Mathematical Expression Result Assigned to a Bash Variable

How to echo result of math expression AND save it to a variable in Bash

How about

wcExp=$(echo 'e(l(5)*.16)' | bc -l)
echo "$wcExp"

How to assign output of float expression to a variable in a shell script?

You need to use quotes around scale=2 and your math expression:

a=10; b=3
c=$(echo "scale=2; $a / $b" | bc)    
echo "$c"

3.33

Execute a input in form of mathematical expression in bash

You can create a simple script for that, let's call test.sh:

#!/bin/bash
x=$1
echo $(($x))

You can replace the line echo $(($x)) with perl -e "print $x" to get floating point output.

And execute it with your input:

$ test.sh "5+50*3/20 + (19*2)/7"
17

With floating point:

#!/bin/bash
x=$1
var=$(perl -e "print $x")
echo $var

The results:

$ test.sh "5+50*3/20 + (19*2)/7"
17.9285714285714

$ test.sh "-105+50*3/20 + (19^2)/7"
-95.0714285714286

Evaluating a mathematical expression stored as a string, into a single number (bash)

How about:

python -c "print $NUM"

By the way, you could just write

BASE="1.e8*1.07**%d"

(In fact, you don't even need the quotes.)

variable error in bash when doing calculation

You received that error because you passed an invalid arithmetic expression into a bash arithetic expansion. Only an arithmetic expression is allowed for this place. What you try to do seems like this:

ref="$(grep -v ">" /data/ref/EN | wc -c)"
sample="$(grep -v ">" /example/SR | wc -l)"

# this is only integer division
#u=$(( sample / ref ))
#z=$(( 100 * u ))

# to do math calculations, you can use bc
u=$(bc <<< "scale=2; $sample/$ref")
z=$(bc <<< "scale=2; 100*$u")

printf "%d, %d, %.2f, %.2f\n" "$ref" "$sample" "$u" "$z"

so hopefully you get an output like this:

41858, 38986, 0.93, 93.00

Notes:

There is no need to cd before executing a grep, it accepts the full path with the target filename as an argument. So without changing directory, you can grep various locations.
In order to save the output of your command (which is only a number) you don't need to save it in a file and cat the file. Just use the syntax var=$( ) and var will be assigned the output of this command substitution.
Have in mind that / will result to 0 for the division 38986/41858 because it's the integer division. If you want to do math calculations with decimals, you can see this post for how to do them using bc.
To print anything, use the shell builtin printf. Here the last two numbers are formatted with 2 decimal points.