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

Percentage Accurate: 99.9% → 99.9%

Time: 4.1s

Alternatives: 7

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 25.9% 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

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

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. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+22} \lor \neg \left(y \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2e+22) (not (<= y 2e+16)))
   (* y (* x (- y)))
   (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2e+22) || !(y <= 2e+16)) {
		tmp = y * (x * -y);
	} 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 <= (-2d+22)) .or. (.not. (y <= 2d+16))) then
        tmp = y * (x * -y)
    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 <= -2e+22) || !(y <= 2e+16)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2e+22) or not (y <= 2e+16):
		tmp = y * (x * -y)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2e+22) || !(y <= 2e+16))
		tmp = Float64(y * Float64(x * Float64(-y)));
	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 <= -2e+22) || ~((y <= 2e+16)))
		tmp = y * (x * -y);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2e+22], N[Not[LessEqual[y, 2e+16]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e22 or 2e16 < y

   1. Initial program 99.9%

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

      1. distribute-lft-out-- 83.5%

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

      2. *-rgt-identity 83.5%

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

      3. associate-*l* 73.8%

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

      4. distribute-lft-out-- 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around inf 90.3%

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

      1. unpow2 90.3%

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

      2. associate-*r* 90.3%

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

      3. mul-1-neg 90.3%

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

      4. distribute-rgt-neg-out 90.3%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

   6. Simplified 99.9%

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

   if -2e22 < y < 2e16

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+22} \lor \neg \left(y \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 3: 92.2% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.9%

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

      1. distribute-lft-out-- 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. associate-*l* 75.4%

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

      4. distribute-lft-out-- 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in y around inf 88.1%

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

      1. unpow2 88.1%

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

      2. mul-1-neg 88.1%

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

      3. distribute-rgt-neg-out 88.1%

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

   6. Simplified 88.1%

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

   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. distribute-lft-out-- 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-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 99.1%

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

4. Final simplification 93.3%

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

Alternative 4: 98.0% accurate, 0.7× speedup

Math

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

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 99.9%

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

      1. distribute-lft-out-- 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. associate-*l* 75.4%

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

      4. distribute-lft-out-- 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in y around inf 88.1%

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

      1. unpow2 88.1%

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

      2. associate-*r* 88.1%

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

      3. mul-1-neg 88.1%

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

      4. distribute-rgt-neg-out 88.1%

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

      5. associate-*l* 97.2%

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

      6. *-commutative 97.2%

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

   6. Simplified 97.2%

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

   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. distribute-lft-out-- 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. distribute-lft-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 99.1%

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

4. Final simplification 98.1%

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

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. Taylor expanded in x around 0 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 64.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x * y), $MachinePrecision], N[(x * (-y)), $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-lft-out-- 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. associate-*l* 97.4%

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

      4. distribute-lft-out-- 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in y around 0 76.2%

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

   if 1 < y

   1. Initial program 99.9%

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

      1. distribute-lft-out-- 70.3%

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

      2. *-rgt-identity 70.3%

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

      3. associate-*l* 59.7%

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

      4. distribute-lft-out-- 89.2%

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

   3. Simplified 89.2%

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

      1. *-un-lft-identity 89.2%

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

      2. distribute-rgt-out-- 89.2%

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

      3. add-sqr-sqrt 0.0%

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

      4. associate-*r* 0.0%

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

   5. Applied egg-rr 0.0%

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

      1. associate-*l* 0.0%

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

      2. add-sqr-sqrt 89.2%

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

      3. flip-- 89.2%

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

      4. metadata-eval 89.2%

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

      5. +-commutative 89.2%

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

      6. associate-*r/ 75.9%

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

      7. associate-/l* 89.1%

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

      8. frac-2neg 89.1%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.6%

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

      11. sqr-neg 0.6%

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

      12. sqrt-prod 0.6%

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

      13. add-sqr-sqrt 0.6%

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

      14. clear-num 0.6%

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

      15. metadata-eval 0.6%

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

      16. +-commutative 0.6%

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

      17. flip-- 0.6%

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

      18. sub-neg 0.6%

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

      19. +-commutative 0.6%

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

      20. add-sqr-sqrt 0.0%

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

      21. sqrt-unprod 87.5%

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

      22. sqr-neg 87.5%

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

      23. sqrt-prod 87.3%

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

      24. add-sqr-sqrt 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. distribute-neg-frac 87.5%

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

      2. metadata-eval 87.5%

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

   9. Simplified 87.5%

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

   10. Taylor expanded in y around 0 33.9%

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

       1. *-commutative 33.9%

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

       2. associate-*r* 33.9%

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

       3. neg-mul-1 33.9%

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

   12. Simplified 33.9%

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

4. Final simplification 64.5%

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

Alternative 7: 56.8% 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. Step-by-step derivation

   1. distribute-lft-out-- 91.7%

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

   2. *-rgt-identity 91.7%

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

   3. associate-*l* 86.9%

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

   4. distribute-lft-out-- 95.1%

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

3. Simplified 95.1%

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

4. Taylor expanded in y around 0 55.3%

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

5. Final simplification 55.3%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023176 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))