Calculating pi using bash

I see a few things that could allow you to improve your program. First, though, I don't consider myself a bash expert, so there may well be better ways of doing these things.

Use a "shebang" line

As this question points out, you should always use a "shebang" line for your bash scripts. So the first line would be:

#!/usr/bin/env bash

Pass values as arguments

Rather than prompting for the SCALE value, it's generally better to use a command line argument. That way, the script can be reused by other shell scripts.

Provide a stopping mechanism

As each term is calculated, eventually, it will be equal to zero given the passed scale. This suggests a mechanism for stopping: check each term for 0 before adding it.

Indent do and while loops

I don't know of a bash style guide (there probably is one!) but I like to see the contents of loops indented to make it easier to read.

Putting it all together

Here's a modification of your script with all of these suggestions implemented:

bashpi.sh

#!/usr/bin/env bash
SCALE=$1
VALUE=2
PI=0
FITNESS=1
DELTA=3
while [ $(echo "scale=$SCALE;$DELTA==0" |bc) != "1" ]
do
    PI=$(echo "scale=$SCALE;$PI+$DELTA" | bc)
    DELTA=$(echo "scale=$SCALE;(4/($VALUE*($VALUE+1)*($VALUE+2)))-(4/(($VALUE+2)*($VALUE+3)*($VALUE+4)))" | bc)
    VALUE=$(($VALUE+4))
    FITNESS=$(($FITNESS+1))
    echo "###############"
    echo "--> $FITNESS // $VALUE"
    echo "$PI"
done

To better understand how this works, you can replace the three echo statements with this one:

echo "DELTA = ${DELTA} --> ${FITNESS} // ${VALUE} : ${PI}"

Sample output

With ./bashpi.sh 4, and the modified echo above I get this output:

DELTA = .1333 --> 2 // 6 : 3
DELTA = .0064 --> 3 // 10 : 3.1333
DELTA = .0012 --> 4 // 14 : 3.1397
DELTA = .0003 --> 5 // 18 : 3.1409
DELTA = .0001 --> 6 // 22 : 3.1412
DELTA = .0001 --> 7 // 26 : 3.1413
DELTA = .0001 --> 8 // 30 : 3.1414
DELTA = 0 --> 9 // 34 : 3.1415

Edward's answer offers the necessary insights to almost all the parts of the original inquiry. I'd like to add my two cents on the "if you have other suggestions on how to improve the code and/or know a better way of calculating pi (with some explanation), I would like to hear" part.

The following script is a compilation of my findings of calculating pi in bash on similar sites:

#!/usr/bin/env bash

: ${SCALE:=${1:-5}}
: ${_PROF_INIT:=-1}
: ${_PROF_START:=0}

function _profiler_calc () { 
    echo "$@" | bc -l | xargs printf "%5.0f\n"
}

function _profiler_ts() {
    date +%s%3N
}

function _profiler_mark() {
    _PROF_START=$(_profiler_ts)
    if [ "x${_PROF_INIT}" == "x-1" ]; then
        _PROF_INIT=${_PROF_START}
    fi
}

function _profiler_elapsed() {
    NOW="$(_profiler_ts)"
    ELAPSED=$(_profiler_calc "${NOW} - ${_PROF_START}")
    printf "%s\n" "$ELAPSED"
}

# http://codereview.stackexchange.com/questions/94266/calculating-pi-using-bash
function calc_edward() {
    local -i SCALE=$1
    local -i VALUE=2
    local -i FITNESS=1
    local DELTA=3
    local PI=0
    if [ $SCALE -gt 10 ]; then
        echo "n/a"
        return
    fi
    while [ $(echo "scale=$SCALE;$DELTA==0" | bc) != "1" ]; do
        PI=$(echo "scale=$SCALE;$PI+$DELTA" | bc)
        DELTA=$(echo "scale=$SCALE;(4/($VALUE*($VALUE+1)*($VALUE+2)))-(4/(($VALUE+2)*($VALUE+3)*($VALUE+4)))" | bc)
        VALUE=$(($VALUE+4))
        FITNESS=$(($FITNESS+1))
    done
    echo $PI
}

# http://superuser.com/questions/275516/how-can-i-generate-pi-to-a-given-number-of-decimal-places-from-a-script
function calc_8088() {
    local -i SCALE=$1
    bc -l <<< "scale=$SCALE; 4*a(1)"
}

# http://stackoverflow.com/questions/23524661/how-can-i-calculate-pi-using-bash-command
function calc_devnull() {
    local -i SCALE=$1
    { echo -n "scale=$SCALE;"; seq 1 2 $((2*SCALE)) | xargs -n1 -I{} echo '(16*(1/5)^{}/{}-4*(1/239)^{}/{})';} | paste -sd-+ | bc -l
}

# http://unix.stackexchange.com/questions/166220/how-do-i-print-pi-3-14159/166582#166582
function calc_pabouk() {
    local -i SCALE=$1
    bc -l <<< "s=$((SCALE+1)); scale=s+2; pi=4*a(1)+5*10^(-s); scale=s-1; pi/1"
}

_profiler_mark
pi=$(calc_edward ${SCALE})
printf "%15s: %$((SCALE+2))s %-$((SCALE+1))s\n" "Edward's Pi" "$pi" "[time:$(_profiler_elapsed)ms]"

_profiler_mark
pi=$(calc_8088 ${SCALE})
printf "%15s: %s %-$((SCALE+1))s\n" "8088's Pi" "$pi" "[time:$(_profiler_elapsed)ms]"

_profiler_mark
pi=$(calc_devnull ${SCALE})
printf "%15s: %s %-$((SCALE+1))s\n" "devnull's Pi" "$pi" "[time:$(_profiler_elapsed)ms]"

_profiler_mark
pi=$(calc_pabouk ${SCALE})
printf "%15s: %s %-$((SCALE+1))s\n" "pabouk's Pi" "$pi" "[time:$(_profiler_elapsed)ms]"

I have added the pointers to the original code inside my script. If you run it, you'll get some interesting output:

$ for s in $(seq 2 4 30); do ./bashpi.sh $s; done
    Edward's Pi: 3.14 [time:   63ms]
      8088's Pi: 3.12 [time:   14ms]
   devnull's Pi: 3.20 [time:   21ms]
    pabouk's Pi: 3.14 [time:   14ms]
    Edward's Pi: 3.141591 [time:  336ms]
      8088's Pi: 3.141592 [time:   14ms]
   devnull's Pi: 3.141595 [time:   30ms]
    pabouk's Pi: 3.141593 [time:   15ms]
    Edward's Pi: 3.1415926488 [time: 2879ms]
      8088's Pi: 3.1415926532 [time:   14ms]
   devnull's Pi: 3.1415926538 [time:   41ms]
    pabouk's Pi: 3.1415926536 [time:   14ms]
    Edward's Pi:              n/a [time:   10ms]
      8088's Pi: 3.14159265358976 [time:   14ms]
   devnull's Pi: 3.14159265358981 [time:   48ms]
    pabouk's Pi: 3.14159265358979 [time:   14ms]
    Edward's Pi:                  n/a [time:   10ms]
      8088's Pi: 3.141592653589793236 [time:   15ms]
   devnull's Pi: 3.141592653589793243 [time:   59ms]
    pabouk's Pi: 3.141592653589793238 [time:   14ms]
    Edward's Pi:                      n/a [time:   11ms]
      8088's Pi: 3.1415926535897932384624 [time:   15ms]
   devnull's Pi: 3.1415926535897932384628 [time:   74ms]
    pabouk's Pi: 3.1415926535897932384626 [time:   14ms]
    Edward's Pi:                          n/a [time:   10ms]
      8088's Pi: 3.14159265358979323846264336 [time:   14ms]
   devnull's Pi: 3.14159265358979323846264340 [time:   88ms]
    pabouk's Pi: 3.14159265358979323846264338 [time:   15ms]
    Edward's Pi:                              n/a [time:   10ms]
      8088's Pi: 3.141592653589793238462643383276 [time:   14ms]
   devnull's Pi: 3.141592653589793238462643383281 [time:  111ms]
    pabouk's Pi: 3.141592653589793238462643383280 [time:   14ms]