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

Percentage Accurate: 67.9% → 99.8%

Time: 16.1s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-*l* 72.0%

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

   2. +-commutative 72.0%

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

   3. +-commutative 72.0%

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

   4. +-commutative 72.0%

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

   5. associate-*l* 72.0%

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

   6. *-commutative 72.0%

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

   7. associate-*r/ 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt1-in 68.3%

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

   10. fma-def 84.7%

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

   11. +-commutative 84.7%

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

   12. +-commutative 84.7%

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

   13. cube-unmult 84.8%

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

   14. +-commutative 84.8%

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

3. Simplified 84.8%

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

   1. associate-*r/ 72.0%

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

   2. *-commutative 72.0%

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

   3. fma-udef 60.7%

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

   4. cube-mult 60.6%

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

   5. distribute-rgt1-in 72.0%

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

   6. *-commutative 72.0%

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

   7. +-commutative 72.0%

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

   8. associate-+r+ 72.0%

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

   9. frac-times 89.3%

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

   10. *-commutative 89.3%

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

   11. associate-/r* 99.7%

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

   12. associate-*r/ 99.8%

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

   13. associate-+r+ 99.8%

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

   14. +-commutative 99.8%

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

   15. associate-+l+ 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3750000000:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3750000000.0)
   (/ (/ y (+ x (+ y 1.0))) (+ y x))
   (if (<= x -8e-150)
     (* (/ y (+ y 1.0)) (/ x (* (+ y x) (+ y x))))
     (/ (/ x (+ y (+ x 1.0))) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3750000000.0) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -8e-150) {
		tmp = (y / (y + 1.0)) * (x / ((y + x) * (y + x)));
	} 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 <= (-3750000000.0d0)) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else if (x <= (-8d-150)) then
        tmp = (y / (y + 1.0d0)) * (x / ((y + x) * (y + x)))
    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 <= -3750000000.0) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -8e-150) {
		tmp = (y / (y + 1.0)) * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3750000000.0:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	elif x <= -8e-150:
		tmp = (y / (y + 1.0)) * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3750000000.0)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	elseif (x <= -8e-150)
		tmp = Float64(Float64(y / Float64(y + 1.0)) * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	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 <= -3750000000.0)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	elseif (x <= -8e-150)
		tmp = (y / (y + 1.0)) * (x / ((y + x) * (y + x)));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3750000000.0], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-150], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.75e9

   1. Initial program 63.7%

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

      1. associate-*l* 63.7%

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

      2. +-commutative 63.7%

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

      3. +-commutative 63.7%

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

      4. +-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. associate-*r/ 83.1%

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

      8. *-commutative 83.1%

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

      9. distribute-rgt1-in 40.1%

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

      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. Step-by-step derivation

      1. associate-*r/ 63.7%

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

      2. *-commutative 63.7%

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

      3. fma-udef 38.5%

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

      4. cube-mult 38.5%

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

      5. distribute-rgt1-in 63.7%

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

      6. *-commutative 63.7%

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

      7. +-commutative 63.7%

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

      8. associate-+r+ 63.7%

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

      9. frac-times 89.6%

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

      10. *-commutative 89.6%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 84.4%

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

   if -3.75e9 < x < -8.00000000000000005e-150

   1. Initial program 82.5%

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

      1. times-frac 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -8.00000000000000005e-150 < x

   1. Initial program 73.5%

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

      1. associate-*l* 73.5%

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

      2. +-commutative 73.5%

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

      3. +-commutative 73.5%

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

      4. +-commutative 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*l* 73.5%

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

      7. times-frac 87.4%

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

      8. +-commutative 87.4%

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

      9. +-commutative 87.4%

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

      10. associate-+l+ 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in y around inf 48.2%

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

      1. associate-*l/ 48.2%

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

      2. *-un-lft-identity 48.2%

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

   6. Applied egg-rr 48.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3750000000:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 3: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2050000.0)
   (* (/ x (* (+ y x) (+ y x))) (/ y (+ y (+ x 1.0))))
   (/ (/ x (/ (+ y x) (/ y (+ y 1.0)))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2050000.0) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	} else {
		tmp = (x / ((y + x) / (y / (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 <= (-2050000.0d0)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0d0)))
    else
        tmp = (x / ((y + x) / (y / (y + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2050000.0) {
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	} else {
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2050000.0:
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)))
	else:
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2050000.0)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) / Float64(y / Float64(y + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2050000.0)
		tmp = (x / ((y + x) * (y + x))) * (y / (y + (x + 1.0)));
	else
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2050000.0], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] / N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e6

   1. Initial program 64.2%

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

      1. times-frac 89.8%

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

      2. +-commutative 89.8%

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

      3. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   if -2.05e6 < x

   1. Initial program 74.7%

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

      1. associate-*l* 74.7%

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

      2. +-commutative 74.7%

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

      3. +-commutative 74.7%

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

      4. +-commutative 74.7%

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

      5. associate-*l* 74.7%

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

      6. *-commutative 74.7%

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

      7. associate-*r/ 85.2%

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

      8. *-commutative 85.2%

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

      9. distribute-rgt1-in 77.6%

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

      10. fma-def 85.2%

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

      11. +-commutative 85.2%

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

      12. +-commutative 85.2%

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

      13. cube-unmult 85.2%

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

      14. +-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. associate-*r/ 74.7%

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

      2. *-commutative 74.7%

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

      3. fma-udef 67.9%

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

      4. cube-mult 67.9%

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

      5. distribute-rgt1-in 74.7%

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

      6. *-commutative 74.7%

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

      7. +-commutative 74.7%

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

      8. associate-+r+ 74.7%

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

      9. frac-times 89.1%

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

      10. *-commutative 89.1%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. *-rgt-identity 99.0%

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

      8. div-inv 99.1%

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

      9. *-rgt-identity 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 80.5%

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

       1. +-commutative 80.5%

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

   12. Simplified 80.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \]

Alternative 4: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2050000.0)
   (* (/ y (* (+ y x) (+ y x))) (/ x (+ y (+ x 1.0))))
   (/ (/ x (/ (+ y x) (/ y (+ y 1.0)))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2050000.0) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	} else {
		tmp = (x / ((y + x) / (y / (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 <= (-2050000.0d0)) then
        tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0d0)))
    else
        tmp = (x / ((y + x) / (y / (y + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2050000.0) {
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	} else {
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2050000.0:
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)))
	else:
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2050000.0)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) * Float64(x / Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) / Float64(y / Float64(y + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2050000.0)
		tmp = (y / ((y + x) * (y + x))) * (x / (y + (x + 1.0)));
	else
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2050000.0], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] / N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05e6

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

      2. +-commutative 64.2%

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

      3. +-commutative 64.2%

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

      4. +-commutative 64.2%

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

      5. *-commutative 64.2%

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

      6. associate-*l* 64.2%

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

      7. times-frac 89.8%

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

      8. +-commutative 89.8%

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

      9. +-commutative 89.8%

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

      10. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   if -2.05e6 < x

   1. Initial program 74.7%

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

      1. associate-*l* 74.7%

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

      2. +-commutative 74.7%

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

      3. +-commutative 74.7%

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

      4. +-commutative 74.7%

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

      5. associate-*l* 74.7%

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

      6. *-commutative 74.7%

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

      7. associate-*r/ 85.2%

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

      8. *-commutative 85.2%

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

      9. distribute-rgt1-in 77.6%

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

      10. fma-def 85.2%

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

      11. +-commutative 85.2%

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

      12. +-commutative 85.2%

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

      13. cube-unmult 85.2%

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

      14. +-commutative 85.2%

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

   3. Simplified 85.2%

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

      1. associate-*r/ 74.7%

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

      2. *-commutative 74.7%

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

      3. fma-udef 67.9%

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

      4. cube-mult 67.9%

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

      5. distribute-rgt1-in 74.7%

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

      6. *-commutative 74.7%

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

      7. +-commutative 74.7%

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

      8. associate-+r+ 74.7%

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

      9. frac-times 89.1%

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

      10. *-commutative 89.1%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. *-rgt-identity 99.0%

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

      8. div-inv 99.1%

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

      9. *-rgt-identity 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 80.5%

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

       1. +-commutative 80.5%

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

   12. Simplified 80.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \]

Alternative 5: 63.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + \left(x + 1\right)}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_0}{y}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (+ x 1.0)))))
   (if (<= y 1.8e-126)
     (/ (/ y (+ x 1.0)) (+ y x))
     (if (<= y 2.9e-7)
       (/ t_0 y)
       (if (<= y 3.65e+148) (/ x (* (+ y x) (+ y x))) (* t_0 (/ 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (x + 1.0));
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.9e-7) {
		tmp = t_0 / y;
	} else if (y <= 3.65e+148) {
		tmp = x / ((y + x) * (y + x));
	} else {
		tmp = t_0 * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + (x + 1.0d0))
    if (y <= 1.8d-126) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 2.9d-7) then
        tmp = t_0 / y
    else if (y <= 3.65d+148) then
        tmp = x / ((y + x) * (y + x))
    else
        tmp = t_0 * (1.0d0 / y)
    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 (y <= 1.8e-126) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 2.9e-7) {
		tmp = t_0 / y;
	} else if (y <= 3.65e+148) {
		tmp = x / ((y + x) * (y + x));
	} else {
		tmp = t_0 * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (x + 1.0))
	tmp = 0
	if y <= 1.8e-126:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 2.9e-7:
		tmp = t_0 / y
	elif y <= 3.65e+148:
		tmp = x / ((y + x) * (y + x))
	else:
		tmp = t_0 * (1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 1.8e-126)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 2.9e-7)
		tmp = Float64(t_0 / y);
	elseif (y <= 3.65e+148)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	else
		tmp = Float64(t_0 * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (x + 1.0));
	tmp = 0.0;
	if (y <= 1.8e-126)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 2.9e-7)
		tmp = t_0 / y;
	elseif (y <= 3.65e+148)
		tmp = x / ((y + x) * (y + x));
	else
		tmp = t_0 * (1.0 / y);
	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[y, 1.8e-126], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e-7], N[(t$95$0 / y), $MachinePrecision], If[LessEqual[y, 3.65e+148], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.8e-126

   1. Initial program 70.1%

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

      1. associate-*l* 70.1%

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

      2. +-commutative 70.1%

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

      3. +-commutative 70.1%

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

      4. +-commutative 70.1%

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

      5. associate-*l* 70.1%

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

      6. *-commutative 70.1%

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

      7. associate-*r/ 84.1%

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

      8. *-commutative 84.1%

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

      9. distribute-rgt1-in 65.4%

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

      10. fma-def 84.1%

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

      11. +-commutative 84.1%

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

      12. +-commutative 84.1%

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

      13. cube-unmult 84.1%

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

      14. +-commutative 84.1%

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

   3. Simplified 84.1%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. fma-udef 56.7%

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

      4. cube-mult 56.7%

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

      5. distribute-rgt1-in 70.1%

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

      6. *-commutative 70.1%

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

      7. +-commutative 70.1%

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

      8. associate-+r+ 70.1%

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

      9. frac-times 87.4%

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

      10. *-commutative 87.4%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

   8. Simplified 66.3%

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

   if 1.8e-126 < y < 2.8999999999999998e-7

   1. Initial program 91.4%

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

      1. associate-*l* 91.4%

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

      2. +-commutative 91.4%

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

      3. +-commutative 91.4%

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

      4. +-commutative 91.4%

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

      5. *-commutative 91.4%

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

      6. associate-*l* 91.4%

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

      7. times-frac 99.3%

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

      8. +-commutative 99.3%

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

      9. +-commutative 99.3%

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

      10. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around inf 34.9%

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

      1. associate-*l/ 35.1%

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

      2. *-un-lft-identity 35.1%

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

   6. Applied egg-rr 35.1%

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

   if 2.8999999999999998e-7 < y < 3.65e148

   1. Initial program 69.5%

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

      1. times-frac 96.3%

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

      2. +-commutative 96.3%

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

      3. associate-+l+ 96.3%

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

   3. Simplified 96.3%

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

   4. Taylor expanded in y around inf 80.6%

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

   if 3.65e148 < y

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. +-commutative 71.2%

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

      3. +-commutative 71.2%

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

      4. +-commutative 71.2%

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

      5. *-commutative 71.2%

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

      6. associate-*l* 71.2%

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

      7. times-frac 85.4%

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

      8. +-commutative 85.4%

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

      9. +-commutative 85.4%

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

      10. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in y around inf 83.7%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + \left(x + 1\right)} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 6: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 29:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.76e-152)
   (/ (/ y (+ x (+ y 1.0))) (+ y x))
   (if (<= y 29.0)
     (* x (/ y (* (+ y x) (+ y x))))
     (/ (/ x (+ y (+ x (+ x 1.0)))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.76e-152) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (y <= 29.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.76d-152) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else if (y <= 29.0d0) then
        tmp = x * (y / ((y + x) * (y + x)))
    else
        tmp = (x / (y + (x + (x + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.76e-152) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (y <= 29.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + (x + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.76e-152:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	elif y <= 29.0:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + (x + (x + 1.0)))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.76e-152)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	elseif (y <= 29.0)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + Float64(x + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.76e-152)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	elseif (y <= 29.0)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + (x + (x + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.76e-152], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 29.0], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.76000000000000001e-152

   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. associate-*l* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-*l* 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*r/ 83.8%

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

      8. *-commutative 83.8%

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

      9. distribute-rgt1-in 64.6%

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

      10. fma-def 83.7%

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

      11. +-commutative 83.7%

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

      12. +-commutative 83.7%

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

      13. cube-unmult 83.8%

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

      14. +-commutative 83.8%

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

   3. Simplified 83.8%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. 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. *-commutative 69.9%

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

      3. fma-udef 56.3%

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

      4. cube-mult 56.3%

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

      5. distribute-rgt1-in 69.9%

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

      6. *-commutative 69.9%

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

      7. +-commutative 69.9%

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

      8. associate-+r+ 69.9%

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

      9. frac-times 87.2%

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

      10. *-commutative 87.2%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 66.6%

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

   if 1.76000000000000001e-152 < y < 29

   1. Initial program 90.1%

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

      1. associate-*l* 90.2%

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

      2. +-commutative 90.2%

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

      3. +-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.1%

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

      7. times-frac 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

      10. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 74.5%

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

      1. +-commutative 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around 0 73.2%

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

   if 29 < y

   1. Initial program 68.5%

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

      1. associate-*l* 68.4%

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

      2. +-commutative 68.4%

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

      3. +-commutative 68.4%

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

      4. +-commutative 68.4%

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

      5. associate-*l* 68.5%

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

      6. *-commutative 68.5%

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

      7. associate-*r/ 81.0%

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

      8. *-commutative 81.0%

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

      9. distribute-rgt1-in 72.6%

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

      10. fma-def 81.0%

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

      11. +-commutative 81.0%

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

      12. +-commutative 81.0%

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

      13. cube-unmult 81.1%

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

      14. +-commutative 81.1%

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

   3. Simplified 81.1%

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

      1. associate-*r/ 68.5%

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

      2. *-commutative 68.5%

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

      3. fma-udef 66.5%

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

      4. cube-mult 66.4%

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

      5. distribute-rgt1-in 68.5%

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

      6. *-commutative 68.5%

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

      7. +-commutative 68.5%

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

      8. associate-+r+ 68.5%

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

      9. frac-times 90.9%

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

      10. *-commutative 90.9%

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

      11. associate-/r* 99.8%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. frac-times 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-+r+ 99.7%

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

      7. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in y around -inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. unsub-neg 69.6%

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

      3. neg-mul-1 69.6%

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

      4. +-commutative 69.6%

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

      5. unsub-neg 69.6%

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

      6. distribute-lft-in 69.6%

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

      7. metadata-eval 69.6%

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

      8. neg-mul-1 69.6%

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

      9. unsub-neg 69.6%

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

   10. Simplified 69.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 29:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + \left(x + 1\right)\right)}}{y + x}\\ \end{array} \]

Alternative 7: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31000000000:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -31000000000.0)
   (/ (/ y (+ x (+ y 1.0))) (+ y x))
   (/ (/ x (/ (+ y x) (/ y (+ y 1.0)))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -31000000000.0) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else {
		tmp = (x / ((y + x) / (y / (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 <= (-31000000000.0d0)) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else
        tmp = (x / ((y + x) / (y / (y + 1.0d0)))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -31000000000.0) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else {
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -31000000000.0:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	else:
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -31000000000.0)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(Float64(y + x) / Float64(y / Float64(y + 1.0)))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -31000000000.0)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	else
		tmp = (x / ((y + x) / (y / (y + 1.0)))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -31000000000.0], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(N[(y + x), $MachinePrecision] / N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e10

   1. Initial program 63.7%

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

      1. associate-*l* 63.7%

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

      2. +-commutative 63.7%

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

      3. +-commutative 63.7%

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

      4. +-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. associate-*r/ 83.1%

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

      8. *-commutative 83.1%

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

      9. distribute-rgt1-in 40.1%

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

      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. Step-by-step derivation

      1. associate-*r/ 63.7%

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

      2. *-commutative 63.7%

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

      3. fma-udef 38.5%

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

      4. cube-mult 38.5%

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

      5. distribute-rgt1-in 63.7%

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

      6. *-commutative 63.7%

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

      7. +-commutative 63.7%

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

      8. associate-+r+ 63.7%

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

      9. frac-times 89.6%

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

      10. *-commutative 89.6%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 84.4%

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

   if -3.1e10 < x

   1. Initial program 74.8%

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

      1. associate-*l* 74.8%

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

      2. +-commutative 74.8%

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

      3. +-commutative 74.8%

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

      4. +-commutative 74.8%

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

      5. associate-*l* 74.8%

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

      6. *-commutative 74.8%

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

      7. associate-*r/ 85.3%

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

      8. *-commutative 85.3%

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

      9. distribute-rgt1-in 77.7%

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

      10. fma-def 85.3%

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

      11. +-commutative 85.3%

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

      12. +-commutative 85.3%

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

      13. cube-unmult 85.3%

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

      14. +-commutative 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 74.8%

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

      2. *-commutative 74.8%

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

      3. fma-udef 68.1%

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

      4. cube-mult 68.0%

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

      5. distribute-rgt1-in 74.8%

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

      6. *-commutative 74.8%

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

      7. +-commutative 74.8%

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

      8. associate-+r+ 74.8%

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

      9. frac-times 89.2%

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

      10. *-commutative 89.2%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.8%

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

      13. associate-+r+ 99.8%

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

      14. +-commutative 99.8%

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

      15. associate-+l+ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. frac-times 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. clear-num 99.0%

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

      4. associate-+r+ 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-+r+ 99.0%

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

      7. *-rgt-identity 99.0%

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

      8. div-inv 99.1%

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

      9. *-rgt-identity 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 80.5%

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

       1. +-commutative 80.5%

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

   12. Simplified 80.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31000000000:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{y + x}{\frac{y}{y + 1}}}}{y + x}\\ \end{array} \]

Alternative 8: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.76e-152)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 45.0)
     (* x (/ y (* (+ y x) (+ y x))))
     (/ (/ x (+ y (+ x 1.0))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.76e-152) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 45.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} 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 (y <= 1.76d-152) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 45.0d0) then
        tmp = x * (y / ((y + x) * (y + x)))
    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 (y <= 1.76e-152) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 45.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.76e-152:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 45.0:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.76e-152)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 45.0)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	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 (y <= 1.76e-152)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 45.0)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.76e-152], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 45.0], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.76000000000000001e-152

   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. associate-*l* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-*l* 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*r/ 83.8%

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

      8. *-commutative 83.8%

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

      9. distribute-rgt1-in 64.6%

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

      10. fma-def 83.7%

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

      11. +-commutative 83.7%

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

      12. +-commutative 83.7%

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

      13. cube-unmult 83.8%

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

      14. +-commutative 83.8%

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

   3. Simplified 83.8%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. 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. *-commutative 69.9%

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

      3. fma-udef 56.3%

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

      4. cube-mult 56.3%

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

      5. distribute-rgt1-in 69.9%

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

      6. *-commutative 69.9%

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

      7. +-commutative 69.9%

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

      8. associate-+r+ 69.9%

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

      9. frac-times 87.2%

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

      10. *-commutative 87.2%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 66.1%

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

      1. +-commutative 66.1%

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

   8. Simplified 66.1%

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

   if 1.76000000000000001e-152 < y < 45

   1. Initial program 90.1%

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

      1. associate-*l* 90.2%

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

      2. +-commutative 90.2%

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

      3. +-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.1%

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

      7. times-frac 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

      10. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 74.5%

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

      1. +-commutative 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around 0 73.2%

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

   if 45 < y

   1. Initial program 68.5%

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

      1. associate-*l* 68.4%

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

      2. +-commutative 68.4%

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

      3. +-commutative 68.4%

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

      4. +-commutative 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-*l* 68.5%

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

      7. times-frac 90.9%

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

      8. +-commutative 90.9%

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

      9. +-commutative 90.9%

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

      10. associate-+l+ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around inf 68.6%

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

      1. associate-*l/ 68.6%

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

      2. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 68.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 9: 66.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 8.5:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.76e-152)
   (/ (/ y (+ x (+ y 1.0))) (+ y x))
   (if (<= y 8.5)
     (* x (/ y (* (+ y x) (+ y x))))
     (/ (/ x (+ y (+ x 1.0))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.76e-152) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (y <= 8.5) {
		tmp = x * (y / ((y + x) * (y + x)));
	} 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 (y <= 1.76d-152) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else if (y <= 8.5d0) then
        tmp = x * (y / ((y + x) * (y + x)))
    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 (y <= 1.76e-152) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (y <= 8.5) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.76e-152:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	elif y <= 8.5:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.76e-152)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	elseif (y <= 8.5)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	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 (y <= 1.76e-152)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	elseif (y <= 8.5)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.76e-152], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.76000000000000001e-152

   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. associate-*l* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-*l* 69.9%

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

      6. *-commutative 69.9%

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

      7. associate-*r/ 83.8%

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

      8. *-commutative 83.8%

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

      9. distribute-rgt1-in 64.6%

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

      10. fma-def 83.7%

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

      11. +-commutative 83.7%

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

      12. +-commutative 83.7%

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

      13. cube-unmult 83.8%

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

      14. +-commutative 83.8%

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

   3. Simplified 83.8%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. 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. *-commutative 69.9%

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

      3. fma-udef 56.3%

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

      4. cube-mult 56.3%

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

      5. distribute-rgt1-in 69.9%

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

      6. *-commutative 69.9%

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

      7. +-commutative 69.9%

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

      8. associate-+r+ 69.9%

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

      9. frac-times 87.2%

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

      10. *-commutative 87.2%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.9%

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

      13. associate-+r+ 99.9%

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

      14. +-commutative 99.9%

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

      15. associate-+l+ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 66.6%

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

   if 1.76000000000000001e-152 < y < 8.5

   1. Initial program 90.1%

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

      1. associate-*l* 90.2%

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

      2. +-commutative 90.2%

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

      3. +-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. associate-*l* 90.1%

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

      7. times-frac 99.4%

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

      8. +-commutative 99.4%

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

      9. +-commutative 99.4%

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

      10. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 74.5%

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

      1. +-commutative 74.5%

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

   6. Simplified 74.5%

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

   7. Taylor expanded in y around 0 73.2%

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

   if 8.5 < y

   1. Initial program 68.5%

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

      1. associate-*l* 68.4%

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

      2. +-commutative 68.4%

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

      3. +-commutative 68.4%

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

      4. +-commutative 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-*l* 68.5%

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

      7. times-frac 90.9%

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

      8. +-commutative 90.9%

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

      9. +-commutative 90.9%

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

      10. associate-+l+ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around inf 68.6%

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

      1. associate-*l/ 68.6%

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

      2. *-un-lft-identity 68.6%

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

   6. Applied egg-rr 68.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;y \leq 8.5:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 10: 55.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -17500000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-168} \lor \neg \left(x \leq 6.8 \cdot 10^{-193}\right):\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -17500000000.0)
   (/ (/ y x) x)
   (if (or (<= x -3.9e-168) (not (<= x 6.8e-193)))
     (* (/ 1.0 y) (/ x y))
     (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -17500000000.0) {
		tmp = (y / x) / x;
	} else if ((x <= -3.9e-168) || !(x <= 6.8e-193)) {
		tmp = (1.0 / y) * (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 (x <= (-17500000000.0d0)) then
        tmp = (y / x) / x
    else if ((x <= (-3.9d-168)) .or. (.not. (x <= 6.8d-193))) then
        tmp = (1.0d0 / y) * (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -17500000000.0) {
		tmp = (y / x) / x;
	} else if ((x <= -3.9e-168) || !(x <= 6.8e-193)) {
		tmp = (1.0 / y) * (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -17500000000.0:
		tmp = (y / x) / x
	elif (x <= -3.9e-168) or not (x <= 6.8e-193):
		tmp = (1.0 / y) * (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -17500000000.0)
		tmp = Float64(Float64(y / x) / x);
	elseif ((x <= -3.9e-168) || !(x <= 6.8e-193))
		tmp = Float64(Float64(1.0 / y) * Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -17500000000.0)
		tmp = (y / x) / x;
	elseif ((x <= -3.9e-168) || ~((x <= 6.8e-193)))
		tmp = (1.0 / y) * (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -17500000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[Or[LessEqual[x, -3.9e-168], N[Not[LessEqual[x, 6.8e-193]], $MachinePrecision]], N[(N[(1.0 / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e10

   1. Initial program 63.7%

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

      1. associate-*l* 63.7%

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

      2. +-commutative 63.7%

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

      3. +-commutative 63.7%

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

      4. +-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. associate-*r/ 83.1%

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

      8. *-commutative 83.1%

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

      9. distribute-rgt1-in 40.1%

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

      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. Taylor expanded in x around inf 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 83.4%

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

   6. Applied egg-rr 83.4%

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

      1. associate-*l/ 83.4%

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

      2. *-lft-identity 83.4%

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

   8. Simplified 83.4%

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

   if -1.75e10 < x < -3.90000000000000012e-168 or 6.8000000000000004e-193 < x

   1. Initial program 79.6%

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

      1. associate-*l* 79.6%

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

      2. +-commutative 79.6%

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

      3. +-commutative 79.6%

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

      4. +-commutative 79.6%

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

      5. *-commutative 79.6%

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

      6. associate-*l* 79.6%

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

      7. times-frac 94.5%

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

      8. +-commutative 94.5%

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

      9. +-commutative 94.5%

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

      10. associate-+l+ 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in y around inf 37.9%

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

   5. Taylor expanded in y around inf 31.0%

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

   if -3.90000000000000012e-168 < x < 6.8000000000000004e-193

   1. Initial program 60.1%

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

      1. times-frac 72.9%

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

      2. +-commutative 72.9%

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

      3. associate-+l+ 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 83.9%

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

      1. +-commutative 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in y around 0 74.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17500000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-168} \lor \neg \left(x \leq 6.8 \cdot 10^{-193}\right):\\ \;\;\;\;\frac{1}{y} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 11: 60.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-126}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.45e+32)
   (/ (/ y x) x)
   (if (<= y 1.55e-126) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+32) {
		tmp = (y / x) / x;
	} else if (y <= 1.55e-126) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.45d+32)) then
        tmp = (y / x) / x
    else if (y <= 1.55d-126) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+32) {
		tmp = (y / x) / x;
	} else if (y <= 1.55e-126) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.45e+32:
		tmp = (y / x) / x
	elif y <= 1.55e-126:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.45e+32)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 1.55e-126)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.45e+32)
		tmp = (y / x) / x;
	elseif (y <= 1.55e-126)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.45e+32], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.55e-126], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4500000000000001e32

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. +-commutative 57.2%

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

      3. +-commutative 57.2%

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

      4. +-commutative 57.2%

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

      5. associate-*l* 57.2%

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

      6. *-commutative 57.2%

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

      7. associate-*r/ 77.8%

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

      8. *-commutative 77.8%

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

      9. distribute-rgt1-in 36.2%

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

      10. fma-def 77.8%

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

      11. +-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. cube-unmult 77.7%

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

      14. +-commutative 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in x around inf 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. unpow2 27.6%

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

      3. times-frac 35.5%

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

   6. Applied egg-rr 35.5%

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

      1. associate-*l/ 35.6%

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

      2. *-lft-identity 35.6%

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

   8. Simplified 35.6%

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

   if -2.4500000000000001e32 < y < 1.5500000000000001e-126

   1. Initial program 76.5%

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

      1. times-frac 88.5%

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

      2. +-commutative 88.5%

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

      3. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 81.0%

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

   if 1.5500000000000001e-126 < y

   1. Initial program 76.8%

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

      1. times-frac 93.8%

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

      2. +-commutative 93.8%

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

      3. associate-+l+ 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in x around 0 55.7%

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

      1. +-commutative 55.7%

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

   6. Simplified 55.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-126}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 12: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.45e+32)
   (/ (/ y x) x)
   (if (<= y 6e-130) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+32) {
		tmp = (y / x) / x;
	} else if (y <= 6e-130) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.45d+32)) then
        tmp = (y / x) / x
    else if (y <= 6d-130) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.45e+32) {
		tmp = (y / x) / x;
	} else if (y <= 6e-130) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.45e+32:
		tmp = (y / x) / x
	elif y <= 6e-130:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.45e+32)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 6e-130)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.45e+32)
		tmp = (y / x) / x;
	elseif (y <= 6e-130)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.45e+32], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 6e-130], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4500000000000001e32

   1. Initial program 57.2%

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

      1. associate-*l* 57.2%

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

      2. +-commutative 57.2%

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

      3. +-commutative 57.2%

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

      4. +-commutative 57.2%

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

      5. associate-*l* 57.2%

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

      6. *-commutative 57.2%

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

      7. associate-*r/ 77.8%

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

      8. *-commutative 77.8%

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

      9. distribute-rgt1-in 36.2%

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

      10. fma-def 77.8%

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

      11. +-commutative 77.8%

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

      12. +-commutative 77.8%

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

      13. cube-unmult 77.7%

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

      14. +-commutative 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in x around inf 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. unpow2 27.6%

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

      3. times-frac 35.5%

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

   6. Applied egg-rr 35.5%

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

      1. associate-*l/ 35.6%

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

      2. *-lft-identity 35.6%

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

   8. Simplified 35.6%

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

   if -2.4500000000000001e32 < y < 5.99999999999999972e-130

   1. Initial program 76.5%

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

      1. times-frac 88.5%

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

      2. +-commutative 88.5%

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

      3. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in y around 0 81.0%

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

   if 5.99999999999999972e-130 < y

   1. Initial program 76.8%

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

      1. associate-*l* 76.8%

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

      2. +-commutative 76.8%

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

      3. +-commutative 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-*l* 76.8%

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

      6. *-commutative 76.8%

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

      7. associate-*r/ 86.2%

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

      8. *-commutative 86.2%

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

      9. distribute-rgt1-in 75.4%

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

      10. fma-def 86.2%

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

      11. +-commutative 86.2%

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

      12. +-commutative 86.2%

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

      13. cube-unmult 86.3%

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

      14. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. associate-*r/ 76.8%

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

      2. *-commutative 76.8%

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

      3. fma-udef 70.1%

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

      4. cube-mult 70.1%

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

      5. distribute-rgt1-in 76.8%

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

      6. *-commutative 76.8%

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

      7. +-commutative 76.8%

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

      8. associate-+r+ 76.8%

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

      9. frac-times 93.8%

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

      10. *-commutative 93.8%

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

      11. associate-/r* 99.7%

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

      12. associate-*r/ 99.7%

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

      13. associate-+r+ 99.7%

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

      14. +-commutative 99.7%

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

      15. associate-+l+ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. frac-times 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. clear-num 99.6%

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

      4. associate-+r+ 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. *-rgt-identity 99.6%

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

      8. div-inv 99.6%

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

      9. *-rgt-identity 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in x around 0 55.7%

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

       1. associate-/r* 55.2%

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

   12. Simplified 55.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 13: 61.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-126) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-126) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-126:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-126)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-126)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-126], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e-126

   1. Initial program 70.1%

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

      1. times-frac 87.4%

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

      2. +-commutative 87.4%

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

      3. associate-+l+ 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in y around 0 63.3%

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

      1. associate-/r* 65.9%

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

      2. +-commutative 65.9%

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

   6. Simplified 65.9%

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

   if 1.8e-126 < y

   1. Initial program 76.8%

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

      1. associate-*l* 76.8%

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

      2. +-commutative 76.8%

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

      3. +-commutative 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-*l* 76.8%

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

      6. *-commutative 76.8%

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

      7. associate-*r/ 86.2%

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

      8. *-commutative 86.2%

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

      9. distribute-rgt1-in 75.4%

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

      10. fma-def 86.2%

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

      11. +-commutative 86.2%

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

      12. +-commutative 86.2%

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

      13. cube-unmult 86.3%

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

      14. +-commutative 86.3%

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

   3. Simplified 86.3%

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

      1. associate-*r/ 76.8%

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

      2. *-commutative 76.8%

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

      3. fma-udef 70.1%

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

      4. cube-mult 70.1%

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

      5. distribute-rgt1-in 76.8%

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

      6. *-commutative 76.8%

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

      7. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 76.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 93.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 93.8%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      12. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      13. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      15. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   6. Taylor expanded in x around 0 55.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   7. Step-by-step derivation

      1. +-commutative 55.7%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   8. Simplified 55.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 14: 61.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-126) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		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 (y <= 1.8d-126) 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 (y <= 1.8e-126) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-126:
		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 (y <= 1.8e-126)
		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 (y <= 1.8e-126)
		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[y, 1.8e-126], 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 y < 1.8e-126

   1. Initial program 70.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.4%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.4%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.4%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 63.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.8e-126 < y

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 76.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 93.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 93.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 93.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 93.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 55.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 55.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 55.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 15: 61.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-126) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y (+ x 1.0))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / (x + 1.0)) / (y + x);
	} 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 (y <= 1.8d-126) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    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 (y <= 1.8e-126) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + (x + 1.0))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-126:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + (x + 1.0))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-126)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	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 (y <= 1.8e-126)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + (x + 1.0))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-126], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $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 y < 1.8e-126

   1. Initial program 70.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 70.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 70.1%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 70.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 84.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 84.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 65.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 84.1%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 84.1%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 70.0%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 70.0%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 56.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 56.7%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 70.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 70.1%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 87.4%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 87.4%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      12. associate-*r/ 99.9%

          \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      13. associate-+r+ 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      15. associate-+l+ 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   6. Taylor expanded in y around 0 66.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   7. Step-by-step derivation

      1. +-commutative 66.3%

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   8. Simplified 66.3%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   if 1.8e-126 < y

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)\right)} \]

      6. associate-*l* 76.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      7. times-frac 93.8%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 93.8%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 93.8%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. associate-+l+ 93.8%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around inf 55.6%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{x}{y + \left(x + 1\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 55.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + \left(x + 1\right)}}{y}} \]

      2. *-un-lft-identity 55.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + \left(x + 1\right)}}}{y} \]

   6. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + \left(x + 1\right)}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y}\\ \end{array} \]

Alternative 16: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6200000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6200000000.0) (/ (/ y x) x) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6200000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6200000000.0d0)) then
        tmp = (y / x) / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6200000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6200000000.0:
		tmp = (y / x) / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6200000000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6200000000.0)
		tmp = (y / x) / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6200000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e9

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      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. Taylor expanded in x around inf 76.8%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 76.8%

         \[\leadsto \frac{\color{blue}{1 \cdot y}}{{x}^{2}} \]

      2. unpow2 76.8%

         \[\leadsto \frac{1 \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 83.4%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x}} \]

   6. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x}} \]
   7. Step-by-step derivation

      1. associate-*l/ 83.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x}}{x}} \]

      2. *-lft-identity 83.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   8. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -6.2e9 < x

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 50.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6200000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 17: 61.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-126) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-126) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-126:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-126)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-126)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-126], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8e-126

   1. Initial program 70.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.4%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 87.4%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 87.4%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in y around 0 63.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 65.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 65.9%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if 1.8e-126 < y

   1. Initial program 76.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 86.2%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 86.2%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 75.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 86.2%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 86.2%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 86.2%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 86.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 86.3%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 76.8%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 76.8%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 70.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 70.1%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 76.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 76.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 93.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 93.8%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      12. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      13. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      14. +-commutative 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      15. associate-+l+ 99.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]
   6. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{x + y}}{x + y} \]

      2. frac-times 99.6%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}}{x + y} \]

      3. *-un-lft-identity 99.6%

         \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}{x + y} \]

      4. associate-+r+ 99.6%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(x + y\right) + 1}}{y} \cdot \left(x + y\right)}}{x + y} \]

      5. +-commutative 99.6%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(y + x\right)} + 1}{y} \cdot \left(x + y\right)}}{x + y} \]

      6. associate-+r+ 99.6%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + \left(x + 1\right)}}{y} \cdot \left(x + y\right)}}{x + y} \]

      7. +-commutative 99.6%

         \[\leadsto \frac{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}}}{x + y} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}}}{x + y} \]
   8. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(x + y\right)}}}{x + y} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}}}{x + y} \]

      3. clear-num 99.6%

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \color{blue}{\frac{1}{\frac{y}{y + \left(x + 1\right)}}}}}{x + y} \]

      4. associate-+r+ 99.6%

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \frac{1}{\frac{y}{\color{blue}{\left(y + x\right) + 1}}}}}{x + y} \]

      5. +-commutative 99.6%

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \frac{1}{\frac{y}{\color{blue}{\left(x + y\right)} + 1}}}}{x + y} \]

      6. associate-+r+ 99.6%

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \frac{1}{\frac{y}{\color{blue}{x + \left(y + 1\right)}}}}}{x + y} \]

      7. *-rgt-identity 99.6%

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \frac{1}{\color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot 1}}}}{x + y} \]

      8. div-inv 99.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)} \cdot 1}}}}{x + y} \]

      9. *-rgt-identity 99.6%

         \[\leadsto \frac{\frac{x}{\frac{x + y}{\color{blue}{\frac{y}{x + \left(y + 1\right)}}}}}{x + y} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{x}{\color{blue}{\frac{x + y}{\frac{y}{x + \left(y + 1\right)}}}}}{x + y} \]

   10. Taylor expanded in x around 0 55.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   11. Step-by-step derivation

       1. associate-/r* 55.2%

          \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   12. Simplified 55.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18: 26.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2600000000:\\ \;\;\;\;\frac{0.5}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2600000000.0) (/ 0.5 (+ y x)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2600000000.0) {
		tmp = 0.5 / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2600000000.0d0)) then
        tmp = 0.5d0 / (y + x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2600000000.0) {
		tmp = 0.5 / (y + x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2600000000.0:
		tmp = 0.5 / (y + x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2600000000.0)
		tmp = Float64(0.5 / Float64(y + x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2600000000.0)
		tmp = 0.5 / (y + x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2600000000.0], N[(0.5 / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6e9

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      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. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 38.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 38.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 63.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 89.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 89.6%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      12. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      13. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      15. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]
   6. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{x + y}}{x + y} \]

      2. frac-times 98.9%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}}{x + y} \]

      3. *-un-lft-identity 98.9%

         \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}{x + y} \]

      4. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(x + y\right) + 1}}{y} \cdot \left(x + y\right)}}{x + y} \]

      5. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(y + x\right)} + 1}{y} \cdot \left(x + y\right)}}{x + y} \]

      6. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + \left(x + 1\right)}}{y} \cdot \left(x + y\right)}}{x + y} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}}}{x + y} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}}}{x + y} \]

   8. Taylor expanded in y around -inf 20.0%

      \[\leadsto \frac{\frac{x}{\color{blue}{y + -1 \cdot \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]
   9. Step-by-step derivation

      1. mul-1-neg 20.0%

         \[\leadsto \frac{\frac{x}{y + \color{blue}{\left(-\left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)\right)}}}{x + y} \]

      2. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]

      3. neg-mul-1 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(1 + x\right)\right)}}{x + y} \]

      4. +-commutative 20.0%

         \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) + \left(-x\right)\right)}}}{x + y} \]

      5. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) - x\right)}}}{x + y} \]

      6. distribute-lft-in 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot x\right)} - x\right)}}{x + y} \]

      7. metadata-eval 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\left(\color{blue}{-1} + -1 \cdot x\right) - x\right)}}{x + y} \]

      8. neg-mul-1 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\left(-1 + \color{blue}{\left(-x\right)}\right) - x\right)}}{x + y} \]

      9. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 - x\right)} - x\right)}}{x + y} \]

   10. Simplified 20.0%

       \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(\left(-1 - x\right) - x\right)}}}{x + y} \]

   11. Taylor expanded in x around inf 6.8%

       \[\leadsto \frac{\color{blue}{0.5}}{x + y} \]

   if -2.6e9 < x

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 50.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2600000000:\\ \;\;\;\;\frac{0.5}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 19: 42.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2100000000.0) (/ y (* x x)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2100000000.0d0)) then
        tmp = y / (x * x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2100000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2100000000.0:
		tmp = y / (x * x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2100000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2100000000.0)
		tmp = y / (x * x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2100000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e9

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      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. Taylor expanded in x around inf 76.8%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 76.8%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   6. Applied egg-rr 76.8%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if -2.1e9 < x

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 50.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2100000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 20: 43.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2100000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2100000000.0) (/ (/ y x) x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2100000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2100000000.0d0)) then
        tmp = (y / x) / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2100000000.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2100000000.0:
		tmp = (y / x) / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2100000000.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2100000000.0)
		tmp = (y / x) / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2100000000.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e9

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      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. Taylor expanded in x around inf 76.8%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 76.8%

         \[\leadsto \frac{\color{blue}{1 \cdot y}}{{x}^{2}} \]

      2. unpow2 76.8%

         \[\leadsto \frac{1 \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 83.4%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x}} \]

   6. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{y}{x}} \]
   7. Step-by-step derivation

      1. associate-*l/ 83.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x}}{x}} \]

      2. *-lft-identity 83.4%

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   8. Simplified 83.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -2.1e9 < x

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 50.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2100000000:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 21: 26.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7500000000.0) (/ 0.5 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7500000000.0) {
		tmp = 0.5 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7500000000.0d0)) then
        tmp = 0.5d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7500000000.0) {
		tmp = 0.5 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7500000000.0:
		tmp = 0.5 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7500000000.0)
		tmp = Float64(0.5 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7500000000.0)
		tmp = 0.5 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7500000000.0], N[(0.5 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e9

   1. Initial program 63.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      2. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      3. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

      4. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

      5. associate-*l* 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

      7. associate-*r/ 83.1%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 40.1%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      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. Step-by-step derivation

      1. associate-*r/ 63.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. *-commutative 63.7%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      3. fma-udef 38.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      4. cube-mult 38.5%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. distribute-rgt1-in 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      6. *-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. +-commutative 63.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      8. associate-+r+ 63.7%

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

      9. frac-times 89.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

      10. *-commutative 89.6%

          \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      11. associate-/r* 99.7%

          \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      12. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      13. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      15. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]
   6. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{x + y}}{x + y} \]

      2. frac-times 98.9%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}}{x + y} \]

      3. *-un-lft-identity 98.9%

         \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}{x + y} \]

      4. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(x + y\right) + 1}}{y} \cdot \left(x + y\right)}}{x + y} \]

      5. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(y + x\right)} + 1}{y} \cdot \left(x + y\right)}}{x + y} \]

      6. associate-+r+ 98.9%

         \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + \left(x + 1\right)}}{y} \cdot \left(x + y\right)}}{x + y} \]

      7. +-commutative 98.9%

         \[\leadsto \frac{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}}}{x + y} \]

   7. Applied egg-rr 98.9%

      \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}}}{x + y} \]

   8. Taylor expanded in y around -inf 20.0%

      \[\leadsto \frac{\frac{x}{\color{blue}{y + -1 \cdot \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]
   9. Step-by-step derivation

      1. mul-1-neg 20.0%

         \[\leadsto \frac{\frac{x}{y + \color{blue}{\left(-\left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)\right)}}}{x + y} \]

      2. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]

      3. neg-mul-1 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(1 + x\right)\right)}}{x + y} \]

      4. +-commutative 20.0%

         \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) + \left(-x\right)\right)}}}{x + y} \]

      5. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) - x\right)}}}{x + y} \]

      6. distribute-lft-in 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot x\right)} - x\right)}}{x + y} \]

      7. metadata-eval 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\left(\color{blue}{-1} + -1 \cdot x\right) - x\right)}}{x + y} \]

      8. neg-mul-1 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\left(-1 + \color{blue}{\left(-x\right)}\right) - x\right)}}{x + y} \]

      9. unsub-neg 20.0%

         \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 - x\right)} - x\right)}}{x + y} \]

   10. Simplified 20.0%

       \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(\left(-1 - x\right) - x\right)}}}{x + y} \]

   11. Taylor expanded in x around inf 6.2%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   if -7.5e9 < x

   1. Initial program 74.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. +-commutative 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(y + x\right)} + 1} \]

      3. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + \left(x + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   4. Taylor expanded in x around 0 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 50.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]

   7. Taylor expanded in y around 0 29.0%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000000:\\ \;\;\;\;\frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 22: 4.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 x))

C

double code(double x, double y) {
	return 0.5 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / x
end function

Java

public static double code(double x, double y) {
	return 0.5 / x;
}

Python

def code(x, y):
	return 0.5 / x

Julia

function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-*l* 72.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   2. +-commutative 72.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   3. +-commutative 72.0%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)\right)} \]

   4. +-commutative 72.0%

      \[\leadsto \frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)\right)} \]

   5. associate-*l* 72.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   6. *-commutative 72.0%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   7. associate-*r/ 84.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   8. *-commutative 84.7%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   9. distribute-rgt1-in 68.3%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   10. fma-def 84.7%

       \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   11. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   12. +-commutative 84.7%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   13. cube-unmult 84.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   14. +-commutative 84.8%

       \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 84.8%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 72.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. *-commutative 72.0%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   3. fma-udef 60.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   4. cube-mult 60.6%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. distribute-rgt1-in 72.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   6. *-commutative 72.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. +-commutative 72.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   8. associate-+r+ 72.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(x + 1\right)\right)}} \]

   9. frac-times 89.3%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + \left(x + 1\right)}} \]

   10. *-commutative 89.3%

       \[\leadsto \color{blue}{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   11. associate-/r* 99.7%

       \[\leadsto \frac{y}{y + \left(x + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   12. associate-*r/ 99.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   13. associate-+r+ 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   14. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   15. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]
6. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{x + y}}{x + y} \]

   2. frac-times 99.0%

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}}{x + y} \]

   3. *-un-lft-identity 99.0%

      \[\leadsto \frac{\frac{\color{blue}{x}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}}{x + y} \]

   4. associate-+r+ 99.0%

      \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(x + y\right) + 1}}{y} \cdot \left(x + y\right)}}{x + y} \]

   5. +-commutative 99.0%

      \[\leadsto \frac{\frac{x}{\frac{\color{blue}{\left(y + x\right)} + 1}{y} \cdot \left(x + y\right)}}{x + y} \]

   6. associate-+r+ 99.0%

      \[\leadsto \frac{\frac{x}{\frac{\color{blue}{y + \left(x + 1\right)}}{y} \cdot \left(x + y\right)}}{x + y} \]

   7. +-commutative 99.0%

      \[\leadsto \frac{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \color{blue}{\left(y + x\right)}}}{x + y} \]

7. Applied egg-rr 99.0%

   \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y + \left(x + 1\right)}{y} \cdot \left(y + x\right)}}}{x + y} \]

8. Taylor expanded in y around -inf 42.5%

   \[\leadsto \frac{\frac{x}{\color{blue}{y + -1 \cdot \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]
9. Step-by-step derivation

   1. mul-1-neg 42.5%

      \[\leadsto \frac{\frac{x}{y + \color{blue}{\left(-\left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)\right)}}}{x + y} \]

   2. unsub-neg 42.5%

      \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(-1 \cdot x + -1 \cdot \left(1 + x\right)\right)}}}{x + y} \]

   3. neg-mul-1 42.5%

      \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-x\right)} + -1 \cdot \left(1 + x\right)\right)}}{x + y} \]

   4. +-commutative 42.5%

      \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) + \left(-x\right)\right)}}}{x + y} \]

   5. unsub-neg 42.5%

      \[\leadsto \frac{\frac{x}{y - \color{blue}{\left(-1 \cdot \left(1 + x\right) - x\right)}}}{x + y} \]

   6. distribute-lft-in 42.5%

      \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot x\right)} - x\right)}}{x + y} \]

   7. metadata-eval 42.5%

      \[\leadsto \frac{\frac{x}{y - \left(\left(\color{blue}{-1} + -1 \cdot x\right) - x\right)}}{x + y} \]

   8. neg-mul-1 42.5%

      \[\leadsto \frac{\frac{x}{y - \left(\left(-1 + \color{blue}{\left(-x\right)}\right) - x\right)}}{x + y} \]

   9. unsub-neg 42.5%

      \[\leadsto \frac{\frac{x}{y - \left(\color{blue}{\left(-1 - x\right)} - x\right)}}{x + y} \]

10. Simplified 42.5%

    \[\leadsto \frac{\frac{x}{\color{blue}{y - \left(\left(-1 - x\right) - x\right)}}}{x + y} \]

11. Taylor expanded in x around inf 4.4%

    \[\leadsto \color{blue}{\frac{0.5}{x}} \]

12. Final simplification 4.4%

    \[\leadsto \frac{0.5}{x} \]

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 2023308 
(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))))