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×

• 24.2% 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

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

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 99.9%

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

Alternative 2: 97.9% 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 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   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 97.7%

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

4. Final simplification 97.6%

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

Alternative 3: 92.3% 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 99.8%

      \[\left(x \cdot y\right) \cdot \left(1 - y\right) \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   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 97.7%

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

4. Final simplification 92.9%

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

Alternative 4: 63.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(N[(x * y), $MachinePrecision] * 1.0), $MachinePrecision], N[((-y) * x), $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. 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 77.7%

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

   if 1 < y

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 33.3

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

   9. Applied rewrites 33.3%

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

Alternative 5: 20.6% accurate, 1.8× speedup

Math

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

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(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. Add Preprocessing

4. Applied rewrites 99.9%

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

6. Applied rewrites 55.7%

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

7. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 23.2

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

9. Applied rewrites 23.2%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))