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

Average Error: 0.1 → 0.1

Time: 5.1s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 5.5

   \[\leadsto \color{blue}{x \cdot \left(y - y \cdot y\right)} \]
   Proof
   (*.f64 x (-.f64 y (*.f64 y y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 x y) (*.f64 x (*.f64 y y)))): 3 points increase in error, 6 points decrease in error
   (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 x y) 1)) (*.f64 x (*.f64 y y))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (*.f64 x y) 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) y))): 7 points increase in error, 34 points decrease in error
   (Rewrite=> distribute-lft-out--_binary64 (*.f64 (*.f64 x y) (-.f64 1 y))): 5 points increase in error, 3 points decrease in error

3. Applied egg-rr 5.5

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	712

\[\begin{array}{l} t_0 := \left(y \cdot x\right) \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+60}:\\ \;\;\;\;x \cdot \left(y - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.4
Cost	648

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

Alternative 3
Error	2.0
Cost	648

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

Alternative 4
Error	0.1
Cost	448

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

Alternative 5
Error	0.1
Cost	448

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

Alternative 6
Error	21.9
Cost	192

\[y \cdot x \]

Reproduce

herbie shell --seed 2022337 
(FPCore (x y)
  :name "Statistics.Distribution.Binomial:$cvariance from math-functions-0.1.5.2"
  :precision binary64
  (* (* x y) (- 1.0 y)))