Real Brainfuck

Real Brainfuck is a brainfuck variant that operates on the real-valued range {0, 1}. The tape starts with all cells set to 0.5. It also has a stack. This is a more usable version of an esolang i designed in the past.

It has the following instructions:

Instruction	Explanation
<	Move data pointer left on tape
>	Move data pointer right on tape
0	Set current cell c to (c + 0) / 2
1	Set current cell c to (c + 1) / 2
N	Set current cell c to 1 - c
[	Randomly choose between 0 and 1 with probability c. if zero jump past matching ]
]	Randomly choose between 0 and 1 with probability c. if not zero jump back to matching [
^	Push cell data to stack
V	Pop item from stack to current cell
&	Pops top two items on the stack, multiplies them together, and pushes the result back
X	Pops top two items on the stack, averages them together, and pushes the result back
,	Obtains real-valued or binary input from an implementation-dependent source and sets the cell to that value
.	Outputs current real value
q	Outputs either 0 or 1 with probability c

Real Brainfuck 2 Variant: operates on all complex real numbers. It has support for abnormal probabilities and the following additional Instructions:

Instruction	Explanation
+	c = c + 1
-	c = c - 1
;	c = c root of c
\	pop two items from stack, a & b, and push c: c = loga(b)
/	c = sin(c)
z	c = c ^ pi
:	pop two items from stack, a & b, and push c: c = a ^ b
f	c = c ^^ 2 (c tetrate 2)
e	set cell to transcendental number e
p	set cell to transcendental number pi
!	c = c! (multiplication factorial)
?	c = c? (addition factorial)
$	c = c$ (exponential factorial)
{	begin ()infinite loop (finishes in finite time, hyperloop)
}	end ()infinite loop (finishes in finite time, hyperloop)