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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 99.9%

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

   2. associate-*r* 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-rgt-in 99.9%

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

   4. *-un-lft-identity 99.9%

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

   5. distribute-lft-neg-in 99.9%

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

   6. *-commutative 99.9%

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

   7. unsub-neg 99.9%

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

   8. *-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 0.4× speedup

Math

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

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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. associate-*r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-in 98.1%

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

   7. Simplified 98.1%

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

   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 0 98.1%

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

4. Final simplification 98.1%

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

Alternative 3: 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-*r* 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

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(1.0 - y) * Float64(x * y))
end

MATLAB

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

Wolfram

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

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

Alternative 5: 55.2% 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. Add Preprocessing

3. Taylor expanded in y around 0 56.3%

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

Alternative 6: 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. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 92.9%

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

   2. associate-*r/ 91.5%

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

   3. metadata-eval 91.5%

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

   4. pow2 91.5%

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

   5. +-commutative 91.5%

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

4. Applied egg-rr 91.5%

   \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \left(1 - {y}^{2}\right)}{y + 1}} \]
5. 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. distribute-rgt-neg-in 83.1%

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

   7. unpow2 83.1%

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

   8. cube-mult 83.2%

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

   9. unsub-neg 83.2%

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

   10. *-rgt-identity 83.2%

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

6. Simplified 83.2%

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

7. Taylor expanded in y around inf 34.0%

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

8. Taylor expanded in y around 0 2.7%

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

   1. associate-*r/ 11.9%

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

   2. frac-2neg 11.9%

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

   3. add-sqr-sqrt 5.3%

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

   4. sqrt-unprod 4.1%

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

   5. sqr-neg 4.1%

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

   6. sqrt-unprod 12.2%

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

   7. add-sqr-sqrt 24.2%

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

10. Applied egg-rr 24.2%

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

    1. distribute-rgt-neg-in 24.2%

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

    2. neg-mul-1 24.2%

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

    3. *-commutative 24.2%

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

    4. associate-/l* 4.4%

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

    5. *-inverses 4.4%

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

    6. *-rgt-identity 4.4%

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

    7. neg-mul-1 4.4%

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

12. Simplified 4.4%

    \[\leadsto \color{blue}{-x} \]
13. 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)))