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

Percentage Accurate: 70.1% → 99.8%

Time: 13.5s

Alternatives: 24

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: 70.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. times-frac 85.6%

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

   2. associate-+l+ 85.6%

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

3. Simplified 85.6%

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

   1. associate-/r* 99.8%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := y \cdot \frac{t_0}{x + y}\\ t_2 := \frac{t_0}{y + 1}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y)))
        (t_1 (* y (/ t_0 (+ x y))))
        (t_2 (/ t_0 (+ y 1.0))))
   (if (<= x -1.9e+15)
     (/ (/ y (+ x y)) x)
     (if (<= x -1.28e-23)
       t_2
       (if (<= x -1.4e-70)
         t_1
         (if (<= x -4.2e-76) (/ x (* y y)) (if (<= x -1.5e-252) t_1 t_2)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y * (t_0 / (x + y));
	double t_2 = t_0 / (y + 1.0);
	double tmp;
	if (x <= -1.9e+15) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.28e-23) {
		tmp = t_2;
	} else if (x <= -1.4e-70) {
		tmp = t_1;
	} else if (x <= -4.2e-76) {
		tmp = x / (y * y);
	} else if (x <= -1.5e-252) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = y * (t_0 / (x + y))
    t_2 = t_0 / (y + 1.0d0)
    if (x <= (-1.9d+15)) then
        tmp = (y / (x + y)) / x
    else if (x <= (-1.28d-23)) then
        tmp = t_2
    else if (x <= (-1.4d-70)) then
        tmp = t_1
    else if (x <= (-4.2d-76)) then
        tmp = x / (y * y)
    else if (x <= (-1.5d-252)) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = y * (t_0 / (x + y));
	double t_2 = t_0 / (y + 1.0);
	double tmp;
	if (x <= -1.9e+15) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.28e-23) {
		tmp = t_2;
	} else if (x <= -1.4e-70) {
		tmp = t_1;
	} else if (x <= -4.2e-76) {
		tmp = x / (y * y);
	} else if (x <= -1.5e-252) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = y * (t_0 / (x + y))
	t_2 = t_0 / (y + 1.0)
	tmp = 0
	if x <= -1.9e+15:
		tmp = (y / (x + y)) / x
	elif x <= -1.28e-23:
		tmp = t_2
	elif x <= -1.4e-70:
		tmp = t_1
	elif x <= -4.2e-76:
		tmp = x / (y * y)
	elif x <= -1.5e-252:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(y * Float64(t_0 / Float64(x + y)))
	t_2 = Float64(t_0 / Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.9e+15)
		tmp = Float64(Float64(y / Float64(x + y)) / x);
	elseif (x <= -1.28e-23)
		tmp = t_2;
	elseif (x <= -1.4e-70)
		tmp = t_1;
	elseif (x <= -4.2e-76)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -1.5e-252)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = y * (t_0 / (x + y));
	t_2 = t_0 / (y + 1.0);
	tmp = 0.0;
	if (x <= -1.9e+15)
		tmp = (y / (x + y)) / x;
	elseif (x <= -1.28e-23)
		tmp = t_2;
	elseif (x <= -1.4e-70)
		tmp = t_1;
	elseif (x <= -4.2e-76)
		tmp = x / (y * y);
	elseif (x <= -1.5e-252)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.9e+15], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.28e-23], t$95$2, If[LessEqual[x, -1.4e-70], t$95$1, If[LessEqual[x, -4.2e-76], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-252], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9e15

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. frac-times 63.5%

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

      2. *-commutative 63.5%

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

      3. frac-times 83.2%

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

      4. clear-num 83.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.9%

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

      7. *-un-lft-identity 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 84.9%

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

   if -1.9e15 < x < -1.28000000000000005e-23 or -1.49999999999999997e-252 < x

   1. Initial program 65.4%

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

      1. times-frac 84.7%

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

      2. associate-+l+ 84.7%

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

   3. Simplified 84.7%

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

      1. *-commutative 84.7%

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

      2. clear-num 84.6%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.2%

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

      5. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 55.8%

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

      1. +-commutative 55.8%

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

   8. Simplified 55.8%

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

   if -1.28000000000000005e-23 < x < -1.4e-70 or -4.19999999999999985e-76 < x < -1.49999999999999997e-252

   1. Initial program 79.2%

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

      1. times-frac 89.4%

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

      2. associate-+l+ 89.4%

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

   3. Simplified 89.4%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 88.4%

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

   if -1.4e-70 < x < -4.19999999999999985e-76

   1. Initial program 99.6%

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

      1. times-frac 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 75.4%

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

      1. unpow2 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 3: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{t_0}{y + 1}\\ t_2 := y \cdot \frac{t_0}{x + y}\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y)))
        (t_1 (/ t_0 (+ y 1.0)))
        (t_2 (* y (/ t_0 (+ x y)))))
   (if (<= x -8.4e+15)
     (/ (/ y (+ x (+ y 1.0))) (+ x y))
     (if (<= x -1.6e-22)
       t_1
       (if (<= x -8e-70)
         t_2
         (if (<= x -5.1e-76) (/ x (* y y)) (if (<= x -3.3e-253) t_2 t_1)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (y + 1.0);
	double t_2 = y * (t_0 / (x + y));
	double tmp;
	if (x <= -8.4e+15) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -1.6e-22) {
		tmp = t_1;
	} else if (x <= -8e-70) {
		tmp = t_2;
	} else if (x <= -5.1e-76) {
		tmp = x / (y * y);
	} else if (x <= -3.3e-253) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = t_0 / (y + 1.0d0)
    t_2 = y * (t_0 / (x + y))
    if (x <= (-8.4d+15)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else if (x <= (-1.6d-22)) then
        tmp = t_1
    else if (x <= (-8d-70)) then
        tmp = t_2
    else if (x <= (-5.1d-76)) then
        tmp = x / (y * y)
    else if (x <= (-3.3d-253)) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (y + 1.0);
	double t_2 = y * (t_0 / (x + y));
	double tmp;
	if (x <= -8.4e+15) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -1.6e-22) {
		tmp = t_1;
	} else if (x <= -8e-70) {
		tmp = t_2;
	} else if (x <= -5.1e-76) {
		tmp = x / (y * y);
	} else if (x <= -3.3e-253) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = t_0 / (y + 1.0)
	t_2 = y * (t_0 / (x + y))
	tmp = 0
	if x <= -8.4e+15:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	elif x <= -1.6e-22:
		tmp = t_1
	elif x <= -8e-70:
		tmp = t_2
	elif x <= -5.1e-76:
		tmp = x / (y * y)
	elif x <= -3.3e-253:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(t_0 / Float64(y + 1.0))
	t_2 = Float64(y * Float64(t_0 / Float64(x + y)))
	tmp = 0.0
	if (x <= -8.4e+15)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	elseif (x <= -1.6e-22)
		tmp = t_1;
	elseif (x <= -8e-70)
		tmp = t_2;
	elseif (x <= -5.1e-76)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -3.3e-253)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = t_0 / (y + 1.0);
	t_2 = y * (t_0 / (x + y));
	tmp = 0.0;
	if (x <= -8.4e+15)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	elseif (x <= -1.6e-22)
		tmp = t_1;
	elseif (x <= -8e-70)
		tmp = t_2;
	elseif (x <= -5.1e-76)
		tmp = x / (y * y);
	elseif (x <= -3.3e-253)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.4e+15], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-22], t$95$1, If[LessEqual[x, -8e-70], t$95$2, If[LessEqual[x, -5.1e-76], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-253], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.4e15

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. *-commutative 83.2%

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

      2. clear-num 83.2%

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

      3. associate-/r* 99.8%

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

      4. frac-times 98.6%

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

      5. *-un-lft-identity 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in x around inf 84.2%

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

      1. expm1-log1p-u 84.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}\right)\right)} \]

      2. expm1-udef 63.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}\right)} - 1} \]

      3. associate-/r* 63.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\frac{x + \left(y + 1\right)}{y}}}{x + y}}\right)} - 1 \]

      4. clear-num 63.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{y}{x + \left(y + 1\right)}}}{x + y}\right)} - 1 \]

      5. +-commutative 63.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{\color{blue}{y + x}}\right)} - 1 \]

   8. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 85.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\right)\right)} \]

      2. expm1-log1p 85.5%

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

   10. Simplified 85.5%

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

   if -8.4e15 < x < -1.59999999999999994e-22 or -3.3000000000000001e-253 < x

   1. Initial program 65.4%

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

      1. times-frac 84.7%

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

      2. associate-+l+ 84.7%

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

   3. Simplified 84.7%

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

      1. *-commutative 84.7%

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

      2. clear-num 84.6%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.2%

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

      5. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 55.8%

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

      1. +-commutative 55.8%

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

   8. Simplified 55.8%

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

   if -1.59999999999999994e-22 < x < -7.99999999999999997e-70 or -5.09999999999999986e-76 < x < -3.3000000000000001e-253

   1. Initial program 79.2%

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

      1. times-frac 89.4%

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

      2. associate-+l+ 89.4%

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

   3. Simplified 89.4%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in y around 0 88.4%

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

   if -7.99999999999999997e-70 < x < -5.09999999999999986e-76

   1. Initial program 99.6%

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

      1. times-frac 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 75.4%

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

      1. unpow2 75.4%

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

   10. Simplified 75.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-253}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 4: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+165}:\\ \;\;\;\;t_0 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (+ x y)) (+ x y))))
   (if (<= x -5e+165)
     (* t_0 (/ y x))
     (if (<= x -5e-20)
       (* (/ y (+ x (+ y 1.0))) (/ x (* (+ x y) (+ x y))))
       (* t_0 (/ y (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -5e+165) {
		tmp = t_0 * (y / x);
	} else if (x <= -5e-20) {
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + y)) / (x + y)
    if (x <= (-5d+165)) then
        tmp = t_0 * (y / x)
    else if (x <= (-5d-20)) then
        tmp = (y / (x + (y + 1.0d0))) * (x / ((x + y) * (x + y)))
    else
        tmp = t_0 * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -5e+165) {
		tmp = t_0 * (y / x);
	} else if (x <= -5e-20) {
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (x + y)) / (x + y)
	tmp = 0
	if x <= -5e+165:
		tmp = t_0 * (y / x)
	elif x <= -5e-20:
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)))
	else:
		tmp = t_0 * (y / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(x + y)) / Float64(x + y))
	tmp = 0.0
	if (x <= -5e+165)
		tmp = Float64(t_0 * Float64(y / x));
	elseif (x <= -5e-20)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(t_0 * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (x + y)) / (x + y);
	tmp = 0.0;
	if (x <= -5e+165)
		tmp = t_0 * (y / x);
	elseif (x <= -5e-20)
		tmp = (y / (x + (y + 1.0))) * (x / ((x + y) * (x + y)));
	else
		tmp = t_0 * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+165], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-20], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e165

   1. Initial program 59.8%

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

      1. times-frac 78.9%

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

      2. associate-+l+ 78.9%

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

   3. Simplified 78.9%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.7%

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

   if -4.9999999999999997e165 < x < -4.9999999999999999e-20

   1. Initial program 70.1%

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

      1. times-frac 91.9%

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

      2. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

   if -4.9999999999999999e-20 < x

   1. Initial program 69.7%

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

      1. times-frac 85.5%

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

      2. associate-+l+ 85.5%

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

   3. Simplified 85.5%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 84.6%

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

      1. +-commutative 84.6%

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

   10. Simplified 84.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 5: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+165}:\\ \;\;\;\;t_0 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (+ x y)) (+ x y))))
   (if (<= x -5e+165)
     (* t_0 (/ y x))
     (if (<= x -2e-45)
       (* (/ y (* (+ x y) (+ x y))) (/ x (+ x (+ y 1.0))))
       (* t_0 (/ y (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -5e+165) {
		tmp = t_0 * (y / x);
	} else if (x <= -2e-45) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + y)) / (x + y)
    if (x <= (-5d+165)) then
        tmp = t_0 * (y / x)
    else if (x <= (-2d-45)) then
        tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0d0)))
    else
        tmp = t_0 * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -5e+165) {
		tmp = t_0 * (y / x);
	} else if (x <= -2e-45) {
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (x + y)) / (x + y)
	tmp = 0
	if x <= -5e+165:
		tmp = t_0 * (y / x)
	elif x <= -2e-45:
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)))
	else:
		tmp = t_0 * (y / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(x + y)) / Float64(x + y))
	tmp = 0.0
	if (x <= -5e+165)
		tmp = Float64(t_0 * Float64(y / x));
	elseif (x <= -2e-45)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(x + y))) * Float64(x / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(t_0 * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (x + y)) / (x + y);
	tmp = 0.0;
	if (x <= -5e+165)
		tmp = t_0 * (y / x);
	elseif (x <= -2e-45)
		tmp = (y / ((x + y) * (x + y))) * (x / (x + (y + 1.0)));
	else
		tmp = t_0 * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+165], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-45], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999997e165

   1. Initial program 59.8%

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

      1. times-frac 78.9%

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

      2. associate-+l+ 78.9%

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

   3. Simplified 78.9%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.7%

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

   if -4.9999999999999997e165 < x < -1.99999999999999997e-45

   1. Initial program 70.7%

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

      1. associate-/r* 84.8%

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

      2. +-commutative 84.8%

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

      3. +-commutative 84.8%

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

      4. +-commutative 84.8%

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

      5. associate-/l/ 70.7%

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

      6. times-frac 92.7%

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

      7. *-commutative 92.7%

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

      8. +-commutative 92.7%

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

      9. +-commutative 92.7%

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

      10. +-commutative 92.7%

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

      11. associate-+l+ 92.7%

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

   3. Simplified 92.7%

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

   if -1.99999999999999997e-45 < x

   1. Initial program 69.5%

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

      1. times-frac 85.2%

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

      2. associate-+l+ 85.2%

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

   3. Simplified 85.2%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 84.3%

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

      1. +-commutative 84.3%

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

   10. Simplified 84.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 6: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{t_0}{x + y}\\ \mathbf{if}\;x \leq -85000000000000:\\ \;\;\;\;t_1 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-253}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x - \left(-1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (/ t_0 (+ x y))))
   (if (<= x -85000000000000.0)
     (* t_1 (/ y x))
     (if (<= x -6.5e-161)
       (* (/ y (+ y 1.0)) (/ x (* (+ x y) (+ x y))))
       (if (<= x -5.5e-253) (* y t_1) (/ t_0 (+ y (- x (- -1.0 x)))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -85000000000000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -6.5e-161) {
		tmp = (y / (y + 1.0)) * (x / ((x + y) * (x + y)));
	} else if (x <= -5.5e-253) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + (x - (-1.0 - x)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = t_0 / (x + y)
    if (x <= (-85000000000000.0d0)) then
        tmp = t_1 * (y / x)
    else if (x <= (-6.5d-161)) then
        tmp = (y / (y + 1.0d0)) * (x / ((x + y) * (x + y)))
    else if (x <= (-5.5d-253)) then
        tmp = y * t_1
    else
        tmp = t_0 / (y + (x - ((-1.0d0) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -85000000000000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -6.5e-161) {
		tmp = (y / (y + 1.0)) * (x / ((x + y) * (x + y)));
	} else if (x <= -5.5e-253) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + (x - (-1.0 - x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = t_0 / (x + y)
	tmp = 0
	if x <= -85000000000000.0:
		tmp = t_1 * (y / x)
	elif x <= -6.5e-161:
		tmp = (y / (y + 1.0)) * (x / ((x + y) * (x + y)))
	elif x <= -5.5e-253:
		tmp = y * t_1
	else:
		tmp = t_0 / (y + (x - (-1.0 - x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(t_0 / Float64(x + y))
	tmp = 0.0
	if (x <= -85000000000000.0)
		tmp = Float64(t_1 * Float64(y / x));
	elseif (x <= -6.5e-161)
		tmp = Float64(Float64(y / Float64(y + 1.0)) * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	elseif (x <= -5.5e-253)
		tmp = Float64(y * t_1);
	else
		tmp = Float64(t_0 / Float64(y + Float64(x - Float64(-1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = t_0 / (x + y);
	tmp = 0.0;
	if (x <= -85000000000000.0)
		tmp = t_1 * (y / x);
	elseif (x <= -6.5e-161)
		tmp = (y / (y + 1.0)) * (x / ((x + y) * (x + y)));
	elseif (x <= -5.5e-253)
		tmp = y * t_1;
	else
		tmp = t_0 / (y + (x - (-1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -85000000000000.0], N[(t$95$1 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.5e-161], N[(N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e-253], N[(y * t$95$1), $MachinePrecision], N[(t$95$0 / N[(y + N[(x - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5e13

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 94.1%

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

   if -8.5e13 < x < -6.50000000000000008e-161

   1. Initial program 84.5%

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

      1. times-frac 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 98.2%

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

      1. +-commutative 98.2%

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

   6. Simplified 98.2%

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

   if -6.50000000000000008e-161 < x < -5.49999999999999974e-253

   1. Initial program 71.6%

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

      1. times-frac 78.2%

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

      2. associate-+l+ 78.2%

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

   3. Simplified 78.2%

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

      1. associate-/r* 100.0%

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

      2. div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in y around 0 95.3%

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

   if -5.49999999999999974e-253 < x

   1. Initial program 64.6%

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

      1. times-frac 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. clear-num 83.3%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.1%

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

      5. *-un-lft-identity 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in y around -inf 53.8%

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

      1. mul-1-neg 53.8%

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

      2. unsub-neg 53.8%

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

      3. neg-mul-1 53.8%

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

      4. +-commutative 53.8%

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

      5. unsub-neg 53.8%

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

      6. distribute-lft-in 53.8%

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

      7. metadata-eval 53.8%

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

      8. neg-mul-1 53.8%

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

      9. unsub-neg 53.8%

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

   8. Simplified 53.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -85000000000000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{y + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-253}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x - \left(-1 - x\right)\right)}\\ \end{array} \]

Alternative 7: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -2500000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-258}:\\ \;\;\;\;y \cdot \frac{t_0}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -6.8e+174)
     (/ (/ y (+ x (+ y 1.0))) (+ x y))
     (if (<= x -2500000.0)
       (* (/ y x) (/ x (* (+ x y) (+ x y))))
       (if (<= x -5e-258) (* y (/ t_0 (+ x y))) (/ t_0 (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -6.8e+174) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -2500000.0) {
		tmp = (y / x) * (x / ((x + y) * (x + y)));
	} else if (x <= -5e-258) {
		tmp = y * (t_0 / (x + y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (x <= (-6.8d+174)) then
        tmp = (y / (x + (y + 1.0d0))) / (x + y)
    else if (x <= (-2500000.0d0)) then
        tmp = (y / x) * (x / ((x + y) * (x + y)))
    else if (x <= (-5d-258)) then
        tmp = y * (t_0 / (x + y))
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -6.8e+174) {
		tmp = (y / (x + (y + 1.0))) / (x + y);
	} else if (x <= -2500000.0) {
		tmp = (y / x) * (x / ((x + y) * (x + y)));
	} else if (x <= -5e-258) {
		tmp = y * (t_0 / (x + y));
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -6.8e+174:
		tmp = (y / (x + (y + 1.0))) / (x + y)
	elif x <= -2500000.0:
		tmp = (y / x) * (x / ((x + y) * (x + y)))
	elif x <= -5e-258:
		tmp = y * (t_0 / (x + y))
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -6.8e+174)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(x + y));
	elseif (x <= -2500000.0)
		tmp = Float64(Float64(y / x) * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	elseif (x <= -5e-258)
		tmp = Float64(y * Float64(t_0 / Float64(x + y)));
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -6.8e+174)
		tmp = (y / (x + (y + 1.0))) / (x + y);
	elseif (x <= -2500000.0)
		tmp = (y / x) * (x / ((x + y) * (x + y)));
	elseif (x <= -5e-258)
		tmp = y * (t_0 / (x + y));
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e+174], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2500000.0], N[(N[(y / x), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-258], N[(y * N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.8000000000000002e174

   1. Initial program 62.5%

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

      1. times-frac 80.0%

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

      2. associate-+l+ 80.0%

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

   3. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. clear-num 80.0%

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

      3. associate-/r* 99.9%

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

      4. frac-times 97.2%

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

      5. *-un-lft-identity 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in x around inf 97.0%

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

      1. expm1-log1p-u 97.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}\right)\right)} \]

      2. expm1-udef 80.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}\right)} - 1} \]

      3. associate-/r* 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{\frac{x + \left(y + 1\right)}{y}}}{x + y}}\right)} - 1 \]

      4. clear-num 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{y}{x + \left(y + 1\right)}}}{x + y}\right)} - 1 \]

      5. +-commutative 80.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{\color{blue}{y + x}}\right)} - 1 \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   10. Simplified 99.7%

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

   if -6.8000000000000002e174 < x < -2.5e6

   1. Initial program 63.8%

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

      1. times-frac 87.1%

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

      2. associate-+l+ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around inf 83.3%

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

   if -2.5e6 < x < -4.9999999999999999e-258

   1. Initial program 79.2%

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

      1. times-frac 90.1%

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

      2. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. +-commutative 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in y around 0 84.5%

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

   if -4.9999999999999999e-258 < 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. times-frac 84.0%

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

      2. associate-+l+ 84.0%

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

   3. Simplified 84.0%

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

      1. *-commutative 84.0%

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

      2. clear-num 83.9%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.1%

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

      5. *-un-lft-identity 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in x around 0 52.8%

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

      1. +-commutative 52.8%

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

   8. Simplified 52.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{x + y}\\ \mathbf{elif}\;x \leq -2500000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-258}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 8: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{t_0}{x + y}\\ \mathbf{if}\;x \leq -2500000:\\ \;\;\;\;t_1 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-255}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + \left(x - \left(-1 - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (/ t_0 (+ x y))))
   (if (<= x -2500000.0)
     (* t_1 (/ y x))
     (if (<= x -1.45e-255) (* y t_1) (/ t_0 (+ y (- x (- -1.0 x))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -2500000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -1.45e-255) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + (x - (-1.0 - x)));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = t_0 / (x + y)
    if (x <= (-2500000.0d0)) then
        tmp = t_1 * (y / x)
    else if (x <= (-1.45d-255)) then
        tmp = y * t_1
    else
        tmp = t_0 / (y + (x - ((-1.0d0) - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -2500000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -1.45e-255) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + (x - (-1.0 - x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = t_0 / (x + y)
	tmp = 0
	if x <= -2500000.0:
		tmp = t_1 * (y / x)
	elif x <= -1.45e-255:
		tmp = y * t_1
	else:
		tmp = t_0 / (y + (x - (-1.0 - x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(t_0 / Float64(x + y))
	tmp = 0.0
	if (x <= -2500000.0)
		tmp = Float64(t_1 * Float64(y / x));
	elseif (x <= -1.45e-255)
		tmp = Float64(y * t_1);
	else
		tmp = Float64(t_0 / Float64(y + Float64(x - Float64(-1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = t_0 / (x + y);
	tmp = 0.0;
	if (x <= -2500000.0)
		tmp = t_1 * (y / x);
	elseif (x <= -1.45e-255)
		tmp = y * t_1;
	else
		tmp = t_0 / (y + (x - (-1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2500000.0], N[(t$95$1 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-255], N[(y * t$95$1), $MachinePrecision], N[(t$95$0 / N[(y + N[(x - N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e6

   1. Initial program 63.2%

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

      1. times-frac 83.8%

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

      2. associate-+l+ 83.8%

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

   3. Simplified 83.8%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 92.5%

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

   if -2.5e6 < x < -1.45000000000000003e-255

   1. Initial program 80.4%

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

      1. times-frac 91.3%

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

      2. associate-+l+ 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. +-commutative 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in y around 0 84.2%

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

   if -1.45000000000000003e-255 < x

   1. Initial program 64.6%

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

      1. times-frac 83.4%

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

      2. associate-+l+ 83.4%

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

   3. Simplified 83.4%

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

      1. *-commutative 83.4%

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

      2. clear-num 83.3%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.1%

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

      5. *-un-lft-identity 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in y around -inf 53.8%

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

      1. mul-1-neg 53.8%

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

      2. unsub-neg 53.8%

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

      3. neg-mul-1 53.8%

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

      4. +-commutative 53.8%

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

      5. unsub-neg 53.8%

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

      6. distribute-lft-in 53.8%

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

      7. metadata-eval 53.8%

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

      8. neg-mul-1 53.8%

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

      9. unsub-neg 53.8%

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

   8. Simplified 53.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-255}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + \left(x - \left(-1 - x\right)\right)}\\ \end{array} \]

Alternative 9: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;t_0 \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x (+ x y)) (+ x y))))
   (if (<= x -3.6e+14) (* t_0 (/ y x)) (* t_0 (/ y (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -3.6e+14) {
		tmp = t_0 * (y / x);
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / (x + y)) / (x + y)
    if (x <= (-3.6d+14)) then
        tmp = t_0 * (y / x)
    else
        tmp = t_0 * (y / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / (x + y)) / (x + y);
	double tmp;
	if (x <= -3.6e+14) {
		tmp = t_0 * (y / x);
	} else {
		tmp = t_0 * (y / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / (x + y)) / (x + y)
	tmp = 0
	if x <= -3.6e+14:
		tmp = t_0 * (y / x)
	else:
		tmp = t_0 * (y / (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / Float64(x + y)) / Float64(x + y))
	tmp = 0.0
	if (x <= -3.6e+14)
		tmp = Float64(t_0 * Float64(y / x));
	else
		tmp = Float64(t_0 * Float64(y / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / (x + y)) / (x + y);
	tmp = 0.0;
	if (x <= -3.6e+14)
		tmp = t_0 * (y / x);
	else
		tmp = t_0 * (y / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.6e+14], N[(t$95$0 * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 94.1%

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

   if -3.6e14 < x

   1. Initial program 69.9%

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

      1. times-frac 86.3%

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

      2. associate-+l+ 86.3%

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

   3. Simplified 86.3%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 85.2%

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

      1. +-commutative 85.2%

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

   10. Simplified 85.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{y + 1}\\ \end{array} \]

Alternative 10: 54.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e+14)
   (/ y (* x x))
   (if (<= x -7.5e-23)
     (/ x (* y y))
     (if (<= x -4e-137)
       (/ y x)
       (if (<= x 4.6e-179) (/ 1.0 (/ y x)) (/ 1.0 (* y (/ y x))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+14) {
		tmp = y / (x * x);
	} else if (x <= -7.5e-23) {
		tmp = x / (y * y);
	} else if (x <= -4e-137) {
		tmp = y / x;
	} else if (x <= 4.6e-179) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.65d+14)) then
        tmp = y / (x * x)
    else if (x <= (-7.5d-23)) then
        tmp = x / (y * y)
    else if (x <= (-4d-137)) then
        tmp = y / x
    else if (x <= 4.6d-179) then
        tmp = 1.0d0 / (y / x)
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65e+14) {
		tmp = y / (x * x);
	} else if (x <= -7.5e-23) {
		tmp = x / (y * y);
	} else if (x <= -4e-137) {
		tmp = y / x;
	} else if (x <= 4.6e-179) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65e+14:
		tmp = y / (x * x)
	elif x <= -7.5e-23:
		tmp = x / (y * y)
	elif x <= -4e-137:
		tmp = y / x
	elif x <= 4.6e-179:
		tmp = 1.0 / (y / x)
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65e+14)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -7.5e-23)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -4e-137)
		tmp = Float64(y / x);
	elseif (x <= 4.6e-179)
		tmp = Float64(1.0 / Float64(y / x));
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e+14)
		tmp = y / (x * x);
	elseif (x <= -7.5e-23)
		tmp = x / (y * y);
	elseif (x <= -4e-137)
		tmp = y / x;
	elseif (x <= 4.6e-179)
		tmp = 1.0 / (y / x);
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65e+14], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-23], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-137], N[(y / x), $MachinePrecision], If[LessEqual[x, 4.6e-179], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.65e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in x around inf 75.2%

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

      1. unpow2 75.2%

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

   6. Simplified 75.2%

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

   if -1.65e14 < x < -7.4999999999999998e-23

   1. Initial program 74.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around inf 90.7%

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

      1. unpow2 90.7%

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

   10. Simplified 90.7%

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

   if -7.4999999999999998e-23 < x < -3.99999999999999991e-137

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

   if -3.99999999999999991e-137 < x < 4.59999999999999975e-179

   1. Initial program 64.2%

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

      1. times-frac 71.2%

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

      2. associate-+l+ 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in x around 0 83.6%

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

      1. associate-/r* 83.6%

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

      2. +-commutative 83.6%

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

   6. Simplified 83.6%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

      3. div-inv 82.5%

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

      4. clear-num 82.5%

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

   8. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

      2. *-commutative 82.5%

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

   10. Simplified 82.5%

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

   11. Taylor expanded in y around 0 73.3%

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

   if 4.59999999999999975e-179 < x

   1. Initial program 68.4%

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

      1. times-frac 91.4%

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

      2. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around 0 40.7%

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

      1. associate-/r* 41.4%

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

      2. +-commutative 41.4%

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

   6. Simplified 41.4%

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

      1. clear-num 40.7%

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

      2. inv-pow 40.6%

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

      3. div-inv 40.6%

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

      4. clear-num 40.6%

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

   8. Applied egg-rr 40.6%

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

      1. unpow-1 40.7%

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

      2. *-commutative 40.7%

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

   10. Simplified 40.7%

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

   11. Taylor expanded in y around inf 37.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{{y}^{2}}{x}}} \]
   12. Step-by-step derivation

       1. unpow2 37.9%

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

       2. associate-*l/ 37.9%

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

       3. *-commutative 37.9%

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

   13. Simplified 37.9%

       \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{y}{x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-179}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 11: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \frac{t_0}{x + y}\\ \mathbf{if}\;x \leq -2500000:\\ \;\;\;\;t_1 \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-252}:\\ \;\;\;\;y \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (/ t_0 (+ x y))))
   (if (<= x -2500000.0)
     (* t_1 (/ y x))
     (if (<= x -5.4e-252) (* y t_1) (/ t_0 (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -2500000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -5.4e-252) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (x + y)
    t_1 = t_0 / (x + y)
    if (x <= (-2500000.0d0)) then
        tmp = t_1 * (y / x)
    else if (x <= (-5.4d-252)) then
        tmp = y * t_1
    else
        tmp = t_0 / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = t_0 / (x + y);
	double tmp;
	if (x <= -2500000.0) {
		tmp = t_1 * (y / x);
	} else if (x <= -5.4e-252) {
		tmp = y * t_1;
	} else {
		tmp = t_0 / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = t_0 / (x + y)
	tmp = 0
	if x <= -2500000.0:
		tmp = t_1 * (y / x)
	elif x <= -5.4e-252:
		tmp = y * t_1
	else:
		tmp = t_0 / (y + 1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(t_0 / Float64(x + y))
	tmp = 0.0
	if (x <= -2500000.0)
		tmp = Float64(t_1 * Float64(y / x));
	elseif (x <= -5.4e-252)
		tmp = Float64(y * t_1);
	else
		tmp = Float64(t_0 / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	t_1 = t_0 / (x + y);
	tmp = 0.0;
	if (x <= -2500000.0)
		tmp = t_1 * (y / x);
	elseif (x <= -5.4e-252)
		tmp = y * t_1;
	else
		tmp = t_0 / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2500000.0], N[(t$95$1 * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.4e-252], N[(y * t$95$1), $MachinePrecision], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e6

   1. Initial program 63.2%

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

      1. times-frac 83.8%

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

      2. associate-+l+ 83.8%

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

   3. Simplified 83.8%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 92.5%

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

   if -2.5e6 < x < -5.39999999999999962e-252

   1. Initial program 81.5%

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

      1. times-frac 92.6%

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

      2. associate-+l+ 92.6%

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

   3. Simplified 92.6%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. +-commutative 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in y around 0 84.0%

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

   if -5.39999999999999962e-252 < x

   1. Initial program 64.1%

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

      1. times-frac 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. clear-num 82.7%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.1%

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

      5. *-un-lft-identity 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in x around 0 53.2%

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

      1. +-commutative 53.2%

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

   8. Simplified 53.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2500000:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-252}:\\ \;\;\;\;y \cdot \frac{\frac{x}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 12: 53.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -1.05e+14)
     (/ y (* x x))
     (if (<= x -1.5e-26)
       t_0
       (if (<= x -2.1e-137) (/ y x) (if (<= x 2e-188) (/ 1.0 (/ y x)) t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.05e+14) {
		tmp = y / (x * x);
	} else if (x <= -1.5e-26) {
		tmp = t_0;
	} else if (x <= -2.1e-137) {
		tmp = y / x;
	} else if (x <= 2e-188) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-1.05d+14)) then
        tmp = y / (x * x)
    else if (x <= (-1.5d-26)) then
        tmp = t_0
    else if (x <= (-2.1d-137)) then
        tmp = y / x
    else if (x <= 2d-188) then
        tmp = 1.0d0 / (y / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.05e+14) {
		tmp = y / (x * x);
	} else if (x <= -1.5e-26) {
		tmp = t_0;
	} else if (x <= -2.1e-137) {
		tmp = y / x;
	} else if (x <= 2e-188) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -1.05e+14:
		tmp = y / (x * x)
	elif x <= -1.5e-26:
		tmp = t_0
	elif x <= -2.1e-137:
		tmp = y / x
	elif x <= 2e-188:
		tmp = 1.0 / (y / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -1.05e+14)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.5e-26)
		tmp = t_0;
	elseif (x <= -2.1e-137)
		tmp = Float64(y / x);
	elseif (x <= 2e-188)
		tmp = Float64(1.0 / Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -1.05e+14)
		tmp = y / (x * x);
	elseif (x <= -1.5e-26)
		tmp = t_0;
	elseif (x <= -2.1e-137)
		tmp = y / x;
	elseif (x <= 2e-188)
		tmp = 1.0 / (y / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+14], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-26], t$95$0, If[LessEqual[x, -2.1e-137], N[(y / x), $MachinePrecision], If[LessEqual[x, 2e-188], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in x around inf 75.2%

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

      1. unpow2 75.2%

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

   6. Simplified 75.2%

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

   if -1.05e14 < x < -1.50000000000000006e-26 or 1.9999999999999999e-188 < x

   1. Initial program 68.0%

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

      1. times-frac 90.7%

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

      2. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around inf 42.3%

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

      1. unpow2 42.3%

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

   10. Simplified 42.3%

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

   if -1.50000000000000006e-26 < x < -2.09999999999999992e-137

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

   if -2.09999999999999992e-137 < x < 1.9999999999999999e-188

   1. Initial program 65.6%

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

      1. times-frac 72.9%

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

      2. associate-+l+ 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in x around 0 85.8%

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

      1. associate-/r* 85.8%

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

      2. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. clear-num 84.6%

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

      2. inv-pow 84.6%

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

      3. div-inv 84.7%

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

      4. clear-num 84.6%

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

   8. Applied egg-rr 84.6%

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

      1. unpow-1 84.6%

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

      2. *-commutative 84.6%

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

   10. Simplified 84.6%

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

   11. Taylor expanded in y around 0 75.0%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-188}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 13: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-25} \lor \neg \left(x \leq -7.5 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+15)
   (/ y (* x x))
   (if (or (<= x -4.8e-25) (not (<= x -7.5e-135)))
     (/ x (* y (+ y 1.0)))
     (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+15) {
		tmp = y / (x * x);
	} else if ((x <= -4.8e-25) || !(x <= -7.5e-135)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.9d+15)) then
        tmp = y / (x * x)
    else if ((x <= (-4.8d-25)) .or. (.not. (x <= (-7.5d-135)))) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+15) {
		tmp = y / (x * x);
	} else if ((x <= -4.8e-25) || !(x <= -7.5e-135)) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e+15:
		tmp = y / (x * x)
	elif (x <= -4.8e-25) or not (x <= -7.5e-135):
		tmp = x / (y * (y + 1.0))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+15)
		tmp = Float64(y / Float64(x * x));
	elseif ((x <= -4.8e-25) || !(x <= -7.5e-135))
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e+15)
		tmp = y / (x * x);
	elseif ((x <= -4.8e-25) || ~((x <= -7.5e-135)))
		tmp = x / (y * (y + 1.0));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+15], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.8e-25], N[Not[LessEqual[x, -7.5e-135]], $MachinePrecision]], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9e15

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in x around inf 75.2%

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

      1. unpow2 75.2%

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

   6. Simplified 75.2%

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

   if -4.9e15 < x < -4.80000000000000018e-25 or -7.5e-135 < x

   1. Initial program 67.1%

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

      1. times-frac 84.0%

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

      2. associate-+l+ 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in x around 0 60.7%

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

   if -4.80000000000000018e-25 < x < -7.5e-135

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-25} \lor \neg \left(x \leq -7.5 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 14: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e+14)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -1.2e-25)
     (/ x (* y (+ y 1.0)))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+14) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1.2e-25) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.4d+14)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-1.2d-25)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.4e+14) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1.2e-25) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.4e+14:
		tmp = (y / x) / (x + 1.0)
	elif x <= -1.2e-25:
		tmp = x / (y * (y + 1.0))
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e+14)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -1.2e-25)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.4e+14)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -1.2e-25)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e+14], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.2e-25], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.4e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. associate-/r* 84.7%

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

      2. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   if -5.4e14 < x < -1.20000000000000005e-25

   1. Initial program 74.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 90.8%

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

   if -1.20000000000000005e-25 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

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

      1. times-frac 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   6. Simplified 59.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 15: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+14)
   (/ (/ y (+ x y)) x)
   (if (<= x -1.8e-23)
     (/ x (* y (+ y 1.0)))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+14) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.8e-23) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d+14)) then
        tmp = (y / (x + y)) / x
    else if (x <= (-1.8d-23)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+14) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.8e-23) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e+14:
		tmp = (y / (x + y)) / x
	elif x <= -1.8e-23:
		tmp = x / (y * (y + 1.0))
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+14)
		tmp = Float64(Float64(y / Float64(x + y)) / x);
	elseif (x <= -1.8e-23)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+14)
		tmp = (y / (x + y)) / x;
	elseif (x <= -1.8e-23)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e+14], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.8e-23], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. frac-times 63.5%

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

      2. *-commutative 63.5%

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

      3. frac-times 83.2%

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

      4. clear-num 83.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.9%

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

      7. *-un-lft-identity 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 84.9%

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

   if -5.5e14 < x < -1.7999999999999999e-23

   1. Initial program 74.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 90.8%

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

   if -1.7999999999999999e-23 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

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

      1. times-frac 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   6. Simplified 59.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e+14)
   (/ (/ y (+ x y)) x)
   (if (<= x -1.5e-26)
     (/ (/ x (+ x y)) (+ y 1.0))
     (if (<= x -1e-134) (/ y x) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+14) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.5e-26) {
		tmp = (x / (x + y)) / (y + 1.0);
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.5d+14)) then
        tmp = (y / (x + y)) / x
    else if (x <= (-1.5d-26)) then
        tmp = (x / (x + y)) / (y + 1.0d0)
    else if (x <= (-1d-134)) then
        tmp = y / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e+14) {
		tmp = (y / (x + y)) / x;
	} else if (x <= -1.5e-26) {
		tmp = (x / (x + y)) / (y + 1.0);
	} else if (x <= -1e-134) {
		tmp = y / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e+14:
		tmp = (y / (x + y)) / x
	elif x <= -1.5e-26:
		tmp = (x / (x + y)) / (y + 1.0)
	elif x <= -1e-134:
		tmp = y / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e+14)
		tmp = Float64(Float64(y / Float64(x + y)) / x);
	elseif (x <= -1.5e-26)
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	elseif (x <= -1e-134)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e+14)
		tmp = (y / (x + y)) / x;
	elseif (x <= -1.5e-26)
		tmp = (x / (x + y)) / (y + 1.0);
	elseif (x <= -1e-134)
		tmp = y / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e+14], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.5e-26], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-134], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.5e14

   1. Initial program 63.5%

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

      1. times-frac 83.2%

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

      2. associate-+l+ 83.2%

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

   3. Simplified 83.2%

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

      1. frac-times 63.5%

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

      2. *-commutative 63.5%

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

      3. frac-times 83.2%

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

      4. clear-num 83.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.9%

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

      7. *-un-lft-identity 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 84.9%

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

   if -7.5e14 < x < -1.50000000000000006e-26

   1. Initial program 74.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. clear-num 99.6%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.7%

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

      5. *-un-lft-identity 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in x around 0 91.1%

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

      1. +-commutative 91.1%

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

   8. Simplified 91.1%

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

   if -1.50000000000000006e-26 < x < -1.00000000000000004e-134

   1. Initial program 86.1%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 45.1%

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

      1. associate-/r* 45.1%

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

      2. +-commutative 45.1%

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

   6. Simplified 45.1%

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

   7. Taylor expanded in x around 0 45.1%

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

   if -1.00000000000000004e-134 < x

   1. Initial program 66.6%

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

      1. times-frac 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. associate-/r* 59.1%

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

      2. +-commutative 59.1%

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

   6. Simplified 59.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-134}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17: 61.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.2e-77)
   (/ y (* x (+ x 1.0)))
   (if (<= y 1.9e+165) (/ x (* y (+ y 1.0))) (/ 1.0 (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-77) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.9e+165) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.2d-77) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 1.9d+165) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.2e-77) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 1.9e+165) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.2e-77:
		tmp = y / (x * (x + 1.0))
	elif y <= 1.9e+165:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.2e-77)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 1.9e+165)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.2e-77)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 1.9e+165)
		tmp = x / (y * (y + 1.0));
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.2e-77], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+165], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.19999999999999994e-77

   1. Initial program 67.4%

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

      1. times-frac 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in y around 0 59.1%

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

   if 9.19999999999999994e-77 < y < 1.89999999999999995e165

   1. Initial program 80.2%

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

      1. times-frac 93.9%

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

      2. associate-+l+ 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in x around 0 65.2%

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

   if 1.89999999999999995e165 < y

   1. Initial program 53.1%

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

      1. times-frac 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. associate-/r* 88.0%

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

      2. +-commutative 88.0%

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

   6. Simplified 88.0%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

      3. div-inv 85.9%

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

      4. clear-num 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. *-commutative 85.9%

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

   10. Simplified 85.9%

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

   11. Taylor expanded in y around inf 80.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{{y}^{2}}{x}}} \]
   12. Step-by-step derivation

       1. unpow2 80.1%

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

       2. associate-*l/ 85.9%

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

       3. *-commutative 85.9%

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

   13. Simplified 85.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 18: 61.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.8e-77)
   (/ y (+ x (* x x)))
   (if (<= y 2.3e+165) (/ x (* y (+ y 1.0))) (/ 1.0 (* y (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-77) {
		tmp = y / (x + (x * x));
	} else if (y <= 2.3e+165) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.8d-77) then
        tmp = y / (x + (x * x))
    else if (y <= 2.3d+165) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = 1.0d0 / (y * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-77) {
		tmp = y / (x + (x * x));
	} else if (y <= 2.3e+165) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = 1.0 / (y * (y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.8e-77:
		tmp = y / (x + (x * x))
	elif y <= 2.3e+165:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = 1.0 / (y * (y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.8e-77)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	elseif (y <= 2.3e+165)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.8e-77)
		tmp = y / (x + (x * x));
	elseif (y <= 2.3e+165)
		tmp = x / (y * (y + 1.0));
	else
		tmp = 1.0 / (y * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.8e-77], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+165], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.7999999999999994e-77

   1. Initial program 67.4%

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

      1. times-frac 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

      1. frac-times 67.4%

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

      2. *-commutative 67.4%

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

      3. frac-times 83.8%

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

      4. clear-num 83.7%

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

      5. associate-/r* 99.7%

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

      6. frac-times 99.3%

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

      7. *-un-lft-identity 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. distribute-rgt-in 59.1%

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

      2. *-lft-identity 59.1%

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

   8. Simplified 59.1%

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

   if 9.7999999999999994e-77 < y < 2.30000000000000016e165

   1. Initial program 80.2%

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

      1. times-frac 93.9%

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

      2. associate-+l+ 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in x around 0 65.2%

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

   if 2.30000000000000016e165 < y

   1. Initial program 53.1%

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

      1. times-frac 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. associate-/r* 88.0%

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

      2. +-commutative 88.0%

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

   6. Simplified 88.0%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

      3. div-inv 85.9%

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

      4. clear-num 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. *-commutative 85.9%

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

   10. Simplified 85.9%

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

   11. Taylor expanded in y around inf 80.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{{y}^{2}}{x}}} \]
   12. Step-by-step derivation

       1. unpow2 80.1%

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

       2. associate-*l/ 85.9%

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

       3. *-commutative 85.9%

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

   13. Simplified 85.9%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 19: 44.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-183) (/ y x) (if (<= y 0.75) (- (/ x y) x) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-183) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-183) then
        tmp = y / x
    else if (y <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-183) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-183:
		tmp = y / x
	elif y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-183)
		tmp = Float64(y / x);
	elseif (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-183)
		tmp = y / x;
	elseif (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-183], N[(y / x), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1000000000000002e-183

   1. Initial program 65.7%

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

      1. times-frac 83.0%

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

      2. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. associate-/r* 58.2%

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

      2. +-commutative 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in x around 0 38.4%

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

   if 2.1000000000000002e-183 < y < 0.75

   1. Initial program 88.3%

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

      1. times-frac 94.9%

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

      2. associate-+l+ 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in x around 0 34.1%

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

      1. associate-/r* 34.2%

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

      2. +-commutative 34.2%

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

   6. Simplified 34.2%

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

   7. Taylor expanded in y around 0 34.2%

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

      1. neg-mul-1 34.2%

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

      2. +-commutative 34.2%

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

      3. unsub-neg 34.2%

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

   9. Simplified 34.2%

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

   if 0.75 < y

   1. Initial program 63.5%

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

      1. times-frac 86.0%

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

      2. associate-+l+ 86.0%

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

   3. Simplified 86.0%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around inf 75.6%

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

      1. unpow2 75.6%

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

   10. Simplified 75.6%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 20: 61.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.3e-77) (/ y (+ x (* x x))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-77) {
		tmp = y / (x + (x * x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.3d-77) then
        tmp = y / (x + (x * x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.3e-77) {
		tmp = y / (x + (x * x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.3e-77:
		tmp = y / (x + (x * x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.3e-77)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.3e-77)
		tmp = y / (x + (x * x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.3e-77], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.29999999999999999e-77

   1. Initial program 67.4%

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

      1. times-frac 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

      1. frac-times 67.4%

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

      2. *-commutative 67.4%

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

      3. frac-times 83.8%

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

      4. clear-num 83.7%

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

      5. associate-/r* 99.7%

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

      6. frac-times 99.3%

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

      7. *-un-lft-identity 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in y around 0 59.1%

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

      1. distribute-rgt-in 59.1%

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

      2. *-lft-identity 59.1%

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

   8. Simplified 59.1%

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

   if 2.29999999999999999e-77 < y

   1. Initial program 70.9%

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

      1. times-frac 89.1%

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

      2. associate-+l+ 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in x around 0 70.3%

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

      1. associate-/r* 71.7%

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

      2. +-commutative 71.7%

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

   6. Simplified 71.7%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 21: 34.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.6e-183) (/ y x) (/ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-183) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.6d-183) then
        tmp = y / x
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-183) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.6e-183:
		tmp = y / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-183)
		tmp = Float64(y / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-183)
		tmp = y / x;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.6e-183], N[(y / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6000000000000001e-183

   1. Initial program 65.7%

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

      1. times-frac 83.0%

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

      2. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in y around 0 56.4%

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

      1. associate-/r* 58.2%

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

      2. +-commutative 58.2%

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

   6. Simplified 58.2%

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

   7. Taylor expanded in x around 0 38.4%

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

   if 1.6000000000000001e-183 < y

   1. Initial program 72.6%

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

      1. times-frac 89.2%

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

      2. associate-+l+ 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

   6. Simplified 63.2%

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

      1. clear-num 62.5%

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

      2. inv-pow 62.5%

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

      3. div-inv 62.5%

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

      4. clear-num 62.5%

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

   8. Applied egg-rr 62.5%

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

      1. unpow-1 62.6%

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

      2. *-commutative 62.6%

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

   10. Simplified 62.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{y}{x} \cdot \left(y + 1\right)}} \]

   11. Taylor expanded in y around 0 30.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]

Alternative 22: 33.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2.2e-183) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-183) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-183) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-183) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-183:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-183)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-183)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-183], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2e-183

   1. Initial program 65.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 83.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 83.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.4%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 58.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

      2. +-commutative 58.2%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x + 1}} \]

   6. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

   7. Taylor expanded in x around 0 38.4%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 2.2e-183 < y

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 89.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 62.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. associate-/r* 63.2%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 63.2%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   6. Simplified 63.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

   7. Taylor expanded in y around 0 29.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 23: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 y))

C

double code(double x, double y) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y) {
	return 1.0 / y;
}

Python

def code(x, y):
	return 1.0 / y

Julia

function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]
4. Step-by-step derivation

   1. *-commutative 85.6%

      \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   2. clear-num 85.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(y + 1\right)}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

   3. associate-/r* 99.7%

      \[\leadsto \frac{1}{\frac{x + \left(y + 1\right)}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. frac-times 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

   5. *-un-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{x + y}}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)}} \]

6. Taylor expanded in x around inf 50.5%

   \[\leadsto \frac{\color{blue}{1}}{\frac{x + \left(y + 1\right)}{y} \cdot \left(x + y\right)} \]

7. Taylor expanded in y around inf 4.0%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

8. Final simplification 4.0%

   \[\leadsto \frac{1}{y} \]

Alternative 24: 25.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 85.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 85.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around 0 50.6%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
5. Step-by-step derivation

   1. associate-/r* 51.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

   2. +-commutative 51.4%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

6. Simplified 51.4%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]

7. Taylor expanded in y around 0 26.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

8. Final simplification 26.6%

   \[\leadsto \frac{x}{y} \]

Developer target: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))