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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 6

Speedup: 1.0×

• 25.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. *-commutative 99.9%

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

   2. flip3-- 68.9%

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

   3. associate-*l/ 68.6%

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

   4. metadata-eval 68.6%

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

   5. metadata-eval 68.6%

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

   6. +-commutative 68.6%

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

   7. distribute-rgt-out 68.6%

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

   8. +-commutative 68.6%

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

   9. fma-define 68.6%

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

   10. +-commutative 68.6%

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

6. Applied egg-rr 68.6%

   \[\leadsto y \cdot \color{blue}{\frac{\left(1 - {y}^{3}\right) \cdot x}{\mathsf{fma}\left(y, y + 1, 1\right)}} \]
7. Step-by-step derivation

   1. *-commutative 68.6%

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

   2. associate-/l* 68.9%

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

   3. metadata-eval 68.9%

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

   4. *-rgt-identity 68.9%

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

   5. fma-undefine 68.9%

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

   6. +-commutative 68.9%

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

   7. metadata-eval 68.9%

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

   8. distribute-lft-in 68.9%

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

   9. *-rgt-identity 68.9%

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

   10. *-rgt-identity 68.9%

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

   11. *-un-lft-identity 68.9%

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

   12. flip3-- 99.9%

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

   13. sub-neg 99.9%

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

   14. *-rgt-identity 99.9%

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

   15. distribute-rgt-out 99.9%

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

   16. *-un-lft-identity 99.9%

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

   17. distribute-lft-neg-out 99.9%

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

   18. *-commutative 99.9%

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

   19. unsub-neg 99.9%

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

   20. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto y \cdot \left(x - y \cdot x\right) \]
10. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+62} \lor \neg \left(y \leq 2.9 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+62) (not (<= y 2.9e+20)))
   (* y (* y (- x)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+62) || !(y <= 2.9e+20)) {
		tmp = y * (y * -x);
	} 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 <= (-5d+62)) .or. (.not. (y <= 2.9d+20))) then
        tmp = y * (y * -x)
    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 <= -5e+62) || !(y <= 2.9e+20)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+62) or not (y <= 2.9e+20):
		tmp = y * (y * -x)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+62) || !(y <= 2.9e+20))
		tmp = Float64(y * Float64(y * Float64(-x)));
	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 <= -5e+62) || ~((y <= 2.9e+20)))
		tmp = y * (y * -x);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+62], N[Not[LessEqual[y, 2.9e+20]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000029e62 or 2.9e20 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

   7. Simplified 99.8%

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

   if -5.00000000000000029e62 < y < 2.9e20

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+62} \lor \neg \left(y \leq 2.9 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y * (y * -x);
	else
		tmp = y * x;
	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[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(y * x), $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. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-out 98.1%

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

   7. Simplified 98.1%

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

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.1%

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

4. Final simplification 98.1%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto y \cdot \left(x \cdot \left(1 - y\right)\right) \]
6. Add Preprocessing

Alternative 5: 55.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 56.3%

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

6. Final simplification 56.3%

   \[\leadsto y \cdot x \]
7. Add Preprocessing

Alternative 6: 2.7% 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* 92.9%

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

3. Simplified 92.9%

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

   1. associate-*r* 99.9%

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

   2. flip-- 92.9%

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

   3. associate-*r/ 91.5%

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

   4. metadata-eval 91.5%

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

   5. pow2 91.5%

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

   6. +-commutative 91.5%

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

6. Applied egg-rr 91.5%

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

   1. associate-*l* 87.7%

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

   2. *-commutative 87.7%

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

   3. associate-/l* 83.1%

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

   4. sub-neg 83.1%

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

   5. distribute-lft-in 83.1%

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

   6. *-rgt-identity 83.1%

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

   7. distribute-rgt-neg-in 83.1%

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

   8. unpow2 83.1%

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

   9. cube-mult 83.2%

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

   10. unsub-neg 83.2%

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

8. Simplified 83.2%

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

9. Taylor expanded in y around inf 34.0%

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

10. Taylor expanded in y around 0 2.7%

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

11. Final simplification 2.7%

    \[\leadsto x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))