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

Average Error: 0.1 → 0.1

Time: 7.5s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 5.4

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

   [Start] 0.1	\[ \left(x \cdot y\right) \cdot \left(1 - y\right) \]
   distribute-lft-out-- [<=] 0.1	\[ \color{blue}{\left(x \cdot y\right) \cdot 1 - \left(x \cdot y\right) \cdot y} \]
   *-rgt-identity [=>] 0.1	\[ \color{blue}{x \cdot y} - \left(x \cdot y\right) \cdot y \]
   associate-*l* [=>] 5.4	\[ x \cdot y - \color{blue}{x \cdot \left(y \cdot y\right)} \]
   distribute-lft-out-- [=>] 5.4	\[ \color{blue}{x \cdot \left(y - y \cdot y\right)} \]

3. Taylor expanded in x around 0 5.4

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

4. Simplified 0.1

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

   [Start] 5.4	\[ \left(y - {y}^{2}\right) \cdot x \]
   *-commutative [=>] 5.4	\[ \color{blue}{x \cdot \left(y - {y}^{2}\right)} \]
   sub-neg [=>] 5.4	\[ x \cdot \color{blue}{\left(y + \left(-{y}^{2}\right)\right)} \]
   unpow2 [=>] 5.4	\[ x \cdot \left(y + \left(-\color{blue}{y \cdot y}\right)\right) \]
   distribute-rgt-out [<=] 5.4	\[ \color{blue}{y \cdot x + \left(-y \cdot y\right) \cdot x} \]
   cancel-sign-sub-inv [<=] 5.4	\[ \color{blue}{y \cdot x - \left(y \cdot y\right) \cdot x} \]
   associate-*l* [=>] 0.1	\[ y \cdot x - \color{blue}{y \cdot \left(y \cdot x\right)} \]
   distribute-lft-out-- [=>] 0.1	\[ \color{blue}{y \cdot \left(x - y \cdot x\right)} \]

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+147} \lor \neg \left(y \leq 20000000000000\right):\\ \;\;\;\;y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \end{array} \]

Alternative 2
Error	7.1
Cost	649

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

Alternative 3
Error	1.8
Cost	649

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

Alternative 4
Error	21.8
Cost	192

\[y \cdot x \]

Alternative 5
Error	61.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))