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

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

Alternatives: 7

Speedup: 1.0×

• 24.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+74} \lor \neg \left(y \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5e+74) (not (<= y 5e+17)))
   (* y (* y (- x)))
   (* x (- y (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5e+74) || !(y <= 5e+17)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5d+74)) .or. (.not. (y <= 5d+17))) then
        tmp = y * (y * -x)
    else
        tmp = x * (y - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5e+74) || !(y <= 5e+17)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5e+74) or not (y <= 5e+17):
		tmp = y * (y * -x)
	else:
		tmp = x * (y - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5e+74) || !(y <= 5e+17))
		tmp = Float64(y * Float64(y * Float64(-x)));
	else
		tmp = Float64(x * Float64(y - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5e+74) || ~((y <= 5e+17)))
		tmp = y * (y * -x);
	else
		tmp = x * (y - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5e+74], N[Not[LessEqual[y, 5e+17]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999963e74 or 5e17 < y

   1. Initial program 99.8%

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

      1. associate-*l* 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. mul-1-neg 85.4%

         \[\leadsto \color{blue}{-x \cdot {y}^{2}} \]

      2. unpow2 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

   6. Simplified 85.4%

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

      1. distribute-rgt-neg-out 85.4%

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

      2. associate-*r* 99.8%

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

      3. add-sqr-sqrt 57.3%

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

      4. sqrt-prod 50.8%

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

      5. sqr-neg 50.8%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 0.4%

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

      8. associate-*r* 0.4%

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

      9. distribute-rgt-neg-in 0.4%

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

      10. *-commutative 0.4%

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

      11. distribute-lft-neg-in 0.4%

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

      12. associate-*l* 0.4%

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

      13. add-sqr-sqrt 0.1%

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

      14. sqrt-unprod 50.8%

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

      15. sqr-neg 50.8%

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

      16. sqrt-prod 57.3%

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

      17. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -4.99999999999999963e74 < y < 5e17

   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* 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+74} \lor \neg \left(y \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+72} \lor \neg \left(y \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2e+72) (not (<= y 5e+17)))
   (* y (* y (- x)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2e+72) || !(y <= 5e+17)) {
		tmp = y * (y * -x);
	} 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 <= (-2d+72)) .or. (.not. (y <= 5d+17))) then
        tmp = y * (y * -x)
    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 <= -2e+72) || !(y <= 5e+17)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2e+72) or not (y <= 5e+17):
		tmp = y * (y * -x)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2e+72) || !(y <= 5e+17))
		tmp = Float64(y * Float64(y * Float64(-x)));
	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 <= -2e+72) || ~((y <= 5e+17)))
		tmp = y * (y * -x);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2e+72], N[Not[LessEqual[y, 5e+17]], $MachinePrecision]], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999989e72 or 5e17 < y

   1. Initial program 99.8%

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

      1. associate-*l* 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in y around inf 85.4%

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

      1. mul-1-neg 85.4%

         \[\leadsto \color{blue}{-x \cdot {y}^{2}} \]

      2. unpow2 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

   6. Simplified 85.4%

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

      1. distribute-rgt-neg-out 85.4%

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

      2. associate-*r* 99.8%

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

      3. add-sqr-sqrt 57.3%

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

      4. sqrt-prod 50.8%

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

      5. sqr-neg 50.8%

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

      6. sqrt-unprod 0.1%

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

      7. add-sqr-sqrt 0.4%

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

      8. associate-*r* 0.4%

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

      9. distribute-rgt-neg-in 0.4%

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

      10. *-commutative 0.4%

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

      11. distribute-lft-neg-in 0.4%

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

      12. associate-*l* 0.4%

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

      13. add-sqr-sqrt 0.1%

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

      14. sqrt-unprod 50.8%

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

      15. sqr-neg 50.8%

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

      16. sqrt-prod 57.3%

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

      17. add-sqr-sqrt 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -1.99999999999999989e72 < y < 5e17

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+72} \lor \neg \left(y \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 3: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 99.7%

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

      1. associate-*l* 87.5%

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

   3. Simplified 87.5%

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

   4. Taylor expanded in y around inf 86.0%

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

      1. mul-1-neg 86.0%

         \[\leadsto \color{blue}{-x \cdot {y}^{2}} \]

      2. unpow2 86.0%

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

      3. distribute-rgt-neg-in 86.0%

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

   6. Simplified 86.0%

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

      1. distribute-rgt-neg-out 86.0%

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

      2. associate-*r* 98.3%

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

      3. add-sqr-sqrt 51.6%

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

      4. sqrt-prod 46.3%

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

      5. sqr-neg 46.3%

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

      6. sqrt-unprod 0.3%

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

      7. add-sqr-sqrt 0.6%

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

      8. associate-*r* 0.5%

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

      9. distribute-rgt-neg-in 0.5%

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

      10. *-commutative 0.5%

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

      11. distribute-lft-neg-in 0.5%

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

      12. associate-*l* 0.6%

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

      13. add-sqr-sqrt 0.3%

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

      14. sqrt-unprod 46.3%

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

      15. sqr-neg 46.3%

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

      16. sqrt-prod 51.6%

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

      17. add-sqr-sqrt 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. 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. Taylor expanded in y around 0 98.1%

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

4. Final simplification 98.2%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 93.6%

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

3. Simplified 93.6%

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

   1. associate-*r* 99.8%

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

   2. flip-- 93.6%

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

   3. associate-*r/ 89.3%

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

   4. metadata-eval 89.3%

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

   5. +-commutative 89.3%

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

5. Applied egg-rr 89.3%

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

   1. associate-*l* 86.7%

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

   2. associate-/l* 89.4%

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

   3. sub-neg 89.4%

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

   4. distribute-rgt-neg-out 89.4%

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

   5. distribute-rgt-in 89.4%

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

   6. *-lft-identity 89.4%

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

   7. distribute-rgt-neg-out 89.4%

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

   8. distribute-lft-neg-in 89.4%

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

   9. unpow3 89.4%

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

   10. unsub-neg 89.4%

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

7. Simplified 89.4%

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

8. Taylor expanded in y around 0 49.3%

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

9. Taylor expanded in y around 0 88.2%

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

    1. +-commutative 88.2%

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

    2. unpow2 88.2%

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

    3. associate-*r* 88.2%

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

    4. *-commutative 88.2%

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

    5. associate-*r* 94.4%

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

    6. distribute-rgt-out 99.9%

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

    7. associate-*l* 99.9%

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

    8. neg-mul-1 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

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 99.8%

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

   1. associate-*l* 93.6%

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

3. Simplified 93.6%

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

   1. associate-*r* 99.8%

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

   2. flip-- 93.6%

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

   3. associate-*r/ 89.3%

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

   4. metadata-eval 89.3%

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

   5. +-commutative 89.3%

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

5. Applied egg-rr 89.3%

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

   1. associate-*l* 86.7%

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

   2. associate-/l* 89.4%

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

   3. sub-neg 89.4%

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

   4. distribute-rgt-neg-out 89.4%

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

   5. distribute-rgt-in 89.4%

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

   6. *-lft-identity 89.4%

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

   7. distribute-rgt-neg-out 89.4%

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

   8. distribute-lft-neg-in 89.4%

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

   9. unpow3 89.4%

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

   10. unsub-neg 89.4%

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

7. Simplified 89.4%

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

8. Taylor expanded in y around 0 49.3%

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

9. Taylor expanded in y around 0 88.2%

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

    1. +-commutative 88.2%

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

    2. unpow2 88.2%

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

    3. associate-*r* 88.2%

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

    4. *-commutative 88.2%

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

    5. associate-*r* 94.4%

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

    6. distribute-rgt-out 99.9%

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

    7. associate-*l* 99.9%

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

    8. neg-mul-1 99.9%

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

11. Simplified 99.9%

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

12. Taylor expanded in y around 0 88.2%

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

    1. unpow2 88.2%

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

    2. *-commutative 88.2%

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

    3. associate-*r* 94.4%

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

    4. associate-*r* 94.4%

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

    5. *-commutative 94.4%

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

    6. *-lft-identity 94.4%

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

    7. distribute-rgt-in 99.8%

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

    8. +-commutative 99.8%

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

    9. associate-*l* 99.9%

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

    10. mul-1-neg 99.9%

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

    11. unsub-neg 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 6: 56.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 93.6%

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

3. Simplified 93.6%

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

4. Taylor expanded in y around 0 54.9%

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

5. Final simplification 54.9%

   \[\leadsto y \cdot x \]

Alternative 7: 2.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 93.6%

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

3. Simplified 93.6%

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

   1. associate-*r* 99.8%

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

   2. flip-- 93.6%

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

   3. associate-*r/ 89.3%

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

   4. metadata-eval 89.3%

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

   5. +-commutative 89.3%

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

5. Applied egg-rr 89.3%

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

   1. associate-*l* 86.7%

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

   2. associate-/l* 89.4%

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

   3. sub-neg 89.4%

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

   4. distribute-rgt-neg-out 89.4%

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

   5. distribute-rgt-in 89.4%

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

   6. *-lft-identity 89.4%

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

   7. distribute-rgt-neg-out 89.4%

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

   8. distribute-lft-neg-in 89.4%

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

   9. unpow3 89.4%

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

   10. unsub-neg 89.4%

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

7. Simplified 89.4%

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

8. Taylor expanded in y around 0 49.3%

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

9. Taylor expanded in y around inf 2.7%

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

10. Final simplification 2.7%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))