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

Percentage Accurate: 99.9% → 99.4%

Time: 6.9s

Alternatives: 7

Speedup: 1.0×

• 24.8% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+170} \lor \neg \left(y \leq 3.1 \cdot 10^{+109}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.3e+170) (not (<= y 3.1e+109)))
   (* y (* y (- x)))
   (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+170) || !(y <= 3.1e+109)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * 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 <= (-1.3d+170)) .or. (.not. (y <= 3.1d+109))) then
        tmp = y * (y * -x)
    else
        tmp = x * (y - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.3e+170) || !(y <= 3.1e+109)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.3e+170) or not (y <= 3.1e+109):
		tmp = y * (y * -x)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.3e+170) || !(y <= 3.1e+109))
		tmp = Float64(y * Float64(y * Float64(-x)));
	else
		tmp = Float64(x * Float64(y - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.3e+170) || ~((y <= 3.1e+109)))
		tmp = y * (y * -x);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.3e+170], N[Not[LessEqual[y, 3.1e+109]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2999999999999999e170 or 3.09999999999999992e109 < y

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 84.8%

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

      2. *-lft-identity 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-*r* 55.0%

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

      5. *-commutative 55.0%

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

      6. distribute-rgt-out-- 70.2%

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

   3. Simplified 70.2%

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

   4. Taylor expanded in y around inf 70.2%

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

      1. unpow2 70.2%

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

      2. associate-*r* 70.2%

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

      3. associate-*r* 99.9%

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

      4. mul-1-neg 99.9%

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

   6. Simplified 99.9%

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

   if -1.2999999999999999e170 < y < 3.09999999999999992e109

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 98.3%

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

      2. *-lft-identity 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r* 98.3%

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

      5. *-commutative 98.3%

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

      6. distribute-rgt-out-- 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+170} \lor \neg \left(y \leq 3.1 \cdot 10^{+109}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 2: 92.2% accurate, 0.7× 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. distribute-rgt-out-- 89.4%

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

      2. *-lft-identity 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*r* 73.7%

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

      5. *-commutative 73.7%

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

      6. distribute-rgt-out-- 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in y around inf 80.4%

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

      1. unpow2 80.4%

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

      2. mul-1-neg 80.4%

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

      3. distribute-rgt-neg-out 80.4%

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

   6. Simplified 80.4%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.0%

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

4. Final simplification 89.4%

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

Alternative 3: 98.0% accurate, 0.7× 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. distribute-rgt-out-- 89.4%

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

      2. *-lft-identity 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*r* 73.7%

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

      5. *-commutative 73.7%

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

      6. distribute-rgt-out-- 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in y around inf 80.4%

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

      1. unpow2 80.4%

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

      2. associate-*r* 80.4%

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

      3. associate-*r* 96.1%

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

      4. mul-1-neg 96.1%

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

   6. Simplified 96.1%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.0%

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

4. Final simplification 97.0%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. distribute-rgt-out-- 94.8%

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

   2. *-lft-identity 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

   6. distribute-rgt-out-- 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in x around 0 92.2%

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

   1. unpow2 92.2%

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

   2. sub-neg 92.2%

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

   3. distribute-rgt-neg-out 92.2%

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

   4. distribute-lft-out 87.2%

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

   5. *-rgt-identity 87.2%

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

   6. associate-*r* 94.8%

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

   7. distribute-lft-in 99.9%

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

   8. sub-neg 99.9%

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

   9. associate-*r* 92.3%

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

   10. *-commutative 92.3%

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

   11. associate-*l* 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

   \[\leadsto y \cdot \left(\left(1 - y\right) \cdot x\right) \]

Alternative 5: 64.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(y * x), $MachinePrecision], N[(y * (-x)), $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. Step-by-step derivation

      1. distribute-rgt-out-- 99.9%

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

      2. *-lft-identity 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 94.7%

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

      5. *-commutative 94.7%

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

      6. distribute-rgt-out-- 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around 0 71.6%

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

   if 1 < y

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 77.4%

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

      2. *-lft-identity 77.4%

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

      3. *-commutative 77.4%

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

      4. associate-*r* 61.2%

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

      5. *-commutative 61.2%

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

      6. distribute-rgt-out-- 83.7%

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

   3. Simplified 83.7%

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

      1. flip-- 42.2%

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

      2. associate-*r/ 42.1%

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

      3. pow2 42.1%

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

      4. pow2 42.1%

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

      5. pow-prod-up 42.2%

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

      6. metadata-eval 42.2%

         \[\leadsto \frac{x \cdot \left(y \cdot y - {y}^{\color{blue}{4}}\right)}{y + y \cdot y} \]

      7. +-commutative 42.2%

         \[\leadsto \frac{x \cdot \left(y \cdot y - {y}^{4}\right)}{\color{blue}{y \cdot y + y}} \]

      8. fma-def 42.2%

         \[\leadsto \frac{x \cdot \left(y \cdot y - {y}^{4}\right)}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot y - {y}^{4}\right)}{\mathsf{fma}\left(y, y, y\right)}} \]
   6. Step-by-step derivation

      1. associate-/l* 42.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(y, y, y\right)}{y \cdot y - {y}^{4}}}} \]

   7. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(y, y, y\right)}{y \cdot y - {y}^{4}}}} \]

   8. Taylor expanded in y around 0 0.9%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{y}}} \]
   9. Step-by-step derivation

      1. frac-2neg 0.9%

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

      2. div-inv 0.9%

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

      3. neg-mul-1 0.9%

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

      4. div-inv 0.9%

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

      5. clear-num 0.9%

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

      6. div-inv 0.9%

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

      7. metadata-eval 0.9%

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

      8. *-commutative 0.9%

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

      9. neg-mul-1 0.9%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 41.2%

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

      12. sqr-neg 41.2%

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

      13. sqrt-prod 28.0%

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

      14. add-sqr-sqrt 28.0%

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

   10. Applied egg-rr 28.0%

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

4. Final simplification 61.7%

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

Alternative 6: 56.9% 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. distribute-rgt-out-- 94.8%

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

   2. *-lft-identity 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

   6. distribute-rgt-out-- 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in y around 0 55.6%

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

5. Final simplification 55.6%

   \[\leadsto y \cdot x \]

Alternative 7: 2.8% 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. distribute-rgt-out-- 94.8%

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

   2. *-lft-identity 94.8%

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

   3. *-commutative 94.8%

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

   4. associate-*r* 87.2%

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

   5. *-commutative 87.2%

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

   6. distribute-rgt-out-- 92.2%

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

3. Simplified 92.2%

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

4. Taylor expanded in x around 0 92.2%

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

   1. unpow2 92.2%

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

   2. sub-neg 92.2%

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

   3. distribute-rgt-neg-out 92.2%

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

   4. distribute-lft-out 87.2%

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

   5. *-rgt-identity 87.2%

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

   6. associate-*r* 94.8%

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

   7. distribute-lft-in 99.9%

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

   8. sub-neg 99.9%

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

   9. associate-*r* 92.3%

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

   10. *-commutative 92.3%

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

   11. associate-*l* 99.9%

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

6. Simplified 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

   3. flip-- 92.2%

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

   4. metadata-eval 92.2%

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

   5. +-commutative 92.2%

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

   6. associate-/r/ 91.4%

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

   7. associate-/l* 89.8%

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

   8. *-commutative 89.8%

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

   9. associate-*l* 85.7%

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

8. Applied egg-rr 85.7%

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

9. Taylor expanded in y around 0 51.3%

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

10. Taylor expanded in y around inf 2.9%

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

11. Final simplification 2.9%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))