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

Percentage Accurate: 69.5% → 99.8%

Time: 16.6s

Alternatives: 17

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: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. associate-+r+ 68.1%

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

   2. *-commutative 68.1%

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

   3. frac-times 84.8%

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

   4. associate-*l/ 79.3%

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

   5. times-frac 99.8%

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

   6. associate-+r+ 99.8%

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

   7. +-commutative 99.8%

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

   8. associate-+l+ 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 68.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= x -5.8e-33)
     (* (/ t_0 (+ y x)) (/ y x))
     (if (<= x -1.5e-162)
       (* (/ x (+ y 1.0)) (/ y (* (+ y x) (+ y x))))
       (if (<= x -1.05e-185)
         (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
         (/ t_0 (+ y (* x 2.0))))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -5.8e-33) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else if (x <= -1.5e-162) {
		tmp = (x / (y + 1.0)) * (y / ((y + x) * (y + x)));
	} else if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	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 + (x + 1.0d0))
    if (x <= (-5.8d-33)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else if (x <= (-1.5d-162)) then
        tmp = (x / (y + 1.0d0)) * (y / ((y + x) * (y + x)))
    else if (x <= (-1.05d-185)) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else
        tmp = t_0 / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -5.8e-33) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else if (x <= -1.5e-162) {
		tmp = (x / (y + 1.0)) * (y / ((y + x) * (y + x)));
	} else if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if x <= -5.8e-33:
		tmp = (t_0 / (y + x)) * (y / x)
	elif x <= -1.5e-162:
		tmp = (x / (y + 1.0)) * (y / ((y + x) * (y + x)))
	elif x <= -1.05e-185:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	else:
		tmp = t_0 / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -5.8e-33)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	elseif (x <= -1.5e-162)
		tmp = Float64(Float64(x / Float64(y + 1.0)) * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	elseif (x <= -1.05e-185)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -5.8e-33)
		tmp = (t_0 / (y + x)) * (y / x);
	elseif (x <= -1.5e-162)
		tmp = (x / (y + 1.0)) * (y / ((y + x) * (y + x)));
	elseif (x <= -1.05e-185)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	else
		tmp = t_0 / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-33], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-162], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-185], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.80000000000000005e-33

   1. Initial program 59.4%

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

      1. associate-+r+ 59.4%

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

      2. *-commutative 59.4%

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

      3. frac-times 78.2%

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

      4. associate-*l/ 78.3%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 89.9%

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

   if -5.80000000000000005e-33 < x < -1.49999999999999999e-162

   1. Initial program 92.7%

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

      1. associate-/r* 92.9%

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

      2. *-commutative 92.9%

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

      3. +-commutative 92.9%

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

      4. +-commutative 92.9%

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

      5. associate-*l/ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-*r/ 99.7%

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

      8. remove-double-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. remove-double-neg 99.7%

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

      12. +-commutative 99.7%

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

      13. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   if -1.49999999999999999e-162 < x < -1.05e-185

   1. Initial program 40.0%

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

      1. associate-+r+ 40.0%

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

      2. *-commutative 40.0%

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

      3. frac-times 41.7%

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

      4. associate-*l/ 40.0%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. clear-num 99.1%

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

      5. associate-+r+ 99.1%

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

      6. +-commutative 99.1%

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

      7. associate-+r+ 99.1%

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

      8. frac-times 99.1%

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

      9. *-un-lft-identity 99.1%

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

      10. +-commutative 99.1%

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

      11. associate-+r+ 99.1%

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

      12. +-commutative 99.1%

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

      13. associate-+r+ 99.1%

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

      14. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around -inf 44.3%

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

      1. mul-1-neg 44.3%

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

      2. unsub-neg 44.3%

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

      3. neg-mul-1 44.3%

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

      4. distribute-lft-in 44.3%

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

      5. metadata-eval 44.3%

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

      6. neg-mul-1 44.3%

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

      7. +-commutative 44.3%

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

      8. +-commutative 44.3%

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

      9. neg-mul-1 44.3%

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

      10. metadata-eval 44.3%

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

      11. sub-neg 44.3%

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

      12. unsub-neg 44.3%

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

      13. sub-neg 44.3%

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

      14. neg-mul-1 44.3%

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

      15. metadata-eval 44.3%

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

      16. +-commutative 44.3%

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

      17. unsub-neg 44.3%

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

   9. Simplified 44.3%

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

   if -1.05e-185 < x

   1. Initial program 69.9%

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

      1. times-frac 87.0%

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

      2. +-commutative 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

      5. times-frac 69.9%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 77.3%

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

      10. fma-def 83.1%

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

      11. +-commutative 83.1%

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

      12. +-commutative 83.1%

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

      13. cube-unmult 83.1%

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

      14. +-commutative 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 69.9%

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

      2. fma-udef 66.2%

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

      3. cube-mult 66.2%

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

      4. distribute-rgt1-in 69.9%

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

      5. associate-+r+ 69.9%

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

      6. *-commutative 69.9%

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

      7. frac-times 87.1%

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

      8. clear-num 87.1%

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

      9. associate-*l/ 87.1%

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

      10. *-un-lft-identity 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+l+ 87.1%

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

      14. pow2 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in x around 0 58.7%

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

      1. *-commutative 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x / t_0) / (y + x)) * (y / x);
	} else {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.35d+154)) then
        tmp = ((x / t_0) / (y + x)) * (y / x)
    else
        tmp = (x / (y + x)) * (y / ((y + x) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x / t_0) / (y + x)) * (y / x);
	} else {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = ((x / t_0) / (y + x)) * (y / x);
	else
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 36.5%

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

      1. associate-+r+ 36.5%

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

      2. *-commutative 36.5%

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

      3. frac-times 60.3%

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

      4. associate-*l/ 60.3%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 93.3%

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

   if -1.35000000000000003e154 < x

   1. Initial program 72.0%

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

      1. associate-+r+ 72.0%

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

      2. associate-*l* 72.0%

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

      3. times-frac 94.3%

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

      4. associate-+r+ 94.3%

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

      5. +-commutative 94.3%

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

      6. associate-+l+ 94.3%

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

   4. Applied egg-rr 94.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{t_0}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= x -3.7e-174) (* (/ t_0 (+ y x)) (/ y x)) (/ t_0 (+ y (* x 2.0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	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 + (x + 1.0d0))
    if (x <= (-3.7d-174)) then
        tmp = (t_0 / (y + x)) * (y / x)
    else
        tmp = t_0 / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (t_0 / (y + x)) * (y / x);
	} else {
		tmp = t_0 / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if x <= -3.7e-174:
		tmp = (t_0 / (y + x)) * (y / x)
	else:
		tmp = t_0 / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (x <= -3.7e-174)
		tmp = Float64(Float64(t_0 / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(t_0 / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (x <= -3.7e-174)
		tmp = (t_0 / (y + x)) * (y / x);
	else
		tmp = t_0 / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e-174], N[(N[(t$95$0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000001e-174

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-commutative 63.2%

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

      3. frac-times 79.2%

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

      4. associate-*l/ 77.8%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 84.9%

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

   if -3.7000000000000001e-174 < x

   1. Initial program 70.2%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. times-frac 70.2%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 77.5%

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

      10. fma-def 83.3%

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

      11. +-commutative 83.3%

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

      12. +-commutative 83.3%

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

      13. cube-unmult 83.3%

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

      14. +-commutative 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 70.2%

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

      2. fma-udef 66.6%

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

      3. cube-mult 66.6%

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

      4. distribute-rgt1-in 70.2%

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

      5. associate-+r+ 70.2%

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

      6. *-commutative 70.2%

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

      7. frac-times 87.2%

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

      8. clear-num 87.2%

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

      9. associate-*l/ 87.3%

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

      10. *-un-lft-identity 87.3%

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

      11. associate-+r+ 87.3%

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

      12. +-commutative 87.3%

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

      13. associate-+l+ 87.3%

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

      14. pow2 87.3%

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in x around 0 59.2%

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

      1. *-commutative 59.2%

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

   9. Simplified 59.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.9e-10)
   (* (/ (/ x (+ y (+ x 1.0))) (+ y x)) (/ y x))
   (* (/ y (+ y x)) (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.9e-10) {
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.9d-10)) then
        tmp = ((x / (y + (x + 1.0d0))) / (y + x)) * (y / x)
    else
        tmp = (y / (y + x)) * ((x / (y + 1.0d0)) / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.9e-10) {
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	} else {
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.9e-10:
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x)
	else:
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.9e-10)
		tmp = Float64(Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x)) * Float64(y / x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.9e-10)
		tmp = ((x / (y + (x + 1.0))) / (y + x)) * (y / x);
	else
		tmp = (y / (y + x)) * ((x / (y + 1.0)) / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.9e-10], N[(N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9e-10

   1. Initial program 58.8%

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

      1. associate-+r+ 58.8%

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

      2. *-commutative 58.8%

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

      3. frac-times 77.9%

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

      4. associate-*l/ 77.9%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0 89.8%

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

   if -3.9e-10 < x

   1. Initial program 70.9%

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

      1. associate-+r+ 70.9%

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

      2. *-commutative 70.9%

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

      3. frac-times 86.9%

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

      4. associate-*l/ 79.7%

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

      5. times-frac 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-185)
   (/ (/ y (+ y x)) (+ x 1.0))
   (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d-185)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-185:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-185)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-185)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-185], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-185

   1. Initial program 64.1%

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

      1. associate-+r+ 64.1%

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

      2. *-commutative 64.1%

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

      3. frac-times 79.7%

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

      4. associate-*l/ 78.4%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. clear-num 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. frac-times 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 77.9%

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

      1. +-commutative 77.9%

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

   9. Simplified 77.9%

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

   if -1.05e-185 < x

   1. Initial program 69.9%

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

      1. times-frac 87.0%

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

      2. +-commutative 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

      5. times-frac 69.9%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 77.3%

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

      10. fma-def 83.1%

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

      11. +-commutative 83.1%

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

      12. +-commutative 83.1%

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

      13. cube-unmult 83.1%

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

      14. +-commutative 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 69.9%

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

      2. fma-udef 66.2%

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

      3. cube-mult 66.2%

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

      4. distribute-rgt1-in 69.9%

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

      5. associate-+r+ 69.9%

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

      6. *-commutative 69.9%

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

      7. frac-times 87.1%

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

      8. clear-num 87.1%

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

      9. associate-*l/ 87.1%

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

      10. *-un-lft-identity 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+l+ 87.1%

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

      14. pow2 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in x around 0 58.7%

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

      1. *-commutative 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-185)
   (/ (/ y (+ y x)) (+ x (+ y (+ y 1.0))))
   (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.05d-185)) then
        tmp = (y / (y + x)) / (x + (y + (y + 1.0d0)))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-185) {
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-185:
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-185)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + Float64(y + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-185)
		tmp = (y / (y + x)) / (x + (y + (y + 1.0)));
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-185], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-185

   1. Initial program 64.1%

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

      1. associate-+r+ 64.1%

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

      2. *-commutative 64.1%

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

      3. frac-times 79.7%

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

      4. associate-*l/ 78.4%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. clear-num 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+r+ 99.7%

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

      8. frac-times 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around -inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

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

      3. neg-mul-1 78.6%

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

      4. distribute-lft-in 78.6%

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

      5. metadata-eval 78.6%

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

      6. neg-mul-1 78.6%

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

      7. +-commutative 78.6%

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

      8. +-commutative 78.6%

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

      9. neg-mul-1 78.6%

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

      10. metadata-eval 78.6%

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

      11. sub-neg 78.6%

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

      12. unsub-neg 78.6%

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

      13. sub-neg 78.6%

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

      14. neg-mul-1 78.6%

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

      15. metadata-eval 78.6%

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

      16. +-commutative 78.6%

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

      17. unsub-neg 78.6%

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

   9. Simplified 78.6%

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

   if -1.05e-185 < x

   1. Initial program 69.9%

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

      1. times-frac 87.0%

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

      2. +-commutative 87.0%

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

      3. +-commutative 87.0%

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

      4. +-commutative 87.0%

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

      5. times-frac 69.9%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 77.3%

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

      10. fma-def 83.1%

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

      11. +-commutative 83.1%

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

      12. +-commutative 83.1%

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

      13. cube-unmult 83.1%

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

      14. +-commutative 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 69.9%

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

      2. fma-udef 66.2%

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

      3. cube-mult 66.2%

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

      4. distribute-rgt1-in 69.9%

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

      5. associate-+r+ 69.9%

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

      6. *-commutative 69.9%

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

      7. frac-times 87.1%

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

      8. clear-num 87.1%

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

      9. associate-*l/ 87.1%

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

      10. *-un-lft-identity 87.1%

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

      11. associate-+r+ 87.1%

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

      12. +-commutative 87.1%

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

      13. associate-+l+ 87.1%

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

      14. pow2 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in x around 0 58.7%

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

      1. *-commutative 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + \left(y + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-161)
   (/ 1.0 (/ x y))
   (if (<= y 7e+161) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-161) {
		tmp = 1.0 / (x / y);
	} else if (y <= 7e+161) {
		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 <= 1.85d-161) then
        tmp = 1.0d0 / (x / y)
    else if (y <= 7d+161) 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 <= 1.85e-161) {
		tmp = 1.0 / (x / y);
	} else if (y <= 7e+161) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e-161:
		tmp = 1.0 / (x / y)
	elif y <= 7e+161:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-161)
		tmp = Float64(1.0 / Float64(x / y));
	elseif (y <= 7e+161)
		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 <= 1.85e-161)
		tmp = 1.0 / (x / y);
	elseif (y <= 7e+161)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e-161], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+161], 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 < 1.8499999999999999e-161

   1. Initial program 69.3%

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

      1. associate-/r* 72.7%

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

      2. *-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. +-commutative 72.7%

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

      5. associate-*l/ 83.7%

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

      6. +-commutative 83.7%

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

      7. associate-*r/ 83.7%

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

      8. remove-double-neg 83.7%

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

      9. +-commutative 83.7%

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

      10. +-commutative 83.7%

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

      11. remove-double-neg 83.7%

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

      12. +-commutative 83.7%

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

      13. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in y around 0 59.8%

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

      1. associate-/r* 61.6%

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

      2. +-commutative 61.6%

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

   7. Simplified 61.6%

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

      1. clear-num 61.0%

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

      2. inv-pow 61.0%

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

      3. div-inv 61.0%

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

      4. clear-num 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. unpow-1 61.0%

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

       2. associate-*r/ 59.5%

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

       3. *-commutative 59.5%

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

   11. Simplified 59.5%

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

   12. Taylor expanded in x around 0 37.2%

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

   if 1.8499999999999999e-161 < y < 6.99999999999999976e161

   1. Initial program 68.1%

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

      1. associate-/r* 74.4%

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

      2. *-commutative 74.4%

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

      3. +-commutative 74.4%

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

      4. +-commutative 74.4%

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

      5. associate-*l/ 92.1%

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

      6. +-commutative 92.1%

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

      7. associate-*r/ 92.2%

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

      8. remove-double-neg 92.2%

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

      9. +-commutative 92.2%

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

      10. +-commutative 92.2%

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

      11. remove-double-neg 92.2%

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

      12. +-commutative 92.2%

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

      13. associate-+l+ 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in x around 0 48.4%

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

      1. +-commutative 48.4%

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

   7. Simplified 48.4%

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

   if 6.99999999999999976e161 < y

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. *-commutative 61.7%

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

      3. +-commutative 61.7%

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

      4. +-commutative 61.7%

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

      5. associate-*l/ 76.1%

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

      6. +-commutative 76.1%

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

      7. associate-*r/ 76.1%

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

      8. remove-double-neg 76.1%

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

      9. +-commutative 76.1%

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

      10. +-commutative 76.1%

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

      11. remove-double-neg 76.1%

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

      12. +-commutative 76.1%

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

      13. associate-+l+ 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. times-frac 85.2%

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

   9. Applied egg-rr 85.2%

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

       1. associate-*l/ 85.2%

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

       2. +-commutative 85.2%

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

       3. *-lft-identity 85.2%

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

       4. +-commutative 85.2%

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

   11. Simplified 85.2%

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

   12. Taylor expanded in y around inf 85.2%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-118)
   (/ y (* x (+ x 1.0)))
   (if (<= y 3.5e+161) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-118) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 3.5e+161) {
		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 <= 3.2d-118) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 3.5d+161) 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 <= 3.2e-118) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 3.5e+161) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-118:
		tmp = y / (x * (x + 1.0))
	elif y <= 3.5e+161:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-118)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 3.5e+161)
		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 <= 3.2e-118)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 3.5e+161)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-118], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+161], 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 < 3.20000000000000004e-118

   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-/r* 72.2%

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

      2. *-commutative 72.2%

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

      3. +-commutative 72.2%

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

      4. +-commutative 72.2%

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

      5. associate-*l/ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-*r/ 84.4%

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

      8. remove-double-neg 84.4%

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

      9. +-commutative 84.4%

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

      10. +-commutative 84.4%

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

      11. remove-double-neg 84.4%

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

      12. +-commutative 84.4%

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

      13. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in y around 0 60.0%

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

   if 3.20000000000000004e-118 < y < 3.49999999999999988e161

   1. Initial program 68.8%

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

      1. associate-/r* 76.1%

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

      2. *-commutative 76.1%

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

      3. +-commutative 76.1%

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

      4. +-commutative 76.1%

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

      5. associate-*l/ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-*r/ 91.1%

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

      8. remove-double-neg 91.1%

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

      9. +-commutative 91.1%

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

      10. +-commutative 91.1%

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

      11. remove-double-neg 91.1%

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

      12. +-commutative 91.1%

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

      13. associate-+l+ 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 49.7%

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

      1. +-commutative 49.7%

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

   7. Simplified 49.7%

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

   if 3.49999999999999988e161 < y

   1. Initial program 61.7%

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

      1. associate-/r* 61.7%

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

      2. *-commutative 61.7%

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

      3. +-commutative 61.7%

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

      4. +-commutative 61.7%

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

      5. associate-*l/ 76.1%

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

      6. +-commutative 76.1%

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

      7. associate-*r/ 76.1%

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

      8. remove-double-neg 76.1%

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

      9. +-commutative 76.1%

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

      10. +-commutative 76.1%

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

      11. remove-double-neg 76.1%

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

      12. +-commutative 76.1%

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

      13. associate-+l+ 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   7. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. times-frac 85.2%

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

   9. Applied egg-rr 85.2%

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

       1. associate-*l/ 85.2%

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

       2. +-commutative 85.2%

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

       3. *-lft-identity 85.2%

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

       4. +-commutative 85.2%

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

   11. Simplified 85.2%

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

   12. Taylor expanded in y around inf 85.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+161}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.46e-160)
   (/ 1.0 (/ x y))
   (if (<= y 0.75) (- (/ x y) x) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.46e-160) {
		tmp = 1.0 / (x / y);
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.46d-160) then
        tmp = 1.0d0 / (x / y)
    else if (y <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.46e-160) {
		tmp = 1.0 / (x / y);
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.46e-160:
		tmp = 1.0 / (x / y)
	elif y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.46e-160)
		tmp = Float64(1.0 / Float64(x / y));
	elseif (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.46e-160)
		tmp = 1.0 / (x / y);
	elseif (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.46e-160], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4600000000000001e-160

   1. Initial program 69.3%

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

      1. associate-/r* 72.7%

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

      2. *-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. +-commutative 72.7%

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

      5. associate-*l/ 83.7%

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

      6. +-commutative 83.7%

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

      7. associate-*r/ 83.7%

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

      8. remove-double-neg 83.7%

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

      9. +-commutative 83.7%

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

      10. +-commutative 83.7%

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

      11. remove-double-neg 83.7%

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

      12. +-commutative 83.7%

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

      13. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in y around 0 59.8%

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

      1. associate-/r* 61.6%

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

      2. +-commutative 61.6%

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

   7. Simplified 61.6%

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

      1. clear-num 61.0%

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

      2. inv-pow 61.0%

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

      3. div-inv 61.0%

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

      4. clear-num 61.0%

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

   9. Applied egg-rr 61.0%

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

       1. unpow-1 61.0%

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

       2. associate-*r/ 59.5%

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

       3. *-commutative 59.5%

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

   11. Simplified 59.5%

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

   12. Taylor expanded in x around 0 37.2%

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

   if 1.4600000000000001e-160 < y < 0.75

   1. Initial program 76.0%

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

      1. associate-/r* 79.2%

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

      2. *-commutative 79.2%

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

      3. +-commutative 79.2%

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

      4. +-commutative 79.2%

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

      5. associate-*l/ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-*r/ 99.6%

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

      8. remove-double-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. remove-double-neg 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 41.3%

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

      1. +-commutative 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in y around 0 41.3%

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

      1. neg-mul-1 41.3%

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

      2. +-commutative 41.3%

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

      3. unsub-neg 41.3%

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

   10. Simplified 41.3%

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

   if 0.75 < y

   1. Initial program 61.6%

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

      1. associate-/r* 66.2%

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

      2. *-commutative 66.2%

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

      3. +-commutative 66.2%

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

      4. +-commutative 66.2%

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

      5. associate-*l/ 81.1%

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

      6. +-commutative 81.1%

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

      7. associate-*r/ 81.2%

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

      8. remove-double-neg 81.2%

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

      9. +-commutative 81.2%

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

      10. +-commutative 81.2%

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

      11. remove-double-neg 81.2%

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

      12. +-commutative 81.2%

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

      13. associate-+l+ 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. +-commutative 64.9%

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

   7. Simplified 64.9%

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

      1. *-un-lft-identity 64.9%

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

      2. times-frac 69.1%

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

   9. Applied egg-rr 69.1%

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

       1. associate-*l/ 69.2%

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

       2. +-commutative 69.2%

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

       3. *-lft-identity 69.2%

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

       4. +-commutative 69.2%

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

   11. Simplified 69.2%

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

   12. Taylor expanded in y around inf 67.3%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.46 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e-174) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d-174)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-174:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-174)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e-174)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-174], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000001e-174

   1. Initial program 63.2%

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

      1. associate-/r* 69.1%

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

      2. *-commutative 69.1%

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

      3. +-commutative 69.1%

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

      4. +-commutative 69.1%

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

      5. associate-*l/ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-*r/ 79.2%

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

      8. remove-double-neg 79.2%

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

      9. +-commutative 79.2%

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

      10. +-commutative 79.2%

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

      11. remove-double-neg 79.2%

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

      12. +-commutative 79.2%

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

      13. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in y around 0 67.7%

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

      1. associate-/r* 79.6%

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

      2. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if -3.7000000000000001e-174 < x

   1. Initial program 70.2%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. times-frac 70.2%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 77.5%

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

      10. fma-def 83.3%

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

      11. +-commutative 83.3%

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

      12. +-commutative 83.3%

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

      13. cube-unmult 83.3%

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

      14. +-commutative 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 70.2%

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

      2. fma-udef 66.6%

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

      3. cube-mult 66.6%

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

      4. distribute-rgt1-in 70.2%

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

      5. associate-+r+ 70.2%

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

      6. *-commutative 70.2%

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

      7. frac-times 87.2%

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

      8. clear-num 87.2%

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

      9. associate-*l/ 87.3%

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

      10. *-un-lft-identity 87.3%

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

      11. associate-+r+ 87.3%

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

      12. +-commutative 87.3%

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

      13. associate-+l+ 87.3%

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

      14. pow2 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. unpow2 87.3%

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

      2. associate-/l* 99.5%

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

      3. associate-/r/ 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in y around inf 57.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e-174) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d-174)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-174:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-174)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e-174)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-174], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000001e-174

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-commutative 63.2%

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

      3. frac-times 79.2%

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

      4. associate-*l/ 77.8%

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

      5. times-frac 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l+ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.8%

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

      3. associate-*r/ 99.8%

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

      4. clear-num 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. frac-times 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+r+ 99.6%

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

      12. +-commutative 99.6%

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

      13. associate-+r+ 99.6%

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

      14. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around 0 79.9%

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

      1. +-commutative 79.9%

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

   9. Simplified 79.9%

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

   if -3.7000000000000001e-174 < x

   1. Initial program 70.2%

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

      1. times-frac 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. +-commutative 87.2%

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

      5. times-frac 70.2%

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

      6. associate-*l/ 83.4%

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

      7. *-commutative 83.4%

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

      8. *-commutative 83.4%

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

      9. distribute-rgt1-in 77.5%

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

      10. fma-def 83.3%

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

      11. +-commutative 83.3%

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

      12. +-commutative 83.3%

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

      13. cube-unmult 83.3%

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

      14. +-commutative 83.3%

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

   3. Simplified 83.3%

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

      1. associate-*r/ 70.2%

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

      2. fma-udef 66.6%

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

      3. cube-mult 66.6%

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

      4. distribute-rgt1-in 70.2%

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

      5. associate-+r+ 70.2%

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

      6. *-commutative 70.2%

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

      7. frac-times 87.2%

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

      8. clear-num 87.2%

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

      9. associate-*l/ 87.3%

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

      10. *-un-lft-identity 87.3%

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

      11. associate-+r+ 87.3%

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

      12. +-commutative 87.3%

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

      13. associate-+l+ 87.3%

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

      14. pow2 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. unpow2 87.3%

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

      2. associate-/l* 99.5%

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

      3. associate-/r/ 99.4%

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

      4. +-commutative 99.4%

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

      5. +-commutative 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in y around inf 57.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.05e-118) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.05e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.05d-118) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.05e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.05e-118:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.05e-118)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.05e-118)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.05e-118], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.04999999999999992e-118

   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-/r* 72.2%

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

      2. *-commutative 72.2%

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

      3. +-commutative 72.2%

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

      4. +-commutative 72.2%

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

      5. associate-*l/ 84.4%

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

      6. +-commutative 84.4%

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

      7. associate-*r/ 84.4%

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

      8. remove-double-neg 84.4%

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

      9. +-commutative 84.4%

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

      10. +-commutative 84.4%

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

      11. remove-double-neg 84.4%

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

      12. +-commutative 84.4%

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

      13. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in y around 0 60.0%

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

   if 3.04999999999999992e-118 < y

   1. Initial program 66.2%

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

      1. associate-/r* 70.8%

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

      2. *-commutative 70.8%

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

      3. +-commutative 70.8%

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

      4. +-commutative 70.8%

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

      5. associate-*l/ 85.5%

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

      6. +-commutative 85.5%

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

      7. associate-*r/ 85.6%

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

      8. remove-double-neg 85.6%

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

      9. +-commutative 85.6%

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

      10. +-commutative 85.6%

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

      11. remove-double-neg 85.6%

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

      12. +-commutative 85.6%

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

      13. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. +-commutative 59.5%

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

   7. Simplified 59.5%

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

      1. *-un-lft-identity 59.5%

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

      2. times-frac 62.7%

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

   9. Applied egg-rr 62.7%

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

       1. associate-*l/ 62.7%

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

       2. +-commutative 62.7%

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

       3. *-lft-identity 62.7%

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

       4. +-commutative 62.7%

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

   11. Simplified 62.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e-174) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d-174)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e-174) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e-174:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e-174)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e-174)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e-174], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000001e-174

   1. Initial program 63.2%

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

      1. associate-/r* 69.1%

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

      2. *-commutative 69.1%

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

      3. +-commutative 69.1%

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

      4. +-commutative 69.1%

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

      5. associate-*l/ 79.1%

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

      6. +-commutative 79.1%

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

      7. associate-*r/ 79.2%

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

      8. remove-double-neg 79.2%

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

      9. +-commutative 79.2%

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

      10. +-commutative 79.2%

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

      11. remove-double-neg 79.2%

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

      12. +-commutative 79.2%

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

      13. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in y around 0 67.7%

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

      1. associate-/r* 79.6%

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

      2. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if -3.7000000000000001e-174 < x

   1. Initial program 70.2%

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

      1. associate-/r* 72.9%

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

      2. *-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. +-commutative 72.9%

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

      5. associate-*l/ 87.2%

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

      6. +-commutative 87.2%

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

      7. associate-*r/ 87.2%

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

      8. remove-double-neg 87.2%

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

      9. +-commutative 87.2%

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

      10. +-commutative 87.2%

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

      11. remove-double-neg 87.2%

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

      12. +-commutative 87.2%

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

      13. associate-+l+ 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in x around 0 54.3%

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

      1. +-commutative 54.3%

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

   7. Simplified 54.3%

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

      1. *-un-lft-identity 54.3%

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

      2. times-frac 57.0%

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

   9. Applied egg-rr 57.0%

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

       1. associate-*l/ 57.1%

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

       2. +-commutative 57.1%

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

       3. *-lft-identity 57.1%

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

       4. +-commutative 57.1%

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

   11. Simplified 57.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.42e-154) (/ 1.0 (/ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.42e-154) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.42d-154) then
        tmp = 1.0d0 / (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.42e-154) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.42e-154:
		tmp = 1.0 / (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.42e-154)
		tmp = Float64(1.0 / Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.42e-154)
		tmp = 1.0 / (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.42e-154], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.42e-154

   1. Initial program 69.3%

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

      1. associate-/r* 72.7%

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

      2. *-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. +-commutative 72.7%

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

      5. associate-*l/ 83.7%

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

      6. +-commutative 83.7%

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

      7. associate-*r/ 83.7%

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

      8. remove-double-neg 83.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 83.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 59.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 61.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 61.6%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]
   8. Step-by-step derivation

      1. clear-num 61.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{\frac{y}{x}}}} \]

      2. inv-pow 61.0%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{\frac{y}{x}}\right)}^{-1}} \]

      3. div-inv 61.0%

         \[\leadsto {\color{blue}{\left(\left(x + 1\right) \cdot \frac{1}{\frac{y}{x}}\right)}}^{-1} \]

      4. clear-num 61.0%

         \[\leadsto {\left(\left(x + 1\right) \cdot \color{blue}{\frac{x}{y}}\right)}^{-1} \]

   9. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(\left(x + 1\right) \cdot \frac{x}{y}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 61.0%

          \[\leadsto \color{blue}{\frac{1}{\left(x + 1\right) \cdot \frac{x}{y}}} \]

       2. associate-*r/ 59.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot x}{y}}} \]

       3. *-commutative 59.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left(x + 1\right)}}{y}} \]

   11. Simplified 59.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x + 1\right)}{y}}} \]

   12. Taylor expanded in x around 0 37.2%

       \[\leadsto \frac{1}{\frac{\color{blue}{x}}{y}} \]

   if 1.42e-154 < y

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 70.1%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.1%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.1%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.8%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 57.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 35.7%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-154}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.5e-160) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-160) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.5d-160) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-160) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.5e-160:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-160)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-160)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-160], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000003e-160

   1. Initial program 69.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 72.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 72.7%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 72.7%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 72.7%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 83.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 83.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 83.7%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 83.7%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 83.7%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 83.7%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 59.8%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 61.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 61.6%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   7. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]
   8. Step-by-step derivation

      1. clear-num 61.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{\frac{y}{x}}}} \]

      2. inv-pow 61.0%

         \[\leadsto \color{blue}{{\left(\frac{x + 1}{\frac{y}{x}}\right)}^{-1}} \]

      3. div-inv 61.0%

         \[\leadsto {\color{blue}{\left(\left(x + 1\right) \cdot \frac{1}{\frac{y}{x}}\right)}}^{-1} \]

      4. clear-num 61.0%

         \[\leadsto {\left(\left(x + 1\right) \cdot \color{blue}{\frac{x}{y}}\right)}^{-1} \]

   9. Applied egg-rr 61.0%

      \[\leadsto \color{blue}{{\left(\left(x + 1\right) \cdot \frac{x}{y}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 61.0%

          \[\leadsto \color{blue}{\frac{1}{\left(x + 1\right) \cdot \frac{x}{y}}} \]

       2. associate-*r/ 59.5%

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x + 1\right) \cdot x}{y}}} \]

       3. *-commutative 59.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left(x + 1\right)}}{y}} \]

   11. Simplified 59.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(x + 1\right)}{y}}} \]

   12. Taylor expanded in x around 0 36.5%

       \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 3.5000000000000003e-160 < y

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. *-commutative 70.1%

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.1%

         \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.1%

         \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      5. associate-*l/ 86.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. +-commutative 86.7%

         \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

      7. associate-*r/ 86.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. remove-double-neg 86.8%

         \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

      9. +-commutative 86.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      10. +-commutative 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

      11. remove-double-neg 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

      12. +-commutative 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      13. associate-+l+ 86.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 57.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 57.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   7. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   8. Taylor expanded in y around 0 35.7%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 71.7%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. *-commutative 71.7%

      \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 71.7%

      \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   4. +-commutative 71.7%

      \[\leadsto \frac{\frac{y \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. associate-*l/ 84.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. +-commutative 84.8%

      \[\leadsto \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot x}{\color{blue}{\left(y + x\right)} + 1} \]

   7. associate-*r/ 84.8%

      \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

   8. remove-double-neg 84.8%

      \[\leadsto \frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{\color{blue}{-\left(-x\right)}}{\left(y + x\right) + 1} \]

   9. +-commutative 84.8%

      \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   10. +-commutative 84.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{-\left(-x\right)}{\left(y + x\right) + 1} \]

   11. remove-double-neg 84.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{\color{blue}{x}}{\left(y + x\right) + 1} \]

   12. +-commutative 84.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

   13. associate-+l+ 84.8%

       \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 84.8%

   \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
6. Step-by-step derivation

   1. +-commutative 43.2%

      \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

7. Simplified 43.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

8. Taylor expanded in y around 0 25.6%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

9. Final simplification 25.6%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Developer target: 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 2024021 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))