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

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 5

Speedup: 1.0×

• 24.2% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \mathsf{fma}\left(x, 1 - y, 0\right), 0\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y (fma x (- 1.0 y) 0.0) 0.0))

C

double code(double x, double y) {
	return fma(y, fma(x, (1.0 - y), 0.0), 0.0);
}

Julia

function code(x, y)
	return fma(y, fma(x, Float64(1.0 - y), 0.0), 0.0)
end

Wolfram

code[x_, y_] := N[(y * N[(x * N[(1.0 - y), $MachinePrecision] + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-rgt-identity N/A

      \[\leadsto \left(y \cdot \left(1 - y\right)\right) \cdot \color{blue}{\left(x + 0\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right) + 0 \cdot \left(y \cdot \left(1 - y\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \left(1 - y\right)} + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   5. distribute-lft-out-- N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y\right)} + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \left(\color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y\right) + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   7. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(x \cdot y\right)\right) \cdot y\right)} + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \left(x \cdot y + \color{blue}{\left(-1 \cdot \left(x \cdot y\right)\right)} \cdot y\right) + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto \color{blue}{y \cdot \left(x + -1 \cdot \left(x \cdot y\right)\right)} + 0 \cdot \left(y \cdot \left(1 - y\right)\right) \]

   10. mul0-lft N/A

       \[\leadsto y \cdot \left(x + -1 \cdot \left(x \cdot y\right)\right) + \color{blue}{0} \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -1 \cdot \left(x \cdot y\right), 0\right)} \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(y, x + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}, 0\right) \]

   13. unsub-neg N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x - x \cdot y}, 0\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x \cdot 1} - x \cdot y, 0\right) \]

   15. distribute-lft-out-- N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x \cdot \left(1 - y\right)}, 0\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(1 - y\right) \cdot x}, 0\right) \]

   17. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(y, \left(1 - y\right) \cdot \color{blue}{\left(x + 0\right)}, 0\right) \]

   18. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{x \cdot \left(1 - y\right) + 0 \cdot \left(1 - y\right)}, 0\right) \]

   19. mul0-lft N/A

       \[\leadsto \mathsf{fma}\left(y, x \cdot \left(1 - y\right) + \color{blue}{0}, 0\right) \]

   20. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(x, 1 - y, 0\right)}, 0\right) \]

   21. --lowering--.f64 99.9

       \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(x, \color{blue}{1 - y}, 0\right), 0\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(x, 1 - y, 0\right), 0\right)} \]
6. Add Preprocessing

Alternative 2: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (* y x) (- 0.0 (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = 0.0 - (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 <= 1.0d0) then
        tmp = y * x
    else
        tmp = 0.0d0 - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = y * x;
	} else {
		tmp = 0.0 - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = y * x
	else:
		tmp = 0.0 - (y * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(y * x), $MachinePrecision], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 79.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0\right)} \]

   5. Simplified 79.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 79.0

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

   7. Applied egg-rr 79.0%

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

   if 1 < y

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. flip3-- N/A

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

      3. frac-2neg N/A

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

      4. associate-*l/ N/A

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

      5. div-inv N/A

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

      6. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. rgt-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. accelerator-lowering-fma.f64 37.4

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

   7. Simplified 37.4%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 31.2

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

   10. Simplified 31.2%

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

   11. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - y\right)} \]

       3. --lowering--.f64 31.2

          \[\leadsto x \cdot \color{blue}{\left(0 - y\right)} \]

   13. Simplified 31.2%

       \[\leadsto x \cdot \color{blue}{\left(0 - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 56.8% 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. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot y + 0} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot x} + 0 \]

   3. accelerator-lowering-fma.f64 59.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0\right)} \]

5. Simplified 59.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 59.4

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

7. Applied egg-rr 59.4%

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

Alternative 5: 2.8% accurate, 14.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. flip3-- N/A

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

   3. frac-2neg N/A

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

   4. associate-*l/ N/A

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

   5. div-inv N/A

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

   6. *-lowering-*.f64 N/A

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

4. Applied egg-rr 69.1%

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

5. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. rgt-mult-inverse N/A

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

   6. distribute-rgt-in N/A

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

   7. *-lft-identity N/A

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

   8. accelerator-lowering-fma.f64 29.5

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

7. Simplified 29.5%

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

8. Taylor expanded in y around 0

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

10. Simplified 2.6%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))