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

Percentage Accurate: 99.9% → 99.9%

Time: 5.0s

Alternatives: 7

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+121} \lor \neg \left(y \leq 10^{+149}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e+121) (not (<= y 1e+149)))
   (* y (* x (- y)))
   (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e+121) || !(y <= 1e+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 <= (-4d+121)) .or. (.not. (y <= 1d+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 <= -4e+121) || !(y <= 1e+149)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e+121) or not (y <= 1e+149):
		tmp = y * (x * -y)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e+121) || !(y <= 1e+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 <= -4e+121) || ~((y <= 1e+149)))
		tmp = y * (x * -y);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e+121], N[Not[LessEqual[y, 1e+149]], $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 < -4.00000000000000015e121 or 1.00000000000000005e149 < y

   1. Initial program 100.0%

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

      1. distribute-rgt-out-- 80.8%

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

      2. *-lft-identity 80.8%

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

      3. *-commutative 80.8%

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

      4. associate-*r* 55.9%

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

      5. *-commutative 55.9%

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

      6. distribute-rgt-out-- 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in y around inf 75.0%

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

      1. unpow2 75.0%

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

      2. associate-*r* 75.0%

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

      3. mul-1-neg 75.0%

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

      4. distribute-rgt-neg-out 75.0%

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

      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 -4.00000000000000015e121 < y < 1.00000000000000005e149

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 96.7%

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

      2. *-lft-identity 96.7%

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

      3. *-commutative 96.7%

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

      4. associate-*r* 96.7%

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

      5. *-commutative 96.7%

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

      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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+121} \lor \neg \left(y \leq 10^{+149}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 2: 92.3% 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 99.8%

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

      1. distribute-rgt-out-- 85.2%

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

      2. *-lft-identity 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-*r* 72.2%

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

      5. *-commutative 72.2%

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

      6. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in y around inf 84.2%

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

      1. unpow2 84.2%

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

      2. mul-1-neg 84.2%

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

      3. distribute-rgt-neg-out 84.2%

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

   6. Simplified 84.2%

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

   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 90.6%

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

Alternative 3: 97.9% 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 99.8%

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

      1. distribute-rgt-out-- 85.2%

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

      2. *-lft-identity 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-*r* 72.2%

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

      5. *-commutative 72.2%

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

      6. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in y around inf 84.2%

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

      1. unpow2 84.2%

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

      2. associate-*r* 84.2%

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

      3. mul-1-neg 84.2%

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

      4. distribute-rgt-neg-out 84.2%

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

      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-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 97.2%

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

Alternative 4: 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 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-- 92.5%

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

   2. *-lft-identity 92.5%

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

   3. *-commutative 92.5%

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

   4. associate-*r* 85.9%

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

   5. *-commutative 85.9%

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

   6. distribute-rgt-out-- 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in x around 0 93.3%

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

   1. *-commutative 93.3%

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

   2. unpow2 93.3%

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

   3. sub-neg 93.3%

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

   4. distribute-rgt-neg-out 93.3%

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

   5. distribute-rgt-in 85.9%

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

   6. associate-*l* 92.5%

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

   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: 63.6% 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* 94.7%

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

      5. *-commutative 94.7%

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

      6. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in y around 0 76.1%

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

   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-- 74.1%

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

      2. *-lft-identity 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*r* 64.1%

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

      5. *-commutative 64.1%

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

      6. distribute-rgt-out-- 89.7%

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

   3. Simplified 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. distribute-rgt-out-- 89.7%

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

      3. associate-*l* 99.8%

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

      4. flip-- 89.6%

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

      5. associate-*r/ 88.3%

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

      6. metadata-eval 88.3%

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

      7. +-commutative 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. associate-*l* 81.9%

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

      2. associate-/l* 81.8%

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

      3. sub-neg 81.8%

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

      4. distribute-rgt-neg-out 81.8%

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

      5. distribute-rgt-in 81.8%

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

      6. *-lft-identity 81.8%

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

      7. distribute-rgt-neg-out 81.8%

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

      8. distribute-lft-neg-in 81.8%

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

      9. unpow3 81.9%

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

      10. unsub-neg 81.9%

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

   7. Simplified 81.9%

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

   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. frac-2neg 0.9%

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

      6. metadata-eval 0.9%

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

      7. neg-mul-1 0.9%

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

      8. add-cbrt-cube 0.9%

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

      9. metadata-eval 0.9%

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

      10. metadata-eval 0.9%

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

      11. rem-cbrt-cube 0.7%

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

      12. cbrt-prod 0.7%

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

      13. neg-mul-1 0.7%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 54.2%

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

      16. sqr-neg 54.2%

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

      17. sqrt-unprod 53.4%

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

      18. add-sqr-sqrt 53.4%

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

      19. rem-cbrt-cube 30.4%

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

      20. /-rgt-identity 30.4%

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

      21. clear-num 30.4%

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

      22. remove-double-div 30.4%

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

   10. Applied egg-rr 30.4%

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

4. Final simplification 62.9%

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

Alternative 7: 56.3% 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-- 92.5%

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

   2. *-lft-identity 92.5%

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

   3. *-commutative 92.5%

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

   4. associate-*r* 85.9%

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

   5. *-commutative 85.9%

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

   6. distribute-rgt-out-- 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in y around 0 54.3%

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

5. Final simplification 54.3%

   \[\leadsto x \cdot y \]

Reproduce

herbie shell --seed 2023264 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))