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

Percentage Accurate: 69.1% → 99.8%

Time: 17.5s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. associate-+r+ 68.6%

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

   2. *-commutative 68.6%

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

   3. frac-times 89.4%

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

   4. associate-*l/ 82.9%

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

   5. times-frac 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-23)
   (* (/ y (+ y x)) (/ (/ x (+ x 1.0)) (+ y x)))
   (if (<= y 1.7e+122)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x (+ y 1.0))))
     (/ (/ x (+ y x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-23) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else if (y <= 1.7e+122) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-23) then
        tmp = (y / (y + x)) * ((x / (x + 1.0d0)) / (y + x))
    else if (y <= 1.7d+122) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0d0)))
    else
        tmp = (x / (y + x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-23) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else if (y <= 1.7e+122) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-23:
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x))
	elif y <= 1.7e+122:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)))
	else:
		tmp = (x / (y + x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-23)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(x + 1.0)) / Float64(y + x)));
	elseif (y <= 1.7e+122)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-23)
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	elseif (y <= 1.7e+122)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	else
		tmp = (x / (y + x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-23], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+122], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1999999999999999e-23

   1. Initial program 67.5%

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

      1. associate-+r+ 67.5%

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

      2. *-commutative 67.5%

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

      3. frac-times 88.6%

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

      4. associate-*l/ 80.6%

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

      5. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 82.5%

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

      1. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   if 2.1999999999999999e-23 < y < 1.7e122

   1. Initial program 81.6%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   if 1.7e122 < y

   1. Initial program 67.4%

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

      1. associate-/r* 70.7%

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

      2. +-commutative 70.7%

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

      3. +-commutative 70.7%

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

      4. +-commutative 70.7%

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

      5. associate-/r* 67.4%

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

      6. associate-*l/ 84.2%

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

      7. *-commutative 84.2%

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

      8. *-commutative 84.2%

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

      9. distribute-rgt1-in 84.0%

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

      10. fma-def 84.2%

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

      11. +-commutative 84.2%

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

      12. +-commutative 84.2%

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

      13. cube-unmult 84.2%

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

      14. +-commutative 84.2%

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

   3. Simplified 84.2%

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

      1. associate-*r/ 67.4%

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

      2. fma-udef 67.4%

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

      3. cube-mult 67.4%

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

      4. distribute-rgt1-in 67.4%

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

      5. associate-+r+ 67.4%

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

      6. *-commutative 67.4%

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

      7. *-commutative 67.4%

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

      8. frac-times 87.5%

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

      9. associate-/r* 99.9%

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

      10. clear-num 99.9%

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

      11. frac-times 100.0%

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

      12. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 93.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y}\\ \end{array} \]

Alternative 3: 67.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e+167)
   (/ (/ y (+ x (+ y 1.0))) x)
   (if (<= x -4.1e-156)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x 1.0)))
     (/ (/ x (+ y x)) (+ y (+ x (- x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+167) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (x <= -4.1e-156) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.1d+167)) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else if (x <= (-4.1d-156)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0d0))
    else
        tmp = (x / (y + x)) / (y + (x + (x - (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+167) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (x <= -4.1e-156) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e+167:
		tmp = (y / (x + (y + 1.0))) / x
	elif x <= -4.1e-156:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0))
	else:
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e+167)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	elseif (x <= -4.1e-156)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e+167)
		tmp = (y / (x + (y + 1.0))) / x;
	elseif (x <= -4.1e-156)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	else
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e+167], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -4.1e-156], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0999999999999999e167

   1. Initial program 53.3%

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

      1. times-frac 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around inf 87.7%

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

      1. associate-*l/ 87.7%

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

      2. *-un-lft-identity 87.7%

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

      3. +-commutative 87.7%

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

   6. Applied egg-rr 87.7%

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

   if -2.0999999999999999e167 < x < -4.1000000000000002e-156

   1. Initial program 76.4%

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

      1. times-frac 98.5%

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

      2. associate-+l+ 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in y around 0 81.3%

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

      1. +-commutative 81.3%

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

   6. Simplified 81.3%

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

   if -4.1000000000000002e-156 < x

   1. Initial program 68.0%

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

      1. associate-/r* 70.3%

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

      2. +-commutative 70.3%

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

      3. +-commutative 70.3%

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

      4. +-commutative 70.3%

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

      5. associate-/r* 68.0%

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

      6. associate-*l/ 82.2%

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

      7. *-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 75.1%

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

      10. fma-def 82.2%

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

      11. +-commutative 82.2%

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

      12. +-commutative 82.2%

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

      13. cube-unmult 82.2%

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

      14. +-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 68.1%

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

      2. fma-udef 61.0%

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

      3. cube-mult 60.9%

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

      4. distribute-rgt1-in 68.0%

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

      5. associate-+r+ 68.0%

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

      6. *-commutative 68.0%

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

      7. *-commutative 68.0%

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

      8. frac-times 85.9%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 98.9%

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

      12. *-un-lft-identity 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in y around -inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. unsub-neg 55.5%

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

      3. neg-mul-1 55.5%

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

      4. +-commutative 55.5%

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

      5. unsub-neg 55.5%

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

      6. distribute-lft-in 55.5%

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

      7. metadata-eval 55.5%

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

      8. neg-mul-1 55.5%

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

      9. unsub-neg 55.5%

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

   8. Simplified 55.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \]

Alternative 4: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+46)
   (* (/ (/ x (+ x (+ y 1.0))) (+ y x)) (/ y x))
   (if (<= x -6e-156)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x 1.0)))
     (/ (/ x (+ y x)) (+ y (+ x (- x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+46) {
		tmp = ((x / (x + (y + 1.0))) / (y + x)) * (y / x);
	} else if (x <= -6e-156) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+46)) then
        tmp = ((x / (x + (y + 1.0d0))) / (y + x)) * (y / x)
    else if (x <= (-6d-156)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0d0))
    else
        tmp = (x / (y + x)) / (y + (x + (x - (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+46) {
		tmp = ((x / (x + (y + 1.0))) / (y + x)) * (y / x);
	} else if (x <= -6e-156) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+46:
		tmp = ((x / (x + (y + 1.0))) / (y + x)) * (y / x)
	elif x <= -6e-156:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0))
	else:
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+46)
		tmp = Float64(Float64(Float64(x / Float64(x + Float64(y + 1.0))) / Float64(y + x)) * Float64(y / x));
	elseif (x <= -6e-156)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+46)
		tmp = ((x / (x + (y + 1.0))) / (y + x)) * (y / x);
	elseif (x <= -6e-156)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	else
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+46], N[(N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6e-156], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000002e46

   1. Initial program 57.1%

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

      1. associate-+r+ 57.1%

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

      2. *-commutative 57.1%

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

      3. frac-times 90.7%

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

      4. associate-*l/ 90.7%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 91.6%

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

   if -5.0000000000000002e46 < x < -6e-156

   1. Initial program 85.1%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 74.8%

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

      1. +-commutative 74.8%

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

   6. Simplified 74.8%

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

   if -6e-156 < x

   1. Initial program 68.0%

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

      1. associate-/r* 70.3%

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

      2. +-commutative 70.3%

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

      3. +-commutative 70.3%

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

      4. +-commutative 70.3%

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

      5. associate-/r* 68.0%

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

      6. associate-*l/ 82.2%

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

      7. *-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 75.1%

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

      10. fma-def 82.2%

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

      11. +-commutative 82.2%

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

      12. +-commutative 82.2%

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

      13. cube-unmult 82.2%

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

      14. +-commutative 82.2%

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

   3. Simplified 82.2%

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

      1. associate-*r/ 68.1%

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

      2. fma-udef 61.0%

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

      3. cube-mult 60.9%

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

      4. distribute-rgt1-in 68.0%

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

      5. associate-+r+ 68.0%

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

      6. *-commutative 68.0%

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

      7. *-commutative 68.0%

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

      8. frac-times 85.9%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 98.9%

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

      12. *-un-lft-identity 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in y around -inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. unsub-neg 55.5%

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

      3. neg-mul-1 55.5%

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

      4. +-commutative 55.5%

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

      5. unsub-neg 55.5%

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

      6. distribute-lft-in 55.5%

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

      7. metadata-eval 55.5%

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

      8. neg-mul-1 55.5%

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

      9. unsub-neg 55.5%

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

   8. Simplified 55.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \]

Alternative 5: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{x}{\left(x + \frac{x}{y}\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e-283)
   (/ (/ y (+ x (+ y 1.0))) x)
   (if (<= y 1.25e-28)
     (/ (/ x (+ (+ x (/ x y)) 1.0)) (+ y x))
     (/ (/ x (+ y x)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e-283) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (y <= 1.25e-28) {
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d-283)) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else if (y <= 1.25d-28) then
        tmp = (x / ((x + (x / y)) + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e-283) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (y <= 1.25e-28) {
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e-283:
		tmp = (y / (x + (y + 1.0))) / x
	elif y <= 1.25e-28:
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x)
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e-283)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	elseif (y <= 1.25e-28)
		tmp = Float64(Float64(x / Float64(Float64(x + Float64(x / y)) + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e-283)
		tmp = (y / (x + (y + 1.0))) / x;
	elseif (y <= 1.25e-28)
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e-283], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.25e-28], N[(N[(x / N[(N[(x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999989e-283

   1. Initial program 65.1%

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

      1. times-frac 91.3%

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

      2. associate-+l+ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around inf 51.4%

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

      1. associate-*l/ 51.5%

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

      2. *-un-lft-identity 51.5%

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

      3. +-commutative 51.5%

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

   6. Applied egg-rr 51.5%

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

   if -1.99999999999999989e-283 < y < 1.25e-28

   1. Initial program 70.7%

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

      1. associate-+r+ 70.7%

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

      2. *-commutative 70.7%

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

      3. frac-times 83.8%

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

      4. associate-*l/ 72.8%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.7%

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

      3. frac-times 98.0%

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

      4. *-un-lft-identity 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. +-commutative 98.0%

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

   8. Applied egg-rr 98.0%

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

   9. Taylor expanded in x around 0 80.8%

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

       1. distribute-lft-in 80.8%

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

       2. *-rgt-identity 80.8%

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

       3. /-rgt-identity 80.8%

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

       4. times-frac 80.8%

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

       5. *-rgt-identity 80.8%

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

       6. *-lft-identity 80.8%

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

   11. Simplified 80.8%

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

   if 1.25e-28 < y

   1. Initial program 74.6%

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

      1. associate-/r* 80.0%

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

      2. +-commutative 80.0%

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

      3. +-commutative 80.0%

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

      4. +-commutative 80.0%

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

      5. associate-/r* 74.6%

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

      6. associate-*l/ 86.4%

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

      7. *-commutative 86.4%

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

      8. *-commutative 86.4%

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

      9. distribute-rgt1-in 82.3%

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

      10. fma-def 86.4%

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

      11. +-commutative 86.4%

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

      12. +-commutative 86.4%

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

      13. cube-unmult 86.5%

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

      14. +-commutative 86.5%

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

   3. Simplified 86.5%

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

      1. associate-*r/ 74.6%

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

      2. fma-udef 72.6%

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

      3. cube-mult 72.6%

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

      4. distribute-rgt1-in 74.6%

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

      5. associate-+r+ 74.6%

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

      6. *-commutative 74.6%

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

      7. *-commutative 74.6%

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

      8. frac-times 93.1%

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

      9. associate-/r* 99.9%

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

      10. clear-num 99.9%

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

      11. frac-times 99.9%

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

      12. *-un-lft-identity 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 76.6%

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

      1. +-commutative 76.6%

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

   8. Simplified 76.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{x}{\left(x + \frac{x}{y}\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 6: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x}{\left(x + \frac{x}{y}\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y - \left(\left(-1 - x\right) - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4e-283)
   (/ (/ y (+ x (+ y 1.0))) x)
   (if (<= y 3.4e-14)
     (/ (/ x (+ (+ x (/ x y)) 1.0)) (+ y x))
     (/ (/ x (+ y x)) (- y (- (- -1.0 x) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-283) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (y <= 3.4e-14) {
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) / (y - ((-1.0 - x) - 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 <= (-1.4d-283)) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else if (y <= 3.4d-14) then
        tmp = (x / ((x + (x / y)) + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + x)) / (y - (((-1.0d0) - x) - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4e-283) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else if (y <= 3.4e-14) {
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + x)) / (y - ((-1.0 - x) - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4e-283:
		tmp = (y / (x + (y + 1.0))) / x
	elif y <= 3.4e-14:
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x)
	else:
		tmp = (x / (y + x)) / (y - ((-1.0 - x) - x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4e-283)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	elseif (y <= 3.4e-14)
		tmp = Float64(Float64(x / Float64(Float64(x + Float64(x / y)) + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y - Float64(Float64(-1.0 - x) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4e-283)
		tmp = (y / (x + (y + 1.0))) / x;
	elseif (y <= 3.4e-14)
		tmp = (x / ((x + (x / y)) + 1.0)) / (y + x);
	else
		tmp = (x / (y + x)) / (y - ((-1.0 - x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4e-283], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.4e-14], N[(N[(x / N[(N[(x + N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y - N[(N[(-1.0 - x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999e-283

   1. Initial program 65.1%

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

      1. times-frac 91.3%

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

      2. associate-+l+ 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in x around inf 51.4%

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

      1. associate-*l/ 51.5%

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

      2. *-un-lft-identity 51.5%

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

      3. +-commutative 51.5%

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

   6. Applied egg-rr 51.5%

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

   if -1.3999999999999999e-283 < y < 3.40000000000000003e-14

   1. Initial program 71.8%

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

      1. associate-+r+ 71.8%

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

      2. *-commutative 71.8%

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

      3. frac-times 84.4%

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

      4. associate-*l/ 73.8%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.7%

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

      3. frac-times 98.1%

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

      4. *-un-lft-identity 98.1%

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

      5. +-commutative 98.1%

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

      6. +-commutative 98.1%

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

      7. +-commutative 98.1%

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

   8. Applied egg-rr 98.1%

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

   9. Taylor expanded in x around 0 79.1%

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

       1. distribute-lft-in 79.1%

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

       2. *-rgt-identity 79.1%

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

       3. /-rgt-identity 79.1%

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

       4. times-frac 79.1%

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

       5. *-rgt-identity 79.1%

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

       6. *-lft-identity 79.1%

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

   11. Simplified 79.1%

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

   if 3.40000000000000003e-14 < y

   1. Initial program 73.1%

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

      1. associate-/r* 78.7%

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

      2. +-commutative 78.7%

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

      3. +-commutative 78.7%

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

      4. +-commutative 78.7%

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

      5. associate-/r* 73.1%

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

      6. associate-*l/ 85.6%

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

      7. *-commutative 85.6%

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

      8. *-commutative 85.6%

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

      9. distribute-rgt1-in 81.2%

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

      10. fma-def 85.6%

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

      11. +-commutative 85.6%

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

      12. +-commutative 85.6%

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

      13. cube-unmult 85.7%

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

      14. +-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. associate-*r/ 73.1%

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

      2. fma-udef 70.9%

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

      3. cube-mult 70.9%

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

      4. distribute-rgt1-in 73.1%

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

      5. associate-+r+ 73.1%

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

      6. *-commutative 73.1%

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

      7. *-commutative 73.1%

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

      8. frac-times 92.7%

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

      9. associate-/r* 99.9%

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

      10. clear-num 99.9%

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

      11. frac-times 100.0%

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

      12. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around -inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unsub-neg 79.8%

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

      3. neg-mul-1 79.8%

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

      4. +-commutative 79.8%

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

      5. unsub-neg 79.8%

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

      6. distribute-lft-in 79.8%

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

      7. metadata-eval 79.8%

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

      8. neg-mul-1 79.8%

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

      9. unsub-neg 79.8%

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

   8. Simplified 79.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-283}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x}{\left(x + \frac{x}{y}\right) + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y - \left(\left(-1 - x\right) - x\right)}\\ \end{array} \]

Alternative 7: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7e+15)
   (* (/ y (+ y x)) (/ (/ x (+ x 1.0)) (+ y x)))
   (/ (/ x (+ y x)) (+ y (+ x (- x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e+15) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7d+15) then
        tmp = (y / (y + x)) * ((x / (x + 1.0d0)) / (y + x))
    else
        tmp = (x / (y + x)) / (y + (x + (x - (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7e+15) {
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e+15:
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x))
	else:
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e+15)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(x + 1.0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + Float64(x - -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e+15)
		tmp = (y / (y + x)) * ((x / (x + 1.0)) / (y + x));
	else
		tmp = (x / (y + x)) / (y + (x + (x - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e+15], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e15

   1. Initial program 68.1%

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

      1. associate-+r+ 68.1%

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

      2. *-commutative 68.1%

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

      3. frac-times 88.8%

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

      4. associate-*l/ 80.9%

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

      5. times-frac 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in y around 0 82.3%

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

      1. +-commutative 82.3%

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

   6. Simplified 82.3%

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

   if 7e15 < 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-/r* 77.4%

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

      2. +-commutative 77.4%

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

      3. +-commutative 77.4%

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

      4. +-commutative 77.4%

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

      5. associate-/r* 71.2%

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

      6. associate-*l/ 84.6%

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

      7. *-commutative 84.6%

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

      8. *-commutative 84.6%

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

      9. distribute-rgt1-in 79.9%

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

      10. fma-def 84.6%

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

      11. +-commutative 84.6%

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

      12. +-commutative 84.6%

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

      13. cube-unmult 84.7%

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

      14. +-commutative 84.7%

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

   3. Simplified 84.7%

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

      1. associate-*r/ 71.2%

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

      2. fma-udef 69.0%

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

      3. cube-mult 69.0%

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

      4. distribute-rgt1-in 71.2%

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

      5. associate-+r+ 71.2%

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

      6. *-commutative 71.2%

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

      7. *-commutative 71.2%

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

      8. frac-times 92.2%

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

      9. associate-/r* 99.9%

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

      10. clear-num 99.9%

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

      11. frac-times 100.0%

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

      12. *-un-lft-identity 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around -inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. unsub-neg 82.8%

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

      3. neg-mul-1 82.8%

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

      4. +-commutative 82.8%

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

      5. unsub-neg 82.8%

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

      6. distribute-lft-in 82.8%

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

      7. metadata-eval 82.8%

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

      8. neg-mul-1 82.8%

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

      9. unsub-neg 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x - -1\right)\right)}\\ \end{array} \]

Alternative 8: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -1.0)
     (/ y (* x x))
     (if (<= x -4.6e-91)
       (- (/ y x) y)
       (if (<= x -1.85e-138)
         t_0
         (if (<= x -7.5e-156)
           (/ y x)
           (if (<= x 1e-104) (/ x (+ y x)) t_0)))))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.6e-91) {
		tmp = (y / x) - y;
	} else if (x <= -1.85e-138) {
		tmp = t_0;
	} else if (x <= -7.5e-156) {
		tmp = y / x;
	} else if (x <= 1e-104) {
		tmp = x / (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-4.6d-91)) then
        tmp = (y / x) - y
    else if (x <= (-1.85d-138)) then
        tmp = t_0
    else if (x <= (-7.5d-156)) then
        tmp = y / x
    else if (x <= 1d-104) then
        tmp = x / (y + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.6e-91) {
		tmp = (y / x) - y;
	} else if (x <= -1.85e-138) {
		tmp = t_0;
	} else if (x <= -7.5e-156) {
		tmp = y / x;
	} else if (x <= 1e-104) {
		tmp = x / (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -4.6e-91:
		tmp = (y / x) - y
	elif x <= -1.85e-138:
		tmp = t_0
	elif x <= -7.5e-156:
		tmp = y / x
	elif x <= 1e-104:
		tmp = x / (y + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -4.6e-91)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= -1.85e-138)
		tmp = t_0;
	elseif (x <= -7.5e-156)
		tmp = Float64(y / x);
	elseif (x <= 1e-104)
		tmp = Float64(x / Float64(y + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -4.6e-91)
		tmp = (y / x) - y;
	elseif (x <= -1.85e-138)
		tmp = t_0;
	elseif (x <= -7.5e-156)
		tmp = y / x;
	elseif (x <= 1e-104)
		tmp = x / (y + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.6e-91], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, -1.85e-138], t$95$0, If[LessEqual[x, -7.5e-156], N[(y / x), $MachinePrecision], If[LessEqual[x, 1e-104], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1

   1. Initial program 61.8%

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

      1. associate-/r* 72.0%

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

      2. +-commutative 72.0%

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

      3. +-commutative 72.0%

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

      4. +-commutative 72.0%

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

      5. associate-/r* 61.8%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 34.3%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x around inf 77.3%

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

      1. unpow2 77.3%

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

   6. Simplified 77.3%

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

   if -1 < x < -4.59999999999999991e-91

   1. Initial program 86.6%

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

      1. 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 43.9%

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

   5. Taylor expanded in x around 0 43.9%

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

      1. neg-mul-1 43.9%

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

      2. +-commutative 43.9%

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

      3. unsub-neg 43.9%

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

   7. Simplified 43.9%

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

   if -4.59999999999999991e-91 < x < -1.84999999999999995e-138 or 9.99999999999999927e-105 < x

   1. Initial program 74.3%

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

      1. associate-/r* 78.4%

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

      2. +-commutative 78.4%

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

      3. +-commutative 78.4%

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

      4. +-commutative 78.4%

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

      5. associate-/r* 74.3%

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

      6. associate-*l/ 87.3%

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

      7. *-commutative 87.3%

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

      8. *-commutative 87.3%

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

      9. distribute-rgt1-in 84.9%

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

      10. fma-def 87.3%

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

      11. +-commutative 87.3%

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

      12. +-commutative 87.3%

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

      13. cube-unmult 87.3%

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

      14. +-commutative 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in y around inf 25.8%

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

      1. unpow2 25.8%

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

   6. Simplified 25.8%

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

   if -1.84999999999999995e-138 < x < -7.49999999999999959e-156

   1. Initial program 52.2%

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

      1. times-frac 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 51.4%

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

   5. Taylor expanded in x around 0 51.4%

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

   if -7.49999999999999959e-156 < x < 9.99999999999999927e-105

   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-+r+ 64.2%

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

      2. *-commutative 64.2%

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

      3. frac-times 79.8%

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

      4. associate-*l/ 64.2%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 82.6%

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

      1. +-commutative 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*r/ 82.6%

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

      2. clear-num 82.6%

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

      3. frac-times 82.6%

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

      4. *-un-lft-identity 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

   9. Taylor expanded in x around 0 65.8%

      \[\leadsto \frac{\color{blue}{x}}{y + x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 9: 53.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -9e-91)
     (- (/ y x) y)
     (if (<= x -8e-140)
       (/ x (* y y))
       (if (<= x -8e-155)
         (/ y x)
         (if (<= x 6.6e-104) (/ x (+ y x)) (/ (/ x y) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -9e-91) {
		tmp = (y / x) - y;
	} else if (x <= -8e-140) {
		tmp = x / (y * y);
	} else if (x <= -8e-155) {
		tmp = y / x;
	} else if (x <= 6.6e-104) {
		tmp = x / (y + x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-9d-91)) then
        tmp = (y / x) - y
    else if (x <= (-8d-140)) then
        tmp = x / (y * y)
    else if (x <= (-8d-155)) then
        tmp = y / x
    else if (x <= 6.6d-104) then
        tmp = x / (y + x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -9e-91) {
		tmp = (y / x) - y;
	} else if (x <= -8e-140) {
		tmp = x / (y * y);
	} else if (x <= -8e-155) {
		tmp = y / x;
	} else if (x <= 6.6e-104) {
		tmp = x / (y + x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -9e-91:
		tmp = (y / x) - y
	elif x <= -8e-140:
		tmp = x / (y * y)
	elif x <= -8e-155:
		tmp = y / x
	elif x <= 6.6e-104:
		tmp = x / (y + x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -9e-91)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= -8e-140)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -8e-155)
		tmp = Float64(y / x);
	elseif (x <= 6.6e-104)
		tmp = Float64(x / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -9e-91)
		tmp = (y / x) - y;
	elseif (x <= -8e-140)
		tmp = x / (y * y);
	elseif (x <= -8e-155)
		tmp = y / x;
	elseif (x <= 6.6e-104)
		tmp = x / (y + x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-91], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, -8e-140], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-155], N[(y / x), $MachinePrecision], If[LessEqual[x, 6.6e-104], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1

   1. Initial program 61.8%

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

      1. associate-/r* 72.0%

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

      2. +-commutative 72.0%

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

      3. +-commutative 72.0%

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

      4. +-commutative 72.0%

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

      5. associate-/r* 61.8%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 34.3%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x around inf 77.3%

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

      1. unpow2 77.3%

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

   6. Simplified 77.3%

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

   if -1 < x < -8.99999999999999952e-91

   1. Initial program 86.6%

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

      1. 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 43.9%

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

   5. Taylor expanded in x around 0 43.9%

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

      1. neg-mul-1 43.9%

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

      2. +-commutative 43.9%

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

      3. unsub-neg 43.9%

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

   7. Simplified 43.9%

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

   if -8.99999999999999952e-91 < x < -7.9999999999999999e-140

   1. Initial program 99.8%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-/r* 99.8%

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

      6. associate-*l/ 88.4%

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

      7. *-commutative 88.4%

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

      8. *-commutative 88.4%

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

      9. distribute-rgt1-in 88.4%

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

      10. fma-def 88.4%

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

      11. +-commutative 88.4%

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

      12. +-commutative 88.4%

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

      13. cube-unmult 88.4%

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

      14. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in y around inf 51.7%

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

      1. unpow2 51.7%

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

   6. Simplified 51.7%

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

   if -7.9999999999999999e-140 < x < -8.00000000000000011e-155

   1. Initial program 52.2%

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

      1. times-frac 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 51.4%

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

   5. Taylor expanded in x around 0 51.4%

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

   if -8.00000000000000011e-155 < x < 6.60000000000000004e-104

   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-+r+ 64.2%

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

      2. *-commutative 64.2%

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

      3. frac-times 79.8%

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

      4. associate-*l/ 64.2%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 82.6%

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

      1. +-commutative 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*r/ 82.6%

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

      2. clear-num 82.6%

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

      3. frac-times 82.6%

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

      4. *-un-lft-identity 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

   9. Taylor expanded in x around 0 65.8%

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

   if 6.60000000000000004e-104 < x

   1. Initial program 71.8%

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

      1. associate-/r* 76.3%

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

      2. +-commutative 76.3%

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

      3. +-commutative 76.3%

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

      4. +-commutative 76.3%

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

      5. associate-/r* 71.8%

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

      6. associate-*l/ 87.2%

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

      7. *-commutative 87.2%

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

      8. *-commutative 87.2%

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

      9. distribute-rgt1-in 84.6%

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

      10. fma-def 87.2%

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

      11. +-commutative 87.2%

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

      12. +-commutative 87.2%

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

      13. cube-unmult 87.2%

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

      14. +-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 71.8%

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

      2. fma-udef 69.3%

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

      3. cube-mult 69.2%

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

      4. distribute-rgt1-in 71.8%

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

      5. associate-+r+ 71.8%

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

      6. *-commutative 71.8%

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

      7. *-commutative 71.8%

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

      8. frac-times 91.8%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 98.1%

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

      12. *-un-lft-identity 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in y around inf 25.2%

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

   7. Taylor expanded in x around 0 24.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 6 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 10: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -5.2e-91)
     (- (/ y x) y)
     (if (<= x -1.25e-138)
       (/ x (* y y))
       (if (<= x -8e-155)
         (/ y x)
         (if (<= x 2.5e-104) (/ x (+ y x)) (/ (/ x y) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.2e-91) {
		tmp = (y / x) - y;
	} else if (x <= -1.25e-138) {
		tmp = x / (y * y);
	} else if (x <= -8e-155) {
		tmp = y / x;
	} else if (x <= 2.5e-104) {
		tmp = x / (y + x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-5.2d-91)) then
        tmp = (y / x) - y
    else if (x <= (-1.25d-138)) then
        tmp = x / (y * y)
    else if (x <= (-8d-155)) then
        tmp = y / x
    else if (x <= 2.5d-104) then
        tmp = x / (y + x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5.2e-91) {
		tmp = (y / x) - y;
	} else if (x <= -1.25e-138) {
		tmp = x / (y * y);
	} else if (x <= -8e-155) {
		tmp = y / x;
	} else if (x <= 2.5e-104) {
		tmp = x / (y + x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -5.2e-91:
		tmp = (y / x) - y
	elif x <= -1.25e-138:
		tmp = x / (y * y)
	elif x <= -8e-155:
		tmp = y / x
	elif x <= 2.5e-104:
		tmp = x / (y + x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -5.2e-91)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= -1.25e-138)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -8e-155)
		tmp = Float64(y / x);
	elseif (x <= 2.5e-104)
		tmp = Float64(x / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -5.2e-91)
		tmp = (y / x) - y;
	elseif (x <= -1.25e-138)
		tmp = x / (y * y);
	elseif (x <= -8e-155)
		tmp = y / x;
	elseif (x <= 2.5e-104)
		tmp = x / (y + x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-91], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, -1.25e-138], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-155], N[(y / x), $MachinePrecision], If[LessEqual[x, 2.5e-104], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1

   1. Initial program 61.8%

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

      1. times-frac 92.3%

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

      2. associate-+l+ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around inf 79.5%

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

   5. Taylor expanded in x around inf 78.1%

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

   if -1 < x < -5.20000000000000028e-91

   1. Initial program 86.6%

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

      1. 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 43.9%

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

   5. Taylor expanded in x around 0 43.9%

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

      1. neg-mul-1 43.9%

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

      2. +-commutative 43.9%

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

      3. unsub-neg 43.9%

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

   7. Simplified 43.9%

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

   if -5.20000000000000028e-91 < x < -1.24999999999999997e-138

   1. Initial program 99.8%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-/r* 99.8%

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

      6. associate-*l/ 88.4%

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

      7. *-commutative 88.4%

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

      8. *-commutative 88.4%

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

      9. distribute-rgt1-in 88.4%

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

      10. fma-def 88.4%

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

      11. +-commutative 88.4%

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

      12. +-commutative 88.4%

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

      13. cube-unmult 88.4%

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

      14. +-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in y around inf 51.7%

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

      1. unpow2 51.7%

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

   6. Simplified 51.7%

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

   if -1.24999999999999997e-138 < x < -8.00000000000000011e-155

   1. Initial program 52.2%

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

      1. times-frac 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 51.4%

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

   5. Taylor expanded in x around 0 51.4%

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

   if -8.00000000000000011e-155 < x < 2.49999999999999989e-104

   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-+r+ 64.2%

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

      2. *-commutative 64.2%

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

      3. frac-times 79.8%

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

      4. associate-*l/ 64.2%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 82.6%

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

      1. +-commutative 82.6%

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

   6. Simplified 82.6%

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

      1. associate-*r/ 82.6%

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

      2. clear-num 82.6%

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

      3. frac-times 82.6%

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

      4. *-un-lft-identity 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

      7. +-commutative 82.6%

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

   8. Applied egg-rr 82.6%

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

   9. Taylor expanded in x around 0 65.8%

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

   if 2.49999999999999989e-104 < x

   1. Initial program 71.8%

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

      1. associate-/r* 76.3%

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

      2. +-commutative 76.3%

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

      3. +-commutative 76.3%

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

      4. +-commutative 76.3%

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

      5. associate-/r* 71.8%

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

      6. associate-*l/ 87.2%

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

      7. *-commutative 87.2%

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

      8. *-commutative 87.2%

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

      9. distribute-rgt1-in 84.6%

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

      10. fma-def 87.2%

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

      11. +-commutative 87.2%

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

      12. +-commutative 87.2%

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

      13. cube-unmult 87.2%

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

      14. +-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r/ 71.8%

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

      2. fma-udef 69.3%

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

      3. cube-mult 69.2%

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

      4. distribute-rgt1-in 71.8%

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

      5. associate-+r+ 71.8%

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

      6. *-commutative 71.8%

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

      7. *-commutative 71.8%

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

      8. frac-times 91.8%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.7%

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

      11. frac-times 98.1%

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

      12. *-un-lft-identity 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in y around inf 25.2%

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

   7. Taylor expanded in x around 0 24.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]
3. Recombined 6 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-155}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 52.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-222}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 52000000000:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -8e+144)
     t_0
     (if (<= y 3.4e-222)
       (/ y x)
       (if (<= y 52000000000.0) (/ x (+ y x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -8e+144) {
		tmp = t_0;
	} else if (y <= 3.4e-222) {
		tmp = y / x;
	} else if (y <= 52000000000.0) {
		tmp = x / (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-8d+144)) then
        tmp = t_0
    else if (y <= 3.4d-222) then
        tmp = y / x
    else if (y <= 52000000000.0d0) then
        tmp = x / (y + x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -8e+144) {
		tmp = t_0;
	} else if (y <= 3.4e-222) {
		tmp = y / x;
	} else if (y <= 52000000000.0) {
		tmp = x / (y + x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -8e+144:
		tmp = t_0
	elif y <= 3.4e-222:
		tmp = y / x
	elif y <= 52000000000.0:
		tmp = x / (y + x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -8e+144)
		tmp = t_0;
	elseif (y <= 3.4e-222)
		tmp = Float64(y / x);
	elseif (y <= 52000000000.0)
		tmp = Float64(x / Float64(y + x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -8e+144)
		tmp = t_0;
	elseif (y <= 3.4e-222)
		tmp = y / x;
	elseif (y <= 52000000000.0)
		tmp = x / (y + x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+144], t$95$0, If[LessEqual[y, 3.4e-222], N[(y / x), $MachinePrecision], If[LessEqual[y, 52000000000.0], N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000019e144 or 5.2e10 < y

   1. Initial program 60.8%

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

      1. associate-/r* 64.3%

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

      2. +-commutative 64.3%

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

      3. +-commutative 64.3%

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

      4. +-commutative 64.3%

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

      5. associate-/r* 60.8%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 46.6%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 82.0%

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

      14. +-commutative 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in y around inf 78.4%

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

      1. unpow2 78.4%

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

   6. Simplified 78.4%

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

   if -8.00000000000000019e144 < y < 3.4000000000000001e-222

   1. Initial program 67.4%

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

      1. times-frac 90.9%

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

      2. associate-+l+ 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 66.7%

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

   5. Taylor expanded in x around 0 41.8%

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

   if 3.4000000000000001e-222 < y < 5.2e10

   1. Initial program 83.4%

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

      1. associate-+r+ 83.4%

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

      2. *-commutative 83.4%

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

      3. frac-times 90.5%

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

      4. associate-*l/ 86.5%

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

      5. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 97.7%

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

      1. +-commutative 97.7%

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

   6. Simplified 97.7%

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

      1. associate-*r/ 97.7%

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

      2. clear-num 97.5%

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

      3. frac-times 95.1%

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

      4. *-un-lft-identity 95.1%

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

      5. +-commutative 95.1%

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

      6. +-commutative 95.1%

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

      7. +-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

   9. Taylor expanded in x around 0 28.6%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+144}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-222}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 52000000000:\\ \;\;\;\;\frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 12: 60.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -1.7e-90) (- (/ y x) y) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.7e-90) {
		tmp = (y / x) - y;
	} 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 <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-1.7d-90)) then
        tmp = (y / x) - y
    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 <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.7e-90) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -1.7e-90:
		tmp = (y / x) - y
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -1.7e-90)
		tmp = Float64(Float64(y / x) - y);
	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 <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -1.7e-90)
		tmp = (y / x) - y;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-90], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 61.8%

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

      1. times-frac 92.3%

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

      2. associate-+l+ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in x around inf 79.5%

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

   5. Taylor expanded in x around inf 78.1%

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

   if -1 < x < -1.69999999999999997e-90

   1. Initial program 86.6%

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

      1. 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. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 43.9%

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

   5. Taylor expanded in x around 0 43.9%

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

      1. neg-mul-1 43.9%

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

      2. +-commutative 43.9%

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

      3. unsub-neg 43.9%

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

   7. Simplified 43.9%

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

   if -1.69999999999999997e-90 < x

   1. Initial program 69.2%

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

      1. times-frac 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. +-commutative 55.0%

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

   6. Simplified 55.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-90}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 13: 60.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+167}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.85e+167)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -6.2e-91) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.85e+167) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -6.2e-91) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.85d+167)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-6.2d-91)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.85e+167) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -6.2e-91) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.85e+167:
		tmp = (y / x) * (1.0 / x)
	elif x <= -6.2e-91:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.85e+167)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -6.2e-91)
		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 (x <= -1.85e+167)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -6.2e-91)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.85e+167], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-91], 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 x < -1.85e167

   1. Initial program 53.3%

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

      1. times-frac 85.9%

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

      2. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around inf 87.7%

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

   5. Taylor expanded in x around inf 87.5%

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

   if -1.85e167 < x < -6.19999999999999962e-91

   1. Initial program 74.9%

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

      1. times-frac 98.2%

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

      2. associate-+l+ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in y around 0 62.5%

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

   if -6.19999999999999962e-91 < x

   1. Initial program 69.2%

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

      1. times-frac 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. +-commutative 55.0%

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

   6. Simplified 55.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+167}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 14: 62.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-82) (/ (/ y (+ x 1.0)) x) (/ (/ x (+ y x)) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-82) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.2d-82) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-82) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e-82:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e-82)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-82)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e-82], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2000000000000001e-82

   1. Initial program 65.3%

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

      1. times-frac 87.3%

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

      2. associate-+l+ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in y around 0 60.2%

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

      1. *-un-lft-identity 60.2%

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

      2. times-frac 61.5%

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

      3. +-commutative 61.5%

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

   6. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.6%

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

      2. *-lft-identity 61.6%

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

   8. Simplified 61.6%

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

   if 4.2000000000000001e-82 < y

   1. Initial program 77.7%

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

      1. associate-/r* 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. associate-/r* 77.7%

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

      6. associate-*l/ 88.9%

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

      7. *-commutative 88.9%

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

      8. *-commutative 88.9%

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

      9. distribute-rgt1-in 84.5%

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

      10. fma-def 88.9%

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

      11. +-commutative 88.9%

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

      12. +-commutative 88.9%

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

      13. cube-unmult 89.0%

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

      14. +-commutative 89.0%

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

   3. Simplified 89.0%

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

      1. associate-*r/ 77.7%

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

      2. fma-udef 74.8%

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

      3. cube-mult 74.8%

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

      4. distribute-rgt1-in 77.7%

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

      5. associate-+r+ 77.7%

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

      6. *-commutative 77.7%

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

      7. *-commutative 77.7%

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

      8. frac-times 94.9%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

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

      11. frac-times 98.8%

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

      12. *-un-lft-identity 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in x around 0 66.4%

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

      1. +-commutative 66.4%

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

   8. Simplified 66.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 15: 62.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-82) (/ (/ y (+ x (+ y 1.0))) x) (/ (/ x (+ y x)) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-82) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-82) then
        tmp = (y / (x + (y + 1.0d0))) / x
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-82) {
		tmp = (y / (x + (y + 1.0))) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-82:
		tmp = (y / (x + (y + 1.0))) / x
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-82)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / x);
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-82)
		tmp = (y / (x + (y + 1.0))) / x;
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-82], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4e-82

   1. Initial program 65.3%

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

      1. times-frac 87.3%

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

      2. associate-+l+ 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in x around inf 61.9%

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

      1. associate-*l/ 61.9%

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

      2. *-un-lft-identity 61.9%

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

      3. +-commutative 61.9%

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

   6. Applied egg-rr 61.9%

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

   if 4e-82 < y

   1. Initial program 77.7%

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

      1. associate-/r* 82.7%

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

      2. +-commutative 82.7%

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

      3. +-commutative 82.7%

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

      4. +-commutative 82.7%

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

      5. associate-/r* 77.7%

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

      6. associate-*l/ 88.9%

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

      7. *-commutative 88.9%

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

      8. *-commutative 88.9%

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

      9. distribute-rgt1-in 84.5%

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

      10. fma-def 88.9%

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

      11. +-commutative 88.9%

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

      12. +-commutative 88.9%

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

      13. cube-unmult 89.0%

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

      14. +-commutative 89.0%

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

   3. Simplified 89.0%

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

      1. associate-*r/ 77.7%

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

      2. fma-udef 74.8%

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

      3. cube-mult 74.8%

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

      4. distribute-rgt1-in 77.7%

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

      5. associate-+r+ 77.7%

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

      6. *-commutative 77.7%

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

      7. *-commutative 77.7%

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

      8. frac-times 94.9%

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

      9. associate-/r* 99.8%

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

      10. clear-num 99.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{\frac{x}{x + y}}{x + y} \]

      11. frac-times 98.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

      12. *-un-lft-identity 98.8%

          \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   6. Taylor expanded in x around 0 66.4%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   7. Step-by-step derivation

      1. +-commutative 66.4%

         \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   8. Simplified 66.4%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 16: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+144} \lor \neg \left(y \leq 4.2 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8e+144) (not (<= y 4.2e-34))) (/ x (* y y)) (/ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8e+144) || !(y <= 4.2e-34)) {
		tmp = x / (y * y);
	} else {
		tmp = 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 <= (-8d+144)) .or. (.not. (y <= 4.2d-34))) then
        tmp = x / (y * y)
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -8e+144) || !(y <= 4.2e-34)) {
		tmp = x / (y * y);
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8e+144) or not (y <= 4.2e-34):
		tmp = x / (y * y)
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8e+144) || !(y <= 4.2e-34))
		tmp = Float64(x / Float64(y * y));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -8e+144) || ~((y <= 4.2e-34)))
		tmp = x / (y * y);
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8e+144], N[Not[LessEqual[y, 4.2e-34]], $MachinePrecision]], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000019e144 or 4.2000000000000002e-34 < y

   1. Initial program 64.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 67.5%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 67.5%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 67.5%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 67.5%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 64.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 83.6%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 83.6%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 83.6%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 51.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 83.6%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 83.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 83.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 83.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 83.6%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 71.5%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.5%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -8.00000000000000019e144 < y < 4.2000000000000002e-34

   1. Initial program 70.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.3%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 90.3%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.3%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 68.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

   5. Taylor expanded in x around 0 42.0%

      \[\leadsto \frac{y}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+144} \lor \neg \left(y \leq 4.2 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 17: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4e-90) (/ (/ y x) (+ x 1.0)) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-90) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d-90)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4e-90) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4e-90:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4e-90)
		tmp = Float64(Float64(y / 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 (x <= -1.4e-90)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4e-90], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e-90

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 94.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 94.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 70.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 70.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 70.9%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   if -1.3999999999999999e-90 < x

   1. Initial program 69.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 86.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 55.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 55.0%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 55.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 18: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 89.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 89.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around inf 54.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]

5. Taylor expanded in y around inf 4.2%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]

Alternative 19: 26.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ y x))

C

double code(double x, double y) {
	return y / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / x
end function

Java

public static double code(double x, double y) {
	return y / x;
}

Python

def code(x, y):
	return y / x

Julia

function code(x, y)
	return Float64(y / x)
end

MATLAB

function tmp = code(x, y)
	tmp = y / x;
end

Wolfram

code[x_, y_] := N[(y / x), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 89.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 89.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 89.4%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around 0 53.9%

   \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

5. Taylor expanded in x around 0 28.8%

   \[\leadsto \frac{y}{\color{blue}{x}} \]

6. Final simplification 28.8%

   \[\leadsto \frac{y}{x} \]

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 2023279 
(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))))