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

Average Error: 0.1 → 0.1

Time: 6.9s

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. Applied egg-rr 0.1

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

3. Taylor expanded in x around 0 5.3

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

4. Simplified 0.1

   \[\leadsto \color{blue}{y \cdot \left(x - y \cdot x\right)} \]
   Proof
   (*.f64 y (-.f64 x (*.f64 y x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 y x) (*.f64 y (*.f64 y x)))): 5 points increase in error, 2 points decrease in error
   (-.f64 (*.f64 y x) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y y) x))): 40 points increase in error, 8 points decrease in error
   (-.f64 (*.f64 y x) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y 2)) x)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 y (pow.f64 y 2)))): 2 points increase in error, 4 points decrease in error
   (*.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 y (neg.f64 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (+.f64 y (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 y 2))))): 0 points increase in error, 0 points decrease in error
   (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (pow.f64 y 2)) y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 -1 (pow.f64 y 2)) y) x)): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	712

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 444220798211483.2:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.1
Cost	648

\[\begin{array}{l} t_0 := x \cdot \left(-y \cdot y\right)\\ \mathbf{if}\;y \leq -1.850651856962209:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0016066581281774998:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.9
Cost	648

\[\begin{array}{l} t_0 := y \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -1.850651856962209:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0016066581281774998:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.0
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))