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

Percentage Accurate: 68.7% → 99.7%

Time: 14.6s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 81.0%

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

   2. +-commutative 81.0%

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

   3. +-commutative 81.0%

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

   4. +-commutative 81.0%

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

   5. *-commutative 81.0%

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

   6. distribute-rgt1-in 64.2%

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

   7. +-commutative 64.2%

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

   8. +-commutative 64.2%

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

   9. cube-unmult 64.2%

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

   10. +-commutative 64.2%

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

3. Simplified 64.2%

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

   1. clear-num 63.8%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

   2. un-div-inv 63.9%

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

   3. cube-mult 63.9%

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

   4. distribute-rgt1-in 80.7%

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

   5. *-commutative 80.7%

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

   6. associate-/l* 82.9%

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

   7. pow2 82.9%

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

   8. associate-+r+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. associate-/r* 86.8%

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

   2. +-commutative 86.8%

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

8. Simplified 86.8%

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

   1. *-un-lft-identity 86.8%

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

   2. +-commutative 86.8%

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

   3. pow2 86.8%

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

   4. times-frac 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 69.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 + y\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{t\_0}{y}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(1 + y\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ 1.0 y))))
   (if (<= y -9e-235)
     (/ (/ 1.0 x) (/ t_0 y))
     (if (<= y 5.2e-132)
       (* x (/ (/ y (+ y x)) (* (+ 1.0 y) (+ y x))))
       (if (<= y 4.6e+96)
         (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
         (* (/ x (+ y x)) (/ (/ y y) (+ 1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = x + (1.0 + y);
	double tmp;
	if (y <= -9e-235) {
		tmp = (1.0 / x) / (t_0 / y);
	} else if (y <= 5.2e-132) {
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	} else if (y <= 4.6e+96) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (1.0d0 + y)
    if (y <= (-9d-235)) then
        tmp = (1.0d0 / x) / (t_0 / y)
    else if (y <= 5.2d-132) then
        tmp = x * ((y / (y + x)) / ((1.0d0 + y) * (y + x)))
    else if (y <= 4.6d+96) then
        tmp = x * (y / (t_0 * ((y + x) * (y + x))))
    else
        tmp = (x / (y + x)) * ((y / y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (1.0 + y);
	double tmp;
	if (y <= -9e-235) {
		tmp = (1.0 / x) / (t_0 / y);
	} else if (y <= 5.2e-132) {
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	} else if (y <= 4.6e+96) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (1.0 + y)
	tmp = 0
	if y <= -9e-235:
		tmp = (1.0 / x) / (t_0 / y)
	elif y <= 5.2e-132:
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)))
	elif y <= 4.6e+96:
		tmp = x * (y / (t_0 * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(1.0 + y))
	tmp = 0.0
	if (y <= -9e-235)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 / y));
	elseif (y <= 5.2e-132)
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(Float64(1.0 + y) * Float64(y + x))));
	elseif (y <= 4.6e+96)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (1.0 + y);
	tmp = 0.0;
	if (y <= -9e-235)
		tmp = (1.0 / x) / (t_0 / y);
	elseif (y <= 5.2e-132)
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	elseif (y <= 4.6e+96)
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	else
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9e-235], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-132], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+96], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.9999999999999996e-235

   1. Initial program 70.4%

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

      1. associate-/l* 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

      4. +-commutative 83.2%

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

      5. *-commutative 83.2%

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

      6. distribute-rgt1-in 57.0%

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

      7. +-commutative 57.0%

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

      8. +-commutative 57.0%

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

      9. cube-unmult 57.0%

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

      10. +-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. clear-num 56.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 56.2%

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

      3. cube-mult 56.2%

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

      4. distribute-rgt1-in 82.4%

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

      5. *-commutative 82.4%

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

      6. associate-/l* 84.0%

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

      7. pow2 84.0%

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

      8. associate-+r+ 84.0%

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

   6. Applied egg-rr 84.0%

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

      1. associate-/r* 89.6%

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

      2. +-commutative 89.6%

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

   8. Simplified 89.6%

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

      1. *-un-lft-identity 89.6%

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

      2. +-commutative 89.6%

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

      3. pow2 89.6%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 51.9%

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

   if -8.9999999999999996e-235 < y < 5.2000000000000002e-132

   1. Initial program 64.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. associate-*l* 78.3%

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

      3. times-frac 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. associate-*r/ 96.3%

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

       2. *-commutative 96.3%

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

       3. associate-*r/ 96.3%

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

       4. *-rgt-identity 96.3%

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

       5. +-commutative 96.3%

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

       6. +-commutative 96.3%

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

       7. *-commutative 96.3%

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

       8. +-commutative 96.3%

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

       9. +-commutative 96.3%

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

   11. Simplified 96.3%

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

   if 5.2000000000000002e-132 < y < 4.6000000000000003e96

   1. Initial program 80.7%

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

      1. associate-/l* 92.6%

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

      2. associate-+l+ 92.6%

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

   3. Simplified 92.6%

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

   if 4.6000000000000003e96 < y

   1. Initial program 53.6%

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

      1. associate-/l* 66.6%

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

      2. associate-+l+ 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in x around 0 66.6%

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

      1. associate-*r/ 53.6%

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

      2. associate-*l* 53.6%

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

      3. +-commutative 53.6%

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

   10. Applied egg-rr 53.6%

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

       1. times-frac 74.4%

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

       2. associate-/r* 84.9%

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

   12. Simplified 84.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-235}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-132}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(1 + y\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(1 + y\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(1 + y\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1e-230)
   (/ (/ 1.0 x) (/ (+ x (+ 1.0 y)) y))
   (if (<= y 3e-133)
     (* x (/ (/ y (+ y x)) (* (+ 1.0 y) (+ y x))))
     (if (<= y 170000.0)
       (* x (/ y (* (+ 1.0 x) (* (+ y x) (+ y x)))))
       (* (/ x (+ y x)) (/ (/ y y) (+ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1e-230) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 3e-133) {
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	} else if (y <= 170000.0) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1d-230)) then
        tmp = (1.0d0 / x) / ((x + (1.0d0 + y)) / y)
    else if (y <= 3d-133) then
        tmp = x * ((y / (y + x)) / ((1.0d0 + y) * (y + x)))
    else if (y <= 170000.0d0) then
        tmp = x * (y / ((1.0d0 + x) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + x)) * ((y / y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1e-230) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 3e-133) {
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	} else if (y <= 170000.0) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1e-230:
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y)
	elif y <= 3e-133:
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)))
	elif y <= 170000.0:
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1e-230)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + Float64(1.0 + y)) / y));
	elseif (y <= 3e-133)
		tmp = Float64(x * Float64(Float64(y / Float64(y + x)) / Float64(Float64(1.0 + y) * Float64(y + x))));
	elseif (y <= 170000.0)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + x) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1e-230)
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	elseif (y <= 3e-133)
		tmp = x * ((y / (y + x)) / ((1.0 + y) * (y + x)));
	elseif (y <= 170000.0)
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1e-230], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-133], N[(x * N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000.0], N[(x * N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.00000000000000005e-230

   1. Initial program 70.4%

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

      1. associate-/l* 83.2%

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

      2. +-commutative 83.2%

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

      3. +-commutative 83.2%

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

      4. +-commutative 83.2%

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

      5. *-commutative 83.2%

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

      6. distribute-rgt1-in 57.0%

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

      7. +-commutative 57.0%

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

      8. +-commutative 57.0%

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

      9. cube-unmult 57.0%

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

      10. +-commutative 57.0%

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

   3. Simplified 57.0%

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

      1. clear-num 56.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 56.2%

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

      3. cube-mult 56.2%

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

      4. distribute-rgt1-in 82.4%

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

      5. *-commutative 82.4%

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

      6. associate-/l* 84.0%

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

      7. pow2 84.0%

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

      8. associate-+r+ 84.0%

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

   6. Applied egg-rr 84.0%

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

      1. associate-/r* 89.6%

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

      2. +-commutative 89.6%

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

   8. Simplified 89.6%

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

      1. *-un-lft-identity 89.6%

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

      2. +-commutative 89.6%

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

      3. pow2 89.6%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 51.9%

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

   if -1.00000000000000005e-230 < y < 3.00000000000000019e-133

   1. Initial program 64.7%

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

      1. associate-/l* 80.4%

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

      2. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

   7. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. associate-*l* 78.3%

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

      3. times-frac 96.3%

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

      4. +-commutative 96.3%

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

      5. +-commutative 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. associate-*r/ 96.3%

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

       2. *-commutative 96.3%

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

       3. associate-*r/ 96.3%

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

       4. *-rgt-identity 96.3%

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

       5. +-commutative 96.3%

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

       6. +-commutative 96.3%

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

       7. *-commutative 96.3%

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

       8. +-commutative 96.3%

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

       9. +-commutative 96.3%

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

   11. Simplified 96.3%

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

   if 3.00000000000000019e-133 < y < 1.7e5

   1. Initial program 88.9%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.7e5 < y

   1. Initial program 56.3%

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

      1. associate-/l* 69.6%

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

      2. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. +-commutative 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in x around 0 66.1%

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

      1. associate-*r/ 52.9%

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

      2. associate-*l* 52.9%

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

      3. +-commutative 52.9%

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

   10. Applied egg-rr 52.9%

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

       1. times-frac 67.4%

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

       2. associate-/r* 75.5%

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

   12. Simplified 75.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \frac{\frac{y}{y + x}}{\left(1 + y\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e-171)
   (/ (/ 1.0 x) (/ (+ x (+ 1.0 y)) y))
   (if (<= y 170000.0)
     (* x (/ y (* (+ 1.0 x) (* (+ y x) (+ y x)))))
     (* (/ x (+ y x)) (/ (/ y y) (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 170000.0) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.6d-171) then
        tmp = (1.0d0 / x) / ((x + (1.0d0 + y)) / y)
    else if (y <= 170000.0d0) then
        tmp = x * (y / ((1.0d0 + x) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + x)) * ((y / y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 170000.0) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e-171:
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y)
	elif y <= 170000.0:
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e-171)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + Float64(1.0 + y)) / y));
	elseif (y <= 170000.0)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + x) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.6e-171)
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	elseif (y <= 170000.0)
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e-171], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 170000.0], N[(x * N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.60000000000000046e-171

   1. Initial program 68.9%

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

      1. associate-/l* 81.9%

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

      2. +-commutative 81.9%

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

      3. +-commutative 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. distribute-rgt1-in 59.1%

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

      7. +-commutative 59.1%

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

      8. +-commutative 59.1%

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

      9. cube-unmult 59.1%

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

      10. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. clear-num 58.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/l* 82.5%

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

      7. pow2 82.5%

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

      8. associate-+r+ 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r* 87.0%

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

      2. +-commutative 87.0%

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

   8. Simplified 87.0%

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

      1. *-un-lft-identity 87.0%

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

      2. +-commutative 87.0%

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

      3. pow2 87.0%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 61.0%

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

   if 5.60000000000000046e-171 < y < 1.7e5

   1. Initial program 83.0%

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

      1. associate-/l* 97.6%

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

      2. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   if 1.7e5 < y

   1. Initial program 56.3%

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

      1. associate-/l* 69.6%

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

      2. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. +-commutative 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in x around 0 66.1%

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

      1. associate-*r/ 52.9%

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

      2. associate-*l* 52.9%

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

      3. +-commutative 52.9%

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

   10. Applied egg-rr 52.9%

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

       1. times-frac 67.4%

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

       2. associate-/r* 75.5%

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

   12. Simplified 75.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 170000:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 0.0035:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e-171)
   (/ (/ 1.0 x) (/ (+ x (+ 1.0 y)) y))
   (if (<= y 0.0035)
     (* x (/ y (* (+ y x) (+ y x))))
     (* (/ x (+ y x)) (/ (/ y y) (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 0.0035) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.6d-171) then
        tmp = (1.0d0 / x) / ((x + (1.0d0 + y)) / y)
    else if (y <= 0.0035d0) then
        tmp = x * (y / ((y + x) * (y + x)))
    else
        tmp = (x / (y + x)) * ((y / y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 0.0035) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e-171:
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y)
	elif y <= 0.0035:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e-171)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + Float64(1.0 + y)) / y));
	elseif (y <= 0.0035)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(Float64(y / y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.6e-171)
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	elseif (y <= 0.0035)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (y + x)) * ((y / y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e-171], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0035], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.60000000000000046e-171

   1. Initial program 68.9%

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

      1. associate-/l* 81.9%

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

      2. +-commutative 81.9%

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

      3. +-commutative 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. distribute-rgt1-in 59.1%

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

      7. +-commutative 59.1%

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

      8. +-commutative 59.1%

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

      9. cube-unmult 59.1%

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

      10. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. clear-num 58.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/l* 82.5%

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

      7. pow2 82.5%

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

      8. associate-+r+ 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r* 87.0%

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

      2. +-commutative 87.0%

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

   8. Simplified 87.0%

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

      1. *-un-lft-identity 87.0%

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

      2. +-commutative 87.0%

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

      3. pow2 87.0%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 61.0%

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

   if 5.60000000000000046e-171 < y < 0.00350000000000000007

   1. Initial program 82.5%

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

      1. associate-/l* 97.5%

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

      2. associate-+l+ 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. +-commutative 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in y around 0 80.5%

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

   if 0.00350000000000000007 < y

   1. Initial program 57.0%

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

      1. associate-/l* 70.1%

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

      2. associate-+l+ 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 68.4%

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

      1. +-commutative 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in x around 0 65.1%

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

      1. associate-*r/ 52.1%

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

      2. associate-*l* 52.1%

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

      3. +-commutative 52.1%

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

   10. Applied egg-rr 52.1%

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

       1. times-frac 66.4%

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

       2. associate-/r* 74.3%

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

   12. Simplified 74.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 0.0035:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 175000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ 1.0 x)))))
   (if (<= y 2.95e-145)
     t_0
     (if (<= y 4.8e-20)
       (/ x y)
       (if (<= y 175000.0) t_0 (* (/ x y) (/ 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = y / (x * (1.0 + x));
	double tmp;
	if (y <= 2.95e-145) {
		tmp = t_0;
	} else if (y <= 4.8e-20) {
		tmp = x / y;
	} else if (y <= 175000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (1.0d0 + x))
    if (y <= 2.95d-145) then
        tmp = t_0
    else if (y <= 4.8d-20) then
        tmp = x / y
    else if (y <= 175000.0d0) then
        tmp = t_0
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x * (1.0 + x));
	double tmp;
	if (y <= 2.95e-145) {
		tmp = t_0;
	} else if (y <= 4.8e-20) {
		tmp = x / y;
	} else if (y <= 175000.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * (1.0 + x))
	tmp = 0
	if y <= 2.95e-145:
		tmp = t_0
	elif y <= 4.8e-20:
		tmp = x / y
	elif y <= 175000.0:
		tmp = t_0
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(1.0 + x)))
	tmp = 0.0
	if (y <= 2.95e-145)
		tmp = t_0;
	elseif (y <= 4.8e-20)
		tmp = Float64(x / y);
	elseif (y <= 175000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * (1.0 + x));
	tmp = 0.0;
	if (y <= 2.95e-145)
		tmp = t_0;
	elseif (y <= 4.8e-20)
		tmp = x / y;
	elseif (y <= 175000.0)
		tmp = t_0;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.95e-145], t$95$0, If[LessEqual[y, 4.8e-20], N[(x / y), $MachinePrecision], If[LessEqual[y, 175000.0], t$95$0, N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9499999999999999e-145 or 4.79999999999999986e-20 < y < 175000

   1. Initial program 70.1%

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

      1. associate-/l* 82.5%

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

      2. associate-+l+ 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in y around 0 62.1%

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

      1. +-commutative 62.1%

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

   7. Simplified 62.1%

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

   if 2.9499999999999999e-145 < y < 4.79999999999999986e-20

   1. Initial program 80.1%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 45.5%

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

      1. +-commutative 45.5%

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

   7. Simplified 45.5%

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

   8. Taylor expanded in y around 0 45.5%

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

   if 175000 < y

   1. Initial program 56.3%

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

      1. associate-/l* 69.6%

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

      2. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 67.2%

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

      1. +-commutative 67.2%

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

   7. Simplified 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 75.0%

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

   9. Applied egg-rr 75.0%

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

   10. Taylor expanded in y around inf 75.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 175000:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 160000:\\ \;\;\;\;\frac{\frac{1}{y + x} \cdot t\_0}{\frac{1 + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 160000.0)
     (/ (* (/ 1.0 (+ y x)) t_0) (/ (+ 1.0 x) y))
     (* t_0 (/ (/ y y) (+ 1.0 y))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 160000.0) {
		tmp = ((1.0 / (y + x)) * t_0) / ((1.0 + x) / y);
	} else {
		tmp = t_0 * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (y <= 160000.0d0) then
        tmp = ((1.0d0 / (y + x)) * t_0) / ((1.0d0 + x) / y)
    else
        tmp = t_0 * ((y / y) / (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 160000.0) {
		tmp = ((1.0 / (y + x)) * t_0) / ((1.0 + x) / y);
	} else {
		tmp = t_0 * ((y / y) / (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 160000.0:
		tmp = ((1.0 / (y + x)) * t_0) / ((1.0 + x) / y)
	else:
		tmp = t_0 * ((y / y) / (1.0 + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 160000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) * t_0) / Float64(Float64(1.0 + x) / y));
	else
		tmp = Float64(t_0 * Float64(Float64(y / y) / Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 160000.0)
		tmp = ((1.0 / (y + x)) * t_0) / ((1.0 + x) / y);
	else
		tmp = t_0 * ((y / y) / (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 160000.0], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(y / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e5

   1. Initial program 71.3%

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

      1. associate-/l* 84.6%

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

      2. +-commutative 84.6%

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

      3. +-commutative 84.6%

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

      4. +-commutative 84.6%

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

      5. *-commutative 84.6%

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

      6. distribute-rgt1-in 62.7%

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

      7. +-commutative 62.7%

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

      8. +-commutative 62.7%

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

      9. cube-unmult 62.7%

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

      10. +-commutative 62.7%

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

   3. Simplified 62.7%

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

      1. clear-num 62.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 62.3%

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

      3. cube-mult 62.3%

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

      4. distribute-rgt1-in 84.2%

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

      5. *-commutative 84.2%

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

      6. associate-/l* 85.1%

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

      7. pow2 85.1%

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

      8. associate-+r+ 85.1%

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

   6. Applied egg-rr 85.1%

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

      1. associate-/r* 88.8%

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

      2. +-commutative 88.8%

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

   8. Simplified 88.8%

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

      1. *-un-lft-identity 88.8%

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

      2. +-commutative 88.8%

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

      3. pow2 88.8%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 92.4%

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

   if 1.6e5 < y

   1. Initial program 56.3%

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

      1. associate-/l* 69.6%

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

      2. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 67.9%

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

      1. +-commutative 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in x around 0 66.1%

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

      1. associate-*r/ 52.9%

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

      2. associate-*l* 52.9%

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

      3. +-commutative 52.9%

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

   10. Applied egg-rr 52.9%

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

       1. times-frac 67.4%

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

       2. associate-/r* 75.5%

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

   12. Simplified 75.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 160000:\\ \;\;\;\;\frac{\frac{1}{y + x} \cdot \frac{x}{y + x}}{\frac{1 + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{\frac{y}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 130000:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e-171)
   (/ (/ 1.0 x) (/ (+ x (+ 1.0 y)) y))
   (if (<= y 130000.0) (* x (/ y (* (+ y x) (+ y x)))) (/ (/ x (+ 1.0 y)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 130000.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.6d-171) then
        tmp = (1.0d0 / x) / ((x + (1.0d0 + y)) / y)
    else if (y <= 130000.0d0) then
        tmp = x * (y / ((y + x) * (y + x)))
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.6e-171) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else if (y <= 130000.0) {
		tmp = x * (y / ((y + x) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.6e-171:
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y)
	elif y <= 130000.0:
		tmp = x * (y / ((y + x) * (y + x)))
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e-171)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + Float64(1.0 + y)) / y));
	elseif (y <= 130000.0)
		tmp = Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.6e-171)
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	elseif (y <= 130000.0)
		tmp = x * (y / ((y + x) * (y + x)));
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e-171], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 130000.0], N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.60000000000000046e-171

   1. Initial program 68.9%

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

      1. associate-/l* 81.9%

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

      2. +-commutative 81.9%

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

      3. +-commutative 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. distribute-rgt1-in 59.1%

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

      7. +-commutative 59.1%

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

      8. +-commutative 59.1%

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

      9. cube-unmult 59.1%

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

      10. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. clear-num 58.5%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/l* 82.5%

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

      7. pow2 82.5%

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

      8. associate-+r+ 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. associate-/r* 87.0%

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

      2. +-commutative 87.0%

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

   8. Simplified 87.0%

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

      1. *-un-lft-identity 87.0%

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

      2. +-commutative 87.0%

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

      3. pow2 87.0%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 61.0%

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

   if 5.60000000000000046e-171 < y < 1.3e5

   1. Initial program 83.0%

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

      1. associate-/l* 97.6%

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

      2. associate-+l+ 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around 0 81.0%

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

      1. +-commutative 81.0%

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

   7. Simplified 81.0%

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

   8. Taylor expanded in y around 0 81.0%

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

   if 1.3e5 < y

   1. Initial program 56.3%

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

      1. associate-/l* 69.6%

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

      2. associate-+l+ 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in x around 0 67.2%

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

      1. +-commutative 67.2%

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

   7. Simplified 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. times-frac 75.0%

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

   9. Applied egg-rr 75.0%

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

       1. associate-*l/ 75.1%

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

       2. *-un-lft-identity 75.1%

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

       3. +-commutative 75.1%

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

   11. Applied egg-rr 75.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{elif}\;y \leq 130000:\\ \;\;\;\;x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 81.0%

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

   2. +-commutative 81.0%

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

   3. +-commutative 81.0%

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

   4. +-commutative 81.0%

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

   5. *-commutative 81.0%

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

   6. distribute-rgt1-in 64.2%

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

   7. +-commutative 64.2%

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

   8. +-commutative 64.2%

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

   9. cube-unmult 64.2%

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

   10. +-commutative 64.2%

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

3. Simplified 64.2%

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

   1. clear-num 63.8%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

   2. un-div-inv 63.9%

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

   3. cube-mult 63.9%

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

   4. distribute-rgt1-in 80.7%

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

   5. *-commutative 80.7%

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

   6. associate-/l* 82.9%

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

   7. pow2 82.9%

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

   8. associate-+r+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. associate-/r* 86.8%

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

   2. +-commutative 86.8%

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

8. Simplified 86.8%

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

   1. *-un-lft-identity 86.8%

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

   2. +-commutative 86.8%

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

   3. pow2 86.8%

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

   4. times-frac 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-/l* 99.7%

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

    2. +-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

Alternative 10: 45.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2e-145)
   (/ y x)
   (if (<= y 3.8e+174) (/ x (* y (+ 1.0 y))) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-145) {
		tmp = y / x;
	} else if (y <= 3.8e+174) {
		tmp = x / (y * (1.0 + y));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-145) then
        tmp = y / x
    else if (y <= 3.8d+174) then
        tmp = x / (y * (1.0d0 + y))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-145) {
		tmp = y / x;
	} else if (y <= 3.8e+174) {
		tmp = x / (y * (1.0 + y));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-145:
		tmp = y / x
	elif y <= 3.8e+174:
		tmp = x / (y * (1.0 + y))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-145)
		tmp = Float64(y / x);
	elseif (y <= 3.8e+174)
		tmp = Float64(x / Float64(y * Float64(1.0 + y)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-145)
		tmp = y / x;
	elseif (y <= 3.8e+174)
		tmp = x / (y * (1.0 + y));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-145], N[(y / x), $MachinePrecision], If[LessEqual[y, 3.8e+174], N[(x / N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999983e-145

   1. Initial program 69.2%

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

      1. associate-/l* 82.0%

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

      2. +-commutative 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. *-commutative 82.0%

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

      6. distribute-rgt1-in 58.7%

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

      7. +-commutative 58.7%

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

      8. +-commutative 58.7%

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

      9. cube-unmult 58.7%

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

      10. +-commutative 58.7%

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

   3. Simplified 58.7%

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

      1. cube-mult 58.7%

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

      2. distribute-rgt1-in 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-/l* 69.2%

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

      5. clear-num 68.7%

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

      6. associate-/l* 71.5%

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

      7. pow2 71.5%

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

      8. associate-+r+ 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in x around 0 83.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(y + \left(2 \cdot \left(1 + y\right) + x \cdot \left(3 + \frac{1}{y}\right)\right)\right) + y \cdot \left(1 + y\right)}{x}}} \]

   8. Taylor expanded in y around 0 39.2%

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

   if 1.99999999999999983e-145 < y < 3.8000000000000002e174

   1. Initial program 68.9%

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

      1. associate-/l* 83.0%

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

      2. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in x around 0 48.6%

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

      1. +-commutative 48.6%

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

   7. Simplified 48.6%

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

   if 3.8000000000000002e174 < y

   1. Initial program 58.7%

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

      1. associate-/l* 73.1%

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

      2. associate-+l+ 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around 0 73.1%

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

      1. +-commutative 73.1%

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

   7. Simplified 73.1%

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

      1. *-un-lft-identity 73.1%

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

      2. times-frac 90.0%

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

   9. Applied egg-rr 90.0%

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

   10. Taylor expanded in y around inf 90.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-145) (/ y x) (if (<= y 1.0) (/ x y) (* (/ x y) (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.95d-145) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.95e-145:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-145)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.95e-145)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.95e-145], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.9499999999999999e-145

   1. Initial program 69.2%

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

      1. associate-/l* 82.0%

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

      2. +-commutative 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. *-commutative 82.0%

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

      6. distribute-rgt1-in 58.7%

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

      7. +-commutative 58.7%

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

      8. +-commutative 58.7%

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

      9. cube-unmult 58.7%

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

      10. +-commutative 58.7%

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

   3. Simplified 58.7%

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

      1. cube-mult 58.7%

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

      2. distribute-rgt1-in 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-/l* 69.2%

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

      5. clear-num 68.7%

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

      6. associate-/l* 71.5%

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

      7. pow2 71.5%

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

      8. associate-+r+ 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in x around 0 83.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(y + \left(2 \cdot \left(1 + y\right) + x \cdot \left(3 + \frac{1}{y}\right)\right)\right) + y \cdot \left(1 + y\right)}{x}}} \]

   8. Taylor expanded in y around 0 39.2%

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

   if 2.9499999999999999e-145 < y < 1

   1. Initial program 82.9%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 39.4%

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

      1. +-commutative 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in y around 0 39.4%

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

   if 1 < y

   1. Initial program 57.0%

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

      1. associate-/l* 70.1%

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

      2. associate-+l+ 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 66.2%

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

      1. +-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. times-frac 73.9%

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

   9. Applied egg-rr 73.9%

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

   10. Taylor expanded in y around inf 73.9%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 81.0%

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

   2. +-commutative 81.0%

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

   3. +-commutative 81.0%

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

   4. +-commutative 81.0%

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

   5. *-commutative 81.0%

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

   6. distribute-rgt1-in 64.2%

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

   7. +-commutative 64.2%

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

   8. +-commutative 64.2%

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

   9. cube-unmult 64.2%

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

   10. +-commutative 64.2%

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

3. Simplified 64.2%

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

   1. clear-num 63.8%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

   2. un-div-inv 63.9%

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

   3. cube-mult 63.9%

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

   4. distribute-rgt1-in 80.7%

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

   5. *-commutative 80.7%

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

   6. associate-/l* 82.9%

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

   7. pow2 82.9%

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

   8. associate-+r+ 82.9%

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

6. Applied egg-rr 82.9%

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

   1. associate-/r* 86.8%

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

   2. +-commutative 86.8%

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

8. Simplified 86.8%

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

   1. *-un-lft-identity 86.8%

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

   2. +-commutative 86.8%

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

   3. pow2 86.8%

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

   4. times-frac 99.7%

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

   5. +-commutative 99.7%

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

   6. +-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 13: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-145)
   (/ (/ 1.0 x) (/ (+ x (+ 1.0 y)) y))
   (/ (/ x (+ 1.0 y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.95d-145) then
        tmp = (1.0d0 / x) / ((x + (1.0d0 + y)) / y)
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.95e-145:
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y)
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-145)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(x + Float64(1.0 + y)) / y));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.95e-145)
		tmp = (1.0 / x) / ((x + (1.0 + y)) / y);
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.95e-145], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9499999999999999e-145

   1. Initial program 69.2%

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

      1. associate-/l* 82.0%

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

      2. +-commutative 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. *-commutative 82.0%

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

      6. distribute-rgt1-in 58.7%

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

      7. +-commutative 58.7%

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

      8. +-commutative 58.7%

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

      9. cube-unmult 58.7%

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

      10. +-commutative 58.7%

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

   3. Simplified 58.7%

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

      1. clear-num 58.1%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      2. un-div-inv 58.2%

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

      3. cube-mult 58.1%

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

      4. distribute-rgt1-in 81.5%

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

      5. *-commutative 81.5%

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

      6. associate-/l* 82.6%

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

      7. pow2 82.6%

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

      8. associate-+r+ 82.6%

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

   6. Applied egg-rr 82.6%

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

      1. associate-/r* 86.9%

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

      2. +-commutative 86.9%

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

   8. Simplified 86.9%

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

      1. *-un-lft-identity 86.9%

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

      2. +-commutative 86.9%

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

      3. pow2 86.9%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 61.6%

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

   if 2.9499999999999999e-145 < y

   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. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. +-commutative 57.8%

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

   7. Simplified 57.8%

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

      1. *-un-lft-identity 57.8%

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

      2. times-frac 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.2%

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

       2. *-un-lft-identity 63.2%

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

       3. +-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x + \left(1 + y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1 + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-145) (/ 1.0 (* x (/ (+ 1.0 x) y))) (/ (/ x (+ 1.0 y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = 1.0 / (x * ((1.0 + x) / y));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.95d-145) then
        tmp = 1.0d0 / (x * ((1.0d0 + x) / y))
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = 1.0 / (x * ((1.0 + x) / y));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.95e-145:
		tmp = 1.0 / (x * ((1.0 + x) / y))
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-145)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(1.0 + x) / y)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.95e-145)
		tmp = 1.0 / (x * ((1.0 + x) / y));
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.95e-145], N[(1.0 / N[(x * N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9499999999999999e-145

   1. Initial program 69.2%

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

      1. associate-/l* 82.0%

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

      2. +-commutative 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. *-commutative 82.0%

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

      6. distribute-rgt1-in 58.7%

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

      7. +-commutative 58.7%

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

      8. +-commutative 58.7%

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

      9. cube-unmult 58.7%

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

      10. +-commutative 58.7%

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

   3. Simplified 58.7%

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

      1. cube-mult 58.7%

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

      2. distribute-rgt1-in 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-/l* 69.2%

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

      5. clear-num 68.7%

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

      6. associate-/l* 71.5%

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

      7. pow2 71.5%

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

      8. associate-+r+ 71.5%

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

   6. Applied egg-rr 71.5%

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

   7. Taylor expanded in y around 0 60.5%

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

      1. associate-/l* 60.8%

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

      2. +-commutative 60.8%

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

   9. Simplified 60.8%

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

   if 2.9499999999999999e-145 < y

   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. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. +-commutative 57.8%

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

   7. Simplified 57.8%

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

      1. *-un-lft-identity 57.8%

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

      2. times-frac 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.2%

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

       2. *-un-lft-identity 63.2%

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

       3. +-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1 + x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.95e-145) (/ y (* x (+ 1.0 x))) (/ (/ x (+ 1.0 y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.95d-145) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / (1.0d0 + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.95e-145) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / (1.0 + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.95e-145:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / (1.0 + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.95e-145)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.95e-145)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / (1.0 + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.95e-145], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9499999999999999e-145

   1. Initial program 69.2%

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

      1. associate-/l* 82.0%

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

      2. associate-+l+ 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in y around 0 61.0%

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

      1. +-commutative 61.0%

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

   7. Simplified 61.0%

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

   if 2.9499999999999999e-145 < y

   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. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. +-commutative 57.8%

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

   7. Simplified 57.8%

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

      1. *-un-lft-identity 57.8%

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

      2. times-frac 63.1%

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

   9. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.2%

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

       2. *-un-lft-identity 63.2%

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

       3. +-commutative 63.2%

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

   11. Applied egg-rr 63.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.8e-172) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-172) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d-172)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-172) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-172:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-172)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-172)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-172], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000011e-172

   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-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. +-commutative 79.1%

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

      4. +-commutative 79.1%

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

      5. *-commutative 79.1%

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

      6. distribute-rgt1-in 48.8%

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

      7. +-commutative 48.8%

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

      8. +-commutative 48.8%

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

      9. cube-unmult 48.8%

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

      10. +-commutative 48.8%

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

   3. Simplified 48.8%

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

      1. cube-mult 48.8%

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

      2. distribute-rgt1-in 79.1%

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

      3. *-commutative 79.1%

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

      4. associate-/l* 71.8%

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

      5. clear-num 71.0%

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

      6. associate-/l* 74.8%

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

      7. pow2 74.8%

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

      8. associate-+r+ 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in x around 0 74.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(y + \left(2 \cdot \left(1 + y\right) + x \cdot \left(3 + \frac{1}{y}\right)\right)\right) + y \cdot \left(1 + y\right)}{x}}} \]

   8. Taylor expanded in y around 0 38.7%

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

   if -2.80000000000000011e-172 < x

   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. associate-/l* 82.3%

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

      2. associate-+l+ 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around 0 54.3%

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

      1. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in y around 0 34.8%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 81.0%

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

   2. associate-+l+ 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in x around 0 44.9%

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

   1. +-commutative 44.9%

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

7. Simplified 44.9%

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

8. Taylor expanded in y around 0 26.8%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 18: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 x))

C

double code(double x, double y) {
	return 0.5 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 / x
end function

Java

public static double code(double x, double y) {
	return 0.5 / x;
}

Python

def code(x, y):
	return 0.5 / x

Julia

function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 81.0%

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

   2. associate-+l+ 81.0%

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

3. Simplified 81.0%

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

5. Taylor expanded in x around 0 53.4%

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

   1. associate-*r* 53.4%

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

   2. +-commutative 53.4%

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

   3. unpow2 53.4%

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

   4. distribute-rgt-in 55.6%

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

7. Simplified 55.6%

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

8. Taylor expanded in x around inf 4.3%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))