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

Percentage Accurate: 99.9% → 99.9%

Time: 17.1s

Alternatives: 7

Speedup: 1.0×

• 25.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+153} \lor \neg \left(y \leq 2.15 \cdot 10^{+16}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1e+153) (not (<= y 2.15e+16)))
   (* y (* y (- x)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1e+153) || !(y <= 2.15e+16)) {
		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 <= (-1d+153)) .or. (.not. (y <= 2.15d+16))) 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 <= -1e+153) || !(y <= 2.15e+16)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1e+153) or not (y <= 2.15e+16):
		tmp = y * (y * -x)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1e+153) || !(y <= 2.15e+16))
		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 <= -1e+153) || ~((y <= 2.15e+16)))
		tmp = y * (y * -x);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1e+153], N[Not[LessEqual[y, 2.15e+16]], $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 < -1e153 or 2.15e16 < y

   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 inf 99.9%

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

      1. neg-mul-1 99.9%

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

   7. Simplified 99.9%

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

   if -1e153 < y < 2.15e16

   1. Initial program 99.9%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+153} \lor \neg \left(y \leq 2.15 \cdot 10^{+16}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% 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* 99.9%

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

   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 inf 98.3%

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

      1. neg-mul-1 98.3%

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

   7. Simplified 98.3%

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

   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 99.3%

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

4. Final simplification 98.8%

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

Alternative 3: 92.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(x * Float64(y * Float64(-y)));
	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 = x * (y * -y);
	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[(x * N[(y * (-y)), $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. associate-*l* 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in y around inf 86.9%

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

      1. neg-mul-1 98.3%

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

   7. Simplified 86.9%

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

   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 99.3%

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

4. Final simplification 93.3%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

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. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

   1. distribute-lft-neg-out 99.9%

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

   2. unsub-neg 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 5: 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. Add Preprocessing

Alternative 6: 56.0% 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 56.0%

   \[\leadsto y \cdot \color{blue}{x} \]
6. Add Preprocessing

Alternative 7: 4.7% accurate, 3.5× 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 Float64(-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* 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
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-- 94.4%

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

   3. associate-*r/ 90.4%

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

   4. metadata-eval 90.4%

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

   5. pow2 90.4%

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

   6. +-commutative 90.4%

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

6. Applied egg-rr 90.4%

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

   1. associate-*l* 87.0%

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

   2. *-commutative 87.0%

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

   3. associate-/l* 85.4%

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

   4. sub-neg 85.4%

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

   5. unpow2 85.4%

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

   6. distribute-rgt-neg-out 85.4%

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

   7. distribute-lft-in 85.4%

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

   8. *-rgt-identity 85.4%

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

   9. associate-*l* 85.4%

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

   10. sqr-neg 85.4%

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

   11. unpow3 85.4%

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

   12. cube-neg 85.4%

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

   13. unsub-neg 85.4%

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

8. Simplified 85.4%

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

9. Taylor expanded in y around inf 35.9%

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

10. Taylor expanded in y around 0 2.9%

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

    1. frac-2neg 2.9%

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

    2. associate-*r/ 16.3%

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

    3. add-sqr-sqrt 7.6%

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

    4. sqrt-unprod 7.4%

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

    5. sqr-neg 7.4%

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

    6. sqrt-unprod 17.7%

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

    7. add-sqr-sqrt 30.3%

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

12. Applied egg-rr 30.3%

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

    1. *-commutative 30.3%

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

    2. associate-/l* 4.8%

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

    3. *-inverses 4.8%

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

    4. distribute-lft-neg-in 4.8%

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

    5. *-commutative 4.8%

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

    6. metadata-eval 4.8%

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

    7. associate-/r/ 4.8%

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

    8. remove-double-div 4.8%

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

14. Simplified 4.8%

    \[\leadsto \color{blue}{-x} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))