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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 5

Speedup: N/A×

• 25.1% 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

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

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. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 0.4× speedup

Math

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

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 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. distribute-rgt-neg-out 98.0%

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

   7. Simplified 98.0%

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

   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. *-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 97.0%

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

4. Final simplification 97.5%

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

Alternative 3: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+101}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+101) (* x (* y (- 1.0 y))) (* y (* y (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+101) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = 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 <= 1d+101) then
        tmp = x * (y * (1.0d0 - y))
    else
        tmp = y * (y * -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+101) {
		tmp = x * (y * (1.0 - y));
	} else {
		tmp = y * (y * -x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+101:
		tmp = x * (y * (1.0 - y))
	else:
		tmp = y * (y * -x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+101)
		tmp = Float64(x * Float64(y * Float64(1.0 - y)));
	else
		tmp = Float64(y * Float64(y * Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+101)
		tmp = x * (y * (1.0 - y));
	else
		tmp = y * (y * -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+101], N[(x * N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.9999999999999998e100

   1. Initial program 99.9%

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

      1. associate-*l* 99.0%

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

   3. Simplified 99.0%

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

   if 9.9999999999999998e100 < 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* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+101}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (* y x) (* y (- x))))

C

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

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = y * x
	else:
		tmp = y * -x
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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 74.9%

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

   if 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* 91.0%

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

   3. Simplified 91.0%

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

      1. associate-*r* 99.7%

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

      2. flip-- 90.9%

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

      3. associate-*r/ 82.0%

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

      4. metadata-eval 82.0%

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

      5. pow2 82.0%

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

      6. +-commutative 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. associate-*l* 77.3%

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

      2. associate-/l* 77.3%

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

      3. sub-neg 77.3%

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

      4. distribute-lft-in 77.3%

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

      5. *-rgt-identity 77.3%

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

      6. distribute-rgt-neg-in 77.3%

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

      7. unpow2 77.3%

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

      8. cube-mult 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in y around 0 0.7%

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

       1. associate-/r/ 0.7%

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

       2. /-rgt-identity 0.7%

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

       3. add-sqr-sqrt 0.2%

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

       4. sqrt-unprod 23.3%

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

       5. sqr-neg 23.3%

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

       6. distribute-rgt-neg-out 23.3%

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

       7. distribute-rgt-neg-out 23.3%

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

       8. sqrt-unprod 20.4%

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

       9. add-sqr-sqrt 36.7%

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

       10. distribute-rgt-neg-out 36.7%

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

       11. distribute-lft-neg-in 36.7%

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

   11. Applied egg-rr 36.7%

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

4. Final simplification 65.7%

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

Alternative 5: 56.6% 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 99.9%

   \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
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 57.0%

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

6. Final simplification 57.0%

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

Reproduce

herbie shell --seed 2024036 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))