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

Percentage Accurate: 99.9% → 99.9%

Time: 7.5s

Alternatives: 8

Speedup: 1.0×

• 24.6% 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154} \lor \neg \left(y \leq 1.5 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35e+154) (not (<= y 1.5e+20)))
   (* y (* x (- y)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35e+154) || !(y <= 1.5e+20)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - 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 <= (-1.35d+154)) .or. (.not. (y <= 1.5d+20))) then
        tmp = y * (x * -y)
    else
        tmp = x * (y * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.35e+154) || !(y <= 1.5e+20)) {
		tmp = y * (x * -y);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35e+154) or not (y <= 1.5e+20):
		tmp = y * (x * -y)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35e+154) || !(y <= 1.5e+20))
		tmp = Float64(y * Float64(x * Float64(-y)));
	else
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.35e+154) || ~((y <= 1.5e+20)))
		tmp = y * (x * -y);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35e+154], N[Not[LessEqual[y, 1.5e+20]], $MachinePrecision]], N[(y * N[(x * (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.35000000000000003e154 or 1.5e20 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 99.9%

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

      1. neg-mul-1 99.9%

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

   7. Simplified 99.9%

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

   if -1.35000000000000003e154 < y < 1.5e20

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+154} \lor \neg \left(y \leq 1.5 \cdot 10^{+20}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 97.9%

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

      1. neg-mul-1 97.9%

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

   7. Simplified 97.9%

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

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.8%

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

4. Final simplification 97.8%

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

Alternative 4: 91.9% accurate, 0.4× 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. associate-*l* 87.8%

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

   3. Simplified 87.8%

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

   5. Taylor expanded in y around inf 85.8%

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

      1. neg-mul-1 97.9%

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

   7. Simplified 85.8%

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

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.8%

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

4. Final simplification 91.6%

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

Alternative 5: 63.3% accurate, 0.8× 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. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 71.8%

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

   if 1 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 0.6%

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

      1. add-sqr-sqrt 0.6%

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

      2. sqrt-prod 0.4%

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

      3. sqr-neg 0.4%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 25.6%

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

      6. distribute-lft-neg-in 25.6%

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

   7. Applied egg-rr 25.6%

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

4. Final simplification 59.5%

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

Alternative 6: 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{y \cdot \left(x \cdot \left(1 - y\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 56.0% 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. *-commutative 99.9%

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

   2. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 52.9%

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

6. Final simplification 52.9%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

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. associate-*l* 93.7%

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

3. Simplified 93.7%

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

   1. associate-*r* 99.9%

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

   2. flip-- 93.7%

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

   3. associate-*r/ 90.1%

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

   4. metadata-eval 90.1%

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

   5. pow2 90.1%

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

   6. +-commutative 90.1%

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

6. Applied egg-rr 90.1%

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

   1. associate-*l* 85.6%

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

   2. *-commutative 85.6%

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

   3. associate-/l* 79.6%

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

   4. sub-neg 79.6%

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

   5. unpow2 79.6%

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

   6. distribute-rgt-neg-out 79.6%

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

   7. distribute-lft-in 79.6%

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

   8. *-rgt-identity 79.6%

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

   9. associate-*l* 79.6%

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

   10. sqr-neg 79.6%

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

   11. unpow3 79.6%

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

   12. cube-neg 79.6%

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

   13. unsub-neg 79.6%

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

8. Simplified 79.6%

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

9. Taylor expanded in y around inf 32.7%

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

10. Taylor expanded in y around 0 3.2%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))