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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 8

Speedup: 1.0×

• 25.4% 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: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = (y * y) * -x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] * (-x)), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.9%

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

      1. distribute-rgt-out-- 83.1%

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

      2. *-lft-identity 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 73.3%

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

      5. *-commutative 73.3%

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

      6. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 88.7%

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

      1. unpow2 88.7%

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

      2. mul-1-neg 88.7%

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

      3. distribute-rgt-neg-out 88.7%

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

   6. Simplified 88.7%

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

   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-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 97.2%

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

4. Final simplification 93.1%

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

Alternative 3: 97.8% 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-rgt-out-- 83.1%

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

      2. *-lft-identity 83.1%

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

      3. *-commutative 83.1%

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

      4. associate-*r* 73.3%

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

      5. *-commutative 73.3%

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

      6. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around inf 88.7%

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

      1. unpow2 88.7%

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

      2. associate-*r* 88.7%

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

      3. mul-1-neg 88.7%

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

      4. distribute-rgt-neg-out 88.7%

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

      5. associate-*l* 98.4%

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

      6. *-commutative 98.4%

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

   6. Simplified 98.4%

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

   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-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 97.2%

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

4. Final simplification 97.8%

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

Alternative 4: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+149}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e+149) (* y (* x (- y))) (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e+149) {
		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 <= (-5d+149)) 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 <= -5e+149) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e+149:
		tmp = y * (x * -y)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e+149)
		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 <= -5e+149)
		tmp = y * (x * -y);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e+149], 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 < -4.9999999999999999e149

   1. Initial program 100.0%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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* 73.6%

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

      5. *-commutative 73.6%

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

      6. distribute-rgt-out-- 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in y around inf 73.6%

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

      1. unpow2 73.6%

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

      2. associate-*r* 73.6%

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

      3. mul-1-neg 73.6%

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

      4. distribute-rgt-neg-out 73.6%

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

      5. associate-*l* 100.0%

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

      6. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -4.9999999999999999e149 < 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-- 90.2%

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

      2. *-lft-identity 90.2%

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

      3. *-commutative 90.2%

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

      4. associate-*r* 89.4%

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

      5. *-commutative 89.4%

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

      6. distribute-rgt-out-- 99.1%

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

   3. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+149}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 5: 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. Step-by-step derivation

   1. distribute-rgt-out-- 91.7%

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

   2. *-lft-identity 91.7%

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

   3. *-commutative 91.7%

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

   4. associate-*r* 87.0%

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

   5. *-commutative 87.0%

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

   6. distribute-rgt-out-- 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in x around 0 95.2%

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

   1. *-commutative 95.2%

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

   2. unpow2 95.2%

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

   3. sub-neg 95.2%

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

   4. distribute-rgt-neg-out 95.2%

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

   5. distribute-rgt-in 87.0%

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

   6. associate-*l* 91.7%

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

   7. distribute-lft-out 99.9%

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

   8. distribute-lft-neg-out 99.9%

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

   9. *-commutative 99.9%

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

   10. unsub-neg 99.9%

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

   11. *-commutative 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 6: 63.8% 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-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* 94.6%

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

      5. *-commutative 94.6%

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

      6. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in y around 0 75.9%

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

   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-rgt-out-- 66.0%

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

      2. *-lft-identity 66.0%

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

      3. *-commutative 66.0%

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

      4. associate-*r* 63.0%

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

      5. *-commutative 63.0%

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

      6. distribute-rgt-out-- 96.8%

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

   3. Simplified 96.8%

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

      1. *-un-lft-identity 96.8%

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

      2. distribute-rgt-out-- 96.8%

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

      3. associate-*l* 99.9%

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

      4. flip-- 96.9%

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

      5. associate-*r/ 89.4%

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

      6. metadata-eval 89.4%

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

      7. +-commutative 89.4%

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

   5. Applied egg-rr 89.4%

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

      1. associate-*l* 83.2%

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

      2. associate-/l* 84.6%

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

      3. sub-neg 84.6%

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

      4. distribute-rgt-neg-out 84.6%

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

      5. distribute-rgt-in 84.6%

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

      6. *-lft-identity 84.6%

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

      7. distribute-rgt-neg-out 84.6%

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

      8. distribute-lft-neg-in 84.6%

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

      9. unpow3 84.7%

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

      10. unsub-neg 84.7%

          \[\leadsto \frac{x}{\frac{y + 1}{\color{blue}{y - {y}^{3}}}} \]

   7. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{x}{\frac{y + 1}{y - {y}^{3}}}} \]

   8. Taylor expanded in y around 0 0.7%

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

      1. div-inv 0.7%

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

      2. add-sqr-sqrt 0.7%

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

      3. sqrt-unprod 0.6%

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

      4. frac-times 0.6%

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

      5. metadata-eval 0.6%

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

      6. metadata-eval 0.6%

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

      7. frac-times 0.6%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 38.5%

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

      10. associate-/r/ 38.5%

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

      11. metadata-eval 38.5%

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

      12. neg-mul-1 38.5%

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

      13. distribute-rgt-neg-in 38.5%

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

      14. distribute-lft-neg-in 38.5%

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

   10. Applied egg-rr 38.5%

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

4. Final simplification 66.8%

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

Alternative 7: 56.6% 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-rgt-out-- 91.7%

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

   2. *-lft-identity 91.7%

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

   3. *-commutative 91.7%

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

   4. associate-*r* 87.0%

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

   5. *-commutative 87.0%

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

   6. distribute-rgt-out-- 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in y around 0 57.7%

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

5. Final simplification 57.7%

   \[\leadsto x \cdot y \]

Alternative 8: 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-- 91.7%

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

   2. *-lft-identity 91.7%

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

   3. *-commutative 91.7%

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

   4. associate-*r* 87.0%

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

   5. *-commutative 87.0%

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

   6. distribute-rgt-out-- 95.2%

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

3. Simplified 95.2%

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

   1. flip-- 50.9%

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

   2. associate-*r/ 46.8%

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

   3. pow2 46.8%

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

   4. pow2 46.8%

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

   5. pow-prod-up 46.7%

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

   6. metadata-eval 46.7%

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

   7. +-commutative 46.7%

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

   8. fma-def 46.7%

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

5. Applied egg-rr 46.7%

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

   1. *-commutative 46.7%

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

   2. associate-/l* 44.9%

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

7. Simplified 44.9%

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

8. Taylor expanded in y around inf 13.0%

   \[\leadsto \frac{y \cdot y - {y}^{4}}{\color{blue}{\frac{{y}^{2}}{x}}} \]
9. Step-by-step derivation

   1. unpow2 13.0%

      \[\leadsto \frac{y \cdot y - {y}^{4}}{\frac{\color{blue}{y \cdot y}}{x}} \]

10. Simplified 13.0%

    \[\leadsto \frac{y \cdot y - {y}^{4}}{\color{blue}{\frac{y \cdot y}{x}}} \]

11. Taylor expanded in y around 0 2.7%

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

12. Final simplification 2.7%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))