Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 2.3min

Alternatives: 6

Speedup: 1.0×

• 25.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) * (1.0 - y)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) * (1.0 - y)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+85} \lor \neg \left(y \leq 5 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2e+85) (not (<= y 5e+70)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2e+85) || !(y <= 5e+70)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2d+85)) .or. (.not. (y <= 5d+70))) then
        tmp = y * (x * -y)
    else
        tmp = x * (y * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2e+85) || !(y <= 5e+70)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2e+85) or not (y <= 5e+70):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2e+85) || !(y <= 5e+70))
		tmp = Float64(y * Float64(x * Float64(-y)));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2e+85) || ~((y <= 5e+70)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2e+85], N[Not[LessEqual[y, 5e+70]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e85 or 5.0000000000000002e70 < y

   1. Initial program 99.9%

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

      1. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-*l* 99.9%

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

      3. flip-- 82.1%

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

      4. associate-*l/ 82.0%

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

      5. clear-num 82.0%

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

      6. un-div-inv 82.0%

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

      7. clear-num 81.9%

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

      8. associate-*l/ 82.0%

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

      9. flip-- 99.9%

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

      10. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around inf 99.9%

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

      1. div-inv 99.9%

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

      2. frac-2neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. remove-double-div 99.9%

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

      5. distribute-lft-neg-in 99.9%

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

      6. associate-*r* 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -2e85 < y < 5.0000000000000002e70

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+85} \lor \neg \left(y \leq 5 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 2: 92.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* x (- (* y y))) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = x * -(y * y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = x * -(y * y)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = x * -(y * y)
	else:
		tmp = x * y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = x * -(y * y);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x * (-N[(y * y), $MachinePrecision])), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

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

      1. associate-*l* 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in y around inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. *-commutative 84.4%

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

      3. unpow2 84.4%

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

      4. distribute-rgt-neg-in 84.4%

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

   6. Simplified 84.4%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.6%

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

4. Final simplification 91.5%

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

Alternative 3: 97.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0))) (* y (* x (- y))) (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (x * -y)
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = y * (x * -y)
	else:
		tmp = x * y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (x * -y);
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

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

      1. associate-*l* 86.2%

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

   3. Simplified 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*l* 99.8%

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

      3. flip-- 86.1%

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

      4. associate-*l/ 86.1%

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

      5. clear-num 86.0%

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

      6. un-div-inv 86.1%

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

      7. clear-num 86.0%

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

      8. associate-*l/ 86.0%

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

      9. flip-- 99.7%

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

      10. *-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 98.0%

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

      1. div-inv 98.0%

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

      2. frac-2neg 98.0%

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

      3. metadata-eval 98.0%

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

      4. remove-double-div 98.1%

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

      5. distribute-lft-neg-in 98.1%

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

      6. associate-*r* 98.1%

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

   8. Applied egg-rr 98.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.6%

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

4. Final simplification 98.3%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (* x y) (- 1.0 y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x * y) * (1.0 - y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]

2. Final simplification 99.9%

   \[\leadsto \left(x \cdot y\right) \cdot \left(1 - y\right) \]

Alternative 5: 56.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

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 99.9%

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

   1. associate-*l* 93.1%

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

3. Simplified 93.1%

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

4. Taylor expanded in y around 0 57.1%

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

5. Final simplification 57.1%

   \[\leadsto x \cdot y \]

Alternative 6: 2.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 93.1%

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

3. Simplified 93.1%

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

   1. *-commutative 93.1%

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

   2. associate-*l* 99.9%

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

   3. flip-- 93.1%

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

   4. associate-*l/ 93.0%

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

   5. clear-num 92.9%

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

   6. un-div-inv 92.9%

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

   7. clear-num 92.8%

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

   8. associate-*l/ 92.8%

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

   9. flip-- 99.7%

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

   10. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. associate-/l* 99.9%

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

   3. flip-- 93.0%

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

   4. metadata-eval 93.0%

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

   5. metadata-eval 93.0%

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

   6. associate-/r/ 92.9%

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

   7. times-frac 81.2%

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

   8. metadata-eval 81.2%

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

7. Applied egg-rr 81.2%

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

8. Taylor expanded in y around inf 32.4%

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

9. Taylor expanded in y around 0 2.7%

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

10. Final simplification 2.7%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))