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

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 6

Speedup: 1.0×

• 26.2% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. *-lft-identity N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-lft-identity N/A

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

   7. distribute-lft-in N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-rgt-identity N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x \cdot y\right) \cdot \left(\mathsf{neg}\left(y\right)\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(x \cdot y\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(x \cdot y\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(y \cdot x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{\left(y \cdot x\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)\right) \]

   17. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, x, \left(\left(-y\right) \cdot x\right) \cdot y\right) \]
6. Add Preprocessing

Alternative 2: 97.9% accurate, 0.6× speedup

Math

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

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(Float64(x * y) * 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 = (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[(N[(x * y), $MachinePrecision] * (-y)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot x} \]

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 28.9

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

   5. Applied rewrites 28.9%

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

   7. Applied rewrites 15.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

4. Final simplification 97.4%

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

Alternative 3: 92.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = (-y * 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 <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (-y * 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 <= -1.0) || !(y <= 1.0)) {
		tmp = (-y * y) * x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(Float64(Float64(-y) * y) * x);
	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 * y) * x;
	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[(N[((-y) * y), $MachinePrecision] * x), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot x} \]

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right)} \cdot x \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot x} \]

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 28.9

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

   5. Applied rewrites 28.9%

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

   7. Applied rewrites 15.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

4. Final simplification 89.4%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

Alternative 5: 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 99.8%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 6: 54.7% 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 99.8%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right)} \cdot x \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot x} \]

   5. unpow2 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 55.1

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

5. Applied rewrites 55.1%

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

7. Applied rewrites 29.3%

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

8. Taylor expanded in y around 0

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

   1. lower-*.f64 58.2

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

10. Applied rewrites 58.2%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))