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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 5

Speedup: 1.0×

• 25.3% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+154} \lor \neg \left(y \leq 9.5 \cdot 10^{+150}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+154) (not (<= y 9.5e+150)))
   (* (* x y) (- y))
   (* (* (- 1.0 y) y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+154) || !(y <= 9.5e+150)) {
		tmp = (x * y) * -y;
	} else {
		tmp = ((1.0 - y) * 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 <= (-1.2d+154)) .or. (.not. (y <= 9.5d+150))) then
        tmp = (x * y) * -y
    else
        tmp = ((1.0d0 - y) * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+154) || !(y <= 9.5e+150)) {
		tmp = (x * y) * -y;
	} else {
		tmp = ((1.0 - y) * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2e+154) or not (y <= 9.5e+150):
		tmp = (x * y) * -y
	else:
		tmp = ((1.0 - y) * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+154) || !(y <= 9.5e+150))
		tmp = Float64(Float64(x * y) * Float64(-y));
	else
		tmp = Float64(Float64(Float64(1.0 - y) * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.2e+154) || ~((y <= 9.5e+150)))
		tmp = (x * y) * -y;
	else
		tmp = ((1.0 - y) * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+154], N[Not[LessEqual[y, 9.5e+150]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000007e154 or 9.5000000000000001e150 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.20000000000000007e154 < y < 9.5000000000000001e150

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft-out-- N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-out-- N/A

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

      7. unsub-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unsub-neg N/A

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

      11. unpow2 N/A

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

      12. *-lft-identity N/A

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

      13. distribute-rgt-out-- N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+154} \lor \neg \left(y \leq 9.5 \cdot 10^{+150}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - y\right) \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = (x * y) * -y;
	} else {
		tmp = (x * y) * 1.0;
	}
	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) * 1.0d0
    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) * 1.0;
	}
	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) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(Float64(x * y) * Float64(-y));
	else
		tmp = Float64(Float64(x * y) * 1.0);
	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) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.5%

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

4. Final simplification 98.7%

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

Alternative 3: 92.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = (-y * y) * x;
	} else {
		tmp = (x * y) * 1.0;
	}
	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) * 1.0d0
    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) * 1.0;
	}
	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) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(Float64(Float64(-y) * y) * x);
	else
		tmp = Float64(Float64(x * y) * 1.0);
	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) * 1.0;
	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[(N[(x * y), $MachinePrecision] * 1.0), $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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -1 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 98.5%

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

4. Final simplification 89.7%

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

Alternative 4: 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]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 55.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

5. Applied rewrites 56.1%

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

Reproduce

herbie shell --seed 2024324 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))