elseif (y <= 2.5e+154)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / 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, -1.45e-69], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e-15], N[(y / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] + N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e+154], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.4499999999999999e-69

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*l* 70.0%

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

      3. times-frac 87.1%

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

      4. +-commutative 87.1%

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

      5. +-commutative 87.1%

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

      6. associate-+r+ 87.1%

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

      7. +-commutative 87.1%

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

      8. associate-+l+ 87.1%

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

   4. Applied egg-rr 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 38.5%

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

   if -1.4499999999999999e-69 < y < 5.80000000000000037e-15

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

      2. associate-*l* 68.2%

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

      3. times-frac 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.6%

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

      4. *-un-lft-identity 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 99.6%

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

      1. associate-+r+ 99.6%

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

      2. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around 0 99.6%

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

   if 5.80000000000000037e-15 < y < 2.50000000000000002e154

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l* 85.5%

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

      3. times-frac 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-+r+ 96.5%

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

      7. +-commutative 96.5%

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

      8. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf 77.4%

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

   if 2.50000000000000002e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(\left(x + 1\right) + y \cdot \frac{1}{x}\right)}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 9e-163)
     (/ (/ y x) t_0)
     (if (<= y 5.8e-15)
       (* x (/ y (* (+ x 1.0) (* (+ x y) (+ x y)))))
       (if (<= y 1.35e+154)
         (/ x (* (+ x y) (+ y (+ x 1.0))))
         (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 9e-163) {
		tmp = (y / x) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 1.35e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / 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 <= 9d-163) then
        tmp = (y / x) / t_0
    else if (y <= 5.8d-15) then
        tmp = x * (y / ((x + 1.0d0) * ((x + y) * (x + y))))
    else if (y <= 1.35d+154) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / 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 <= 9e-163) {
		tmp = (y / x) / t_0;
	} else if (y <= 5.8e-15) {
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	} else if (y <= 1.35e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 9e-163:
		tmp = (y / x) / t_0
	elif y <= 5.8e-15:
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))))
	elif y <= 1.35e+154:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 9e-163)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 5.8e-15)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * Float64(x + y)))));
	elseif (y <= 1.35e+154)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 9e-163)
		tmp = (y / x) / t_0;
	elseif (y <= 5.8e-15)
		tmp = x * (y / ((x + 1.0) * ((x + y) * (x + y))));
	elseif (y <= 1.35e+154)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / 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, 9e-163], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 5.8e-15], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 8.9999999999999995e-163

   1. Initial program 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-*l* 65.4%

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

      3. times-frac 93.0%

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

      4. +-commutative 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-+r+ 93.0%

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

      7. +-commutative 93.0%

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

      8. associate-+l+ 93.0%

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

   4. Applied egg-rr 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 54.4%

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

   if 8.9999999999999995e-163 < y < 5.80000000000000037e-15

   1. Initial program 91.3%

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

      1. associate-/l* 96.0%

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

      2. associate-+l+ 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y around 0 96.0%

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

      1. +-commutative 96.0%

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

   7. Simplified 96.0%

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

   if 5.80000000000000037e-15 < y < 1.35000000000000003e154

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l* 85.5%

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

      3. times-frac 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-+r+ 96.5%

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

      7. +-commutative 96.5%

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

      8. associate-+l+ 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in y around inf 77.4%

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

   if 1.35000000000000003e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1.4e-169)
     (/ (/ y x) t_0)
     (if (<= y 1.4e+154) (/ x (* (+ x y) (+ y (+ x 1.0)))) (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.4e-169) {
		tmp = (y / x) / t_0;
	} else if (y <= 1.4e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / 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 <= 1.4d-169) then
        tmp = (y / x) / t_0
    else if (y <= 1.4d+154) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / 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 <= 1.4e-169) {
		tmp = (y / x) / t_0;
	} else if (y <= 1.4e+154) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 1.4e-169:
		tmp = (y / x) / t_0
	elif y <= 1.4e+154:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.4e-169)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 1.4e+154)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 1.4e-169)
		tmp = (y / x) / t_0;
	elseif (y <= 1.4e+154)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / 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, 1.4e-169], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.4e+154], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.39999999999999994e-169

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 66.0%

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

      2. associate-*l* 66.0%

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

      3. times-frac 92.9%

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

      4. +-commutative 92.9%

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

      5. +-commutative 92.9%

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

      6. associate-+r+ 92.9%

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

      7. +-commutative 92.9%

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

      8. associate-+l+ 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-rgt-identity 99.7%

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

      2. *-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+l+ 99.7%

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

      7. *-rgt-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 54.7%

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

   if 1.39999999999999994e-169 < y < 1.4e154

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-*l* 85.4%

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

      3. times-frac 98.0%

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

      4. +-commutative 98.0%

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

      5. +-commutative 98.0%

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

      6. associate-+r+ 98.0%

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

      7. +-commutative 98.0%

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

      8. associate-+l+ 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in y around inf 77.2%

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

   if 1.4e154 < y

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*l* 53.1%

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

      3. times-frac 76.1%

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

      4. +-commutative 76.1%

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

      5. +-commutative 76.1%

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

      6. associate-+r+ 76.1%

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

      7. +-commutative 76.1%

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

      8. associate-+l+ 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. *-commutative 76.1%

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

      2. associate-/r* 99.9%

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

      3. clear-num 99.9%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. *-rgt-identity 98.5%

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

      2. *-commutative 98.5%

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

      3. times-frac 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. *-rgt-identity 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 84.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + \left(y + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-169}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-85)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 6e+103) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d-85) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 6d+103) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-85) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 6e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-85:
		tmp = (y / x) / (x + 1.0)
	elif y <= 6e+103:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 6e+103)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-85)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 6e+103)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+103], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.20000000000000023e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 55.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 5.20000000000000023e-85 < y < 6e103

   1. Initial program 92.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 92.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 6e103 < y

   1. Initial program 56.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 77.9%

      \[\leadsto x \cdot \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 77.9%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{1 + y}} \]

      2. +-commutative 77.9%

         \[\leadsto x \cdot \frac{\frac{1}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 77.9%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*r/ 84.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{y}}{y + 1}} \]

      2. frac-2neg 84.2%

         \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 84.2%

          \[\leadsto \frac{-\color{blue}{\frac{x \cdot 1}{y}}}{-\left(y + 1\right)} \]

       2. *-rgt-identity 84.2%

          \[\leadsto \frac{-\frac{\color{blue}{x}}{y}}{-\left(y + 1\right)} \]

       3. +-commutative 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{-\color{blue}{\left(1 + y\right)}} \]

       4. mul-1-neg 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot \left(1 + y\right)}} \]

       5. distribute-lft-in 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot 1 + -1 \cdot y}} \]

       6. metadata-eval 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1} + -1 \cdot y} \]

       7. neg-mul-1 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{-1 + \color{blue}{\left(-y\right)}} \]

       8. unsub-neg 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 - y}} \]

   11. Simplified 84.2%

       \[\leadsto \color{blue}{\frac{-\frac{x}{y}}{-1 - y}} \]

   12. Taylor expanded in y around inf 84.2%

       \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot y}} \]
   13. Step-by-step derivation

       1. neg-mul-1 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-y}} \]

   14. Simplified 84.2%

       \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.9e-85)
   (/ y (* x (+ x 1.0)))
   (if (<= y 6.1e+103) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-85) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 6.1e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.9d-85) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 6.1d+103) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.9e-85) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 6.1e+103) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.9e-85:
		tmp = y / (x * (x + 1.0))
	elif y <= 6.1e+103:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.9e-85)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 6.1e+103)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.9e-85)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 6.1e+103)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.9e-85], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.1e+103], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.90000000000000015e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 4.90000000000000015e-85 < y < 6.1000000000000002e103

   1. Initial program 92.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 92.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 92.5%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 92.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   if 6.1000000000000002e103 < y

   1. Initial program 56.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.6%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 77.9%

      \[\leadsto x \cdot \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 77.9%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{1 + y}} \]

      2. +-commutative 77.9%

         \[\leadsto x \cdot \frac{\frac{1}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 77.9%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*r/ 84.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{y}}{y + 1}} \]

      2. frac-2neg 84.2%

         \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]

   9. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 84.2%

          \[\leadsto \frac{-\color{blue}{\frac{x \cdot 1}{y}}}{-\left(y + 1\right)} \]

       2. *-rgt-identity 84.2%

          \[\leadsto \frac{-\frac{\color{blue}{x}}{y}}{-\left(y + 1\right)} \]

       3. +-commutative 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{-\color{blue}{\left(1 + y\right)}} \]

       4. mul-1-neg 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot \left(1 + y\right)}} \]

       5. distribute-lft-in 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot 1 + -1 \cdot y}} \]

       6. metadata-eval 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1} + -1 \cdot y} \]

       7. neg-mul-1 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{-1 + \color{blue}{\left(-y\right)}} \]

       8. unsub-neg 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 - y}} \]

   11. Simplified 84.2%

       \[\leadsto \color{blue}{\frac{-\frac{x}{y}}{-1 - y}} \]

   12. Taylor expanded in y around inf 84.2%

       \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot y}} \]
   13. Step-by-step derivation

       1. neg-mul-1 84.2%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-y}} \]

   14. Simplified 84.2%

       \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.8e-100) (/ (/ y x) (+ x (+ y 1.0))) (/ (/ x (+ x y)) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-100) {
		tmp = (y / x) / (x + (y + 1.0));
	} else {
		tmp = (x / (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 <= 6.8d-100) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-100) {
		tmp = (y / x) / (x + (y + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.8e-100:
		tmp = (y / x) / (x + (y + 1.0))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-100)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-100)
		tmp = (y / x) / (x + (y + 1.0));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.8e-100], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999953e-100

   1. Initial program 66.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 66.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 66.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.5%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 93.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{y}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \cdot \frac{y}{y + x} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + x}}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{y}}} \]

      4. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}}} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \frac{y + x}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{x + y}}{y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{x}{x + y} \cdot 1\right)} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{x}{x + y} \cdot 1\right)}}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      3. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{1}{y + \left(x + 1\right)} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}}} \]

      4. associate-+r+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{1}{x + \left(y + 1\right)} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y}} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{x + \left(y + 1\right)} \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

      2. *-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}}{x + \left(y + 1\right)} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

   11. Taylor expanded in x around inf 55.9%

       \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + \left(y + 1\right)} \]

   if 6.79999999999999953e-100 < y

   1. Initial program 74.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 74.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 74.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 88.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 88.5%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 88.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 88.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 88.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 88.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 88.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{y}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \cdot \frac{y}{y + x} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + x}}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{y}}} \]

      4. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}}} \]

      5. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}} \]

      6. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \frac{y + x}{y}} \]

      7. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{x + y}}{y}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]

   7. Taylor expanded in x around 0 72.7%

      \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\color{blue}{1 + y}} \]
   8. Step-by-step derivation

      1. +-commutative 72.7%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\color{blue}{y + 1}} \]

   9. Simplified 72.7%

      \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 5e-85) (/ (/ y x) t_0) (/ (/ x y) t_0))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 5e-85) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / 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 <= 5d-85) then
        tmp = (y / x) / t_0
    else
        tmp = (x / y) / 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 <= 5e-85) {
		tmp = (y / x) / t_0;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 5e-85:
		tmp = (y / x) / t_0
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 5e-85)
		tmp = Float64(Float64(y / x) / t_0);
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 5e-85)
		tmp = (y / x) / t_0;
	else
		tmp = (x / y) / 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, 5e-85], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 66.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 66.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 93.5%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 93.5%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 93.5%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 93.5%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{y}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \cdot \frac{y}{y + x} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + x}}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{y}}} \]

      4. frac-times 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}}} \]

      5. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \frac{y + x}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{x + y}}{y}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\left(\frac{x}{x + y} \cdot 1\right)} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{x}{x + y} \cdot 1\right)}}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      3. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{1}{y + \left(x + 1\right)} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}}} \]

      4. associate-+r+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{1}{x + \left(y + 1\right)} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y}} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{x + \left(y + 1\right)} \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

      2. *-lft-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}}{x + \left(y + 1\right)} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

   11. Taylor expanded in x around inf 56.1%

       \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + \left(y + 1\right)} \]

   if 5.0000000000000002e-85 < y

   1. Initial program 73.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 73.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 73.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 88.3%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 88.3%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{y}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \cdot \frac{y}{y + x} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + x}}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{y}}} \]

      4. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}}} \]

      5. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}} \]

      6. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \frac{y + x}{y}} \]

      7. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{x + y}}{y}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto \frac{\color{blue}{\left(\frac{x}{x + y} \cdot 1\right)} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{x}{x + y} \cdot 1\right)}}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      3. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{1}{y + \left(x + 1\right)} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}}} \]

      4. associate-+r+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{1}{x + \left(y + 1\right)} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y}} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{x + \left(y + 1\right)} \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

      2. *-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}}{x + \left(y + 1\right)} \]

   10. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

   11. Taylor expanded in x around 0 73.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + \left(y + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-85) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ x (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-85) {
		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 <= 5.2d-85) 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 <= 5.2e-85) {
		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 <= 5.2e-85:
		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 <= 5.2e-85)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(x + Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-85)
		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, 5.2e-85], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000023e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 55.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 5.20000000000000023e-85 < y

   1. Initial program 73.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 73.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. associate-*l* 73.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      3. times-frac 88.3%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. +-commutative 88.3%

         \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      5. +-commutative 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      6. associate-+r+ 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-commutative 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

      8. associate-+l+ 88.3%

         \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

   4. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 88.3%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot \frac{y}{y + x}} \]

      2. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x}}{y + \left(1 + x\right)}} \cdot \frac{y}{y + x} \]

      3. clear-num 99.8%

         \[\leadsto \frac{\frac{x}{y + x}}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{1}{\frac{y + x}{y}}} \]

      4. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}}} \]

      5. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}} \cdot 1}{\left(y + \left(1 + x\right)\right) \cdot \frac{y + x}{y}} \]

      6. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \frac{y + x}{y}} \]

      7. +-commutative 99.1%

         \[\leadsto \frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{\color{blue}{x + y}}{y}} \]

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}}} \]
   7. Step-by-step derivation

      1. *-rgt-identity 99.1%

         \[\leadsto \frac{\color{blue}{\left(\frac{x}{x + y} \cdot 1\right)} \cdot 1}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{x}{x + y} \cdot 1\right)}}{\left(y + \left(x + 1\right)\right) \cdot \frac{x + y}{y}} \]

      3. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{1}{y + \left(x + 1\right)} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}}} \]

      4. associate-+r+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      5. +-commutative 99.7%

         \[\leadsto \frac{1}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      6. associate-+l+ 99.7%

         \[\leadsto \frac{1}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{\frac{x}{x + y} \cdot 1}{\frac{x + y}{y}} \]

      7. *-rgt-identity 99.7%

         \[\leadsto \frac{1}{x + \left(y + 1\right)} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + y}{y}} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{1}{x + \left(y + 1\right)} \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

      2. *-lft-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}}{x + \left(y + 1\right)} \]

   10. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{x + y}}{\frac{x + y}{y}}}{x + \left(y + 1\right)}} \]

   11. Taylor expanded in x around 0 73.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + \left(y + 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-85) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-85) {
		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 <= 5.2d-85) 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 <= 5.2e-85) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-85:
		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 <= 5.2e-85)
		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 (y <= 5.2e-85)
		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, 5.2e-85], 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 y < 5.20000000000000023e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 55.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 55.8%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 5.20000000000000023e-85 < y

   1. Initial program 73.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.6%

      \[\leadsto x \cdot \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 69.7%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{1 + y}} \]

      2. +-commutative 69.7%

         \[\leadsto x \cdot \frac{\frac{1}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 69.7%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*r/ 72.8%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{y}}{y + 1}} \]

      2. frac-2neg 72.8%

         \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]

   9. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{\frac{-x \cdot \frac{1}{y}}{-\left(y + 1\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 72.9%

          \[\leadsto \frac{-\color{blue}{\frac{x \cdot 1}{y}}}{-\left(y + 1\right)} \]

       2. *-rgt-identity 72.9%

          \[\leadsto \frac{-\frac{\color{blue}{x}}{y}}{-\left(y + 1\right)} \]

       3. +-commutative 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{-\color{blue}{\left(1 + y\right)}} \]

       4. mul-1-neg 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot \left(1 + y\right)}} \]

       5. distribute-lft-in 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 \cdot 1 + -1 \cdot y}} \]

       6. metadata-eval 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1} + -1 \cdot y} \]

       7. neg-mul-1 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{-1 + \color{blue}{\left(-y\right)}} \]

       8. unsub-neg 72.9%

          \[\leadsto \frac{-\frac{x}{y}}{\color{blue}{-1 - y}} \]

   11. Simplified 72.9%

       \[\leadsto \color{blue}{\frac{-\frac{x}{y}}{-1 - y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-85) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-85) {
		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 <= 5.2d-85) 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 <= 5.2e-85) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-85:
		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 <= 5.2e-85)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-85)
		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, 5.2e-85], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000023e-85

   1. Initial program 66.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 51.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   if 5.20000000000000023e-85 < y

   1. Initial program 73.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 69.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 69.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 27.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.1) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001

   1. Initial program 62.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 58.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{x}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   4. Taylor expanded in x around inf 57.9%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{x} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. Taylor expanded in y around inf 6.6%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -1.1000000000000001 < x

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 82.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 82.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 62.5%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 35.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 48.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot \left(y + 1\right)} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x (* y (+ y 1.0))))

C

double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * (y + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return x / (y * (y + 1.0));
}

Python

def code(x, y):
	return x / (y * (y + 1.0))

Julia

function code(x, y)
	return Float64(x / Float64(y * Float64(y + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = x / (y * (y + 1.0));
end

Wolfram

code[x_, y_] := N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 80.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 80.6%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 80.6%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 52.0%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 52.0%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
8. Add Preprocessing

Alternative 21: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 37.4%

   \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{x}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

4. Taylor expanded in x around inf 31.5%

   \[\leadsto \frac{x \cdot y}{\left(\color{blue}{x} \cdot x\right) \cdot \left(\left(x + y\right) + 1\right)} \]

5. Taylor expanded in y around inf 4.4%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Alternative 22: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 69.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 69.0%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. associate-*l* 69.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   3. times-frac 91.9%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. +-commutative 91.9%

      \[\leadsto \frac{y}{\color{blue}{y + x}} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   5. +-commutative 91.9%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   6. associate-+r+ 91.9%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   7. +-commutative 91.9%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \]

   8. associate-+l+ 91.9%

      \[\leadsto \frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \]

4. Applied egg-rr 91.9%

   \[\leadsto \color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}} \]

5. Taylor expanded in x around 0 52.0%

   \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 3.5%

   \[\leadsto \color{blue}{1} \]
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 2024118 
(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))))