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

Percentage Accurate: 99.9% → 99.9%

Time: 4.1s

Alternatives: 5

Speedup: 1.0×

• 25.0% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * x + N[(y * N[(y * (-x)), $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. associate-*l* 95.2%

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

3. Simplified 95.2%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 92.5%

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

   4. *-commutative 92.5%

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

   5. *-un-lft-identity 92.5%

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

   6. *-commutative 92.5%

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

   7. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, x, y \cdot \left(y \cdot \left(-x\right)\right)\right) \]

Alternative 2: 94.2% accurate, 1.0× speedup

Math

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

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(x * Float64(y * Float64(1.0 - y)))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(y * 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. associate-*l* 95.2%

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

3. Simplified 95.2%

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

4. Final simplification 95.2%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

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(Float64(y * x) * Float64(1.0 - y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 56.5% 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. associate-*l* 95.2%

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

3. Simplified 95.2%

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

4. Taylor expanded in y around 0 60.0%

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

5. Final simplification 60.0%

   \[\leadsto y \cdot x \]

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* 95.2%

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

3. Simplified 95.2%

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

   1. associate-*r* 99.9%

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

   2. flip-- 95.1%

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

   3. associate-*r/ 91.5%

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

   4. metadata-eval 91.5%

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

   5. pow2 91.5%

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

   6. +-commutative 91.5%

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

5. Applied egg-rr 91.5%

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

   1. associate-*l* 87.8%

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

   2. associate-/l* 89.0%

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

   3. sub-neg 89.0%

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

   4. distribute-lft-in 89.0%

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

   5. *-rgt-identity 89.0%

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

   6. distribute-rgt-neg-in 89.0%

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

   7. unpow2 89.0%

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

   8. cube-mult 89.1%

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

   9. unsub-neg 89.1%

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

7. Simplified 89.1%

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

8. Taylor expanded in y around 0 50.3%

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

9. Taylor expanded in y around inf 2.7%

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

10. Final simplification 2.7%

    \[\leadsto x \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))