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

Percentage Accurate: 99.9% → 99.8%

Time: 6.2s

Alternatives: 5

Speedup: 1.0×

• 24.8% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+156} \lor \neg \left(y \leq 10^{+41}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.2e+156) (not (<= y 1e+41)))
   (* y (* y (- x)))
   (* x (* y (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.2e+156) || !(y <= 1e+41)) {
		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 <= (-1.2d+156)) .or. (.not. (y <= 1d+41))) 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 <= -1.2e+156) || !(y <= 1e+41)) {
		tmp = y * (y * -x);
	} else {
		tmp = x * (y * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.2e+156) or not (y <= 1e+41):
		tmp = y * (y * -x)
	else:
		tmp = x * (y * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.2e+156) || !(y <= 1e+41))
		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 <= -1.2e+156) || ~((y <= 1e+41)))
		tmp = y * (y * -x);
	else
		tmp = x * (y * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.2e+156], N[Not[LessEqual[y, 1e+41]], $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 < -1.2000000000000001e156 or 1.00000000000000001e41 < 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%

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

      1. mul-1-neg 99.9%

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

      2. distribute-rgt-neg-out 99.9%

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

   7. Simplified 99.9%

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

   if -1.2000000000000001e156 < y < 1.00000000000000001e41

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+156} \lor \neg \left(y \leq 10^{+41}\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(1 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.4× speedup

Math

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

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 96.9%

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

      1. mul-1-neg 96.9%

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

      2. distribute-rgt-neg-out 96.9%

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

   7. Simplified 96.9%

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

   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 98.0%

      \[\leadsto y \cdot \color{blue}{x} \]
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: 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 4: 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 62.3%

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

6. Final simplification 62.3%

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

Alternative 5: 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* 95.6%

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

3. Simplified 95.6%

   \[\leadsto \color{blue}{x \cdot \left(y \cdot \left(1 - y\right)\right)} \]
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-- 95.5%

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

   3. associate-*r/ 93.4%

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

   4. metadata-eval 93.4%

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

   5. pow2 93.4%

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

   6. +-commutative 93.4%

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

6. Applied egg-rr 93.4%

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

   1. associate-*l* 90.4%

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

   2. *-commutative 90.4%

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

   3. associate-/l* 88.4%

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

   4. sub-neg 88.4%

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

   5. distribute-lft-in 88.4%

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

   6. *-rgt-identity 88.4%

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

   7. distribute-rgt-neg-in 88.4%

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

   8. unpow2 88.4%

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

   9. cube-mult 88.4%

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

   10. unsub-neg 88.4%

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

8. Simplified 88.4%

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

9. Taylor expanded in y around inf 34.9%

   \[\leadsto \left(y - {y}^{3}\right) \cdot \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. *-commutative 34.9%

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

    2. clear-num 33.7%

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

    3. flip-- 20.9%

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

    4. frac-times 14.1%

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

    5. pow-sqr 14.0%

       \[\leadsto \frac{1 \cdot \left(y \cdot y - \color{blue}{{y}^{\left(2 \cdot 3\right)}}\right)}{\frac{y}{x} \cdot \left(y + {y}^{3}\right)} \]

    6. metadata-eval 14.0%

       \[\leadsto \frac{1 \cdot \left(y \cdot y - {y}^{\color{blue}{6}}\right)}{\frac{y}{x} \cdot \left(y + {y}^{3}\right)} \]

    7. *-un-lft-identity 14.0%

       \[\leadsto \frac{\color{blue}{y \cdot y - {y}^{6}}}{\frac{y}{x} \cdot \left(y + {y}^{3}\right)} \]

    8. pow2 14.0%

       \[\leadsto \frac{\color{blue}{{y}^{2}} - {y}^{6}}{\frac{y}{x} \cdot \left(y + {y}^{3}\right)} \]

11. Applied egg-rr 14.0%

    \[\leadsto \color{blue}{\frac{{y}^{2} - {y}^{6}}{\frac{y}{x} \cdot \left(y + {y}^{3}\right)}} \]
12. Step-by-step derivation

    1. associate-*l/ 9.3%

       \[\leadsto \frac{{y}^{2} - {y}^{6}}{\color{blue}{\frac{y \cdot \left(y + {y}^{3}\right)}{x}}} \]

13. Simplified 9.3%

    \[\leadsto \color{blue}{\frac{{y}^{2} - {y}^{6}}{\frac{y \cdot \left(y + {y}^{3}\right)}{x}}} \]

14. Taylor expanded in y around 0 2.7%

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

15. Final simplification 2.7%

    \[\leadsto x \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))