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Insane calculations in bash
A few weeks ago I wrote an article about various methods of arithmetic calculation in shell scripts and in bash in particular, but it was all leading up to today's article, which I think is more interesting technically.

A while back, Zach Holman (who I hadn't heard of before, but who is apparently a bigwig at GitHub) implemented a kind of cute little hack, called "spark". It's a little shell utility, spark, which gets a list of numbers as its input and uses Unicode block characters to print a little bar graph of the numbers on the output. For example, the invocation:

  spark 2,4,6,8

  ▃▄▆▇

Let's begin by discussing the (very simple) mathematics that underlies drawing bar graphs. Suppose you want to generate a set of bars for the numbers $1, $9, $20. And suppose you can actually generate bars of integer heights only, say integers from 0–7:

  0    1 ▁  2 ▂  3 ▃  4 ▄  5 ▅  6 ▆  7 ▇

Absolute scaling

A reasonable thing to try is to say that since your largest number is $20, we will set 7 bar units = $20. Then 0.35 bar units = $1, and 3.45 bar units = $9. We'll call these the "natural heights" for the bars.

Unfortunately we can't render the bars at their natural heights; we can only render them at integer heights, so we have to round off. 0.35 bar units rounds off to 0, so we will represent $1 as no bar at all. 3.45 bar units rounds off, badly, to 3, but that's the way it goes; if you try to squeeze the numbers from 1 to 20 into the range 0 to 7, something has to give. Anyway, this gives

     (1,9,20) → ( ▃▇)

$$x\rightarrow {n\cdot x \over max } $$

I'll call this method "absolute scaling", because big numbers turn into big bars. (It fails for negative numbers, but we'll assume that the numbers are non-negative.)

     (0…20) → (  ▁▁▁▂▂▂▃▃▄▄▄▅▅▅▆▆▆▇▇)

     (0…20) → ( ▁▁▂▂▂▃▃▃▄▄▄▅▅▅▆▆▆▇▇▇)
     (1,9,20)      → (▁▄▇)

Relative scaling

     (18,19,20) → (▆▇▇)

     (18,19,20) → (▇▇▇)

     (18,19,20) → (▃▅▇)

In general, if we put the baseline at b, the natural height for a bar representing number x is:

$$x\rightarrow {n\cdot (x-b) \over (max-b) } $$

A reasonable choice of b would be the minimum input value, or perhaps a bit less than the minimum input value.

The shell sucks

	$ x=3.5; echo $((x + 1))
	bash: 3.5: syntax error: invalid arithmetic operator (error token is ".5")

My first response to this was to replace all the uses of $((…)) with bc, which, as I explained in the previous article, does not share this problem. M. Holman rejected this, saying that calling out to bc all the time made the program too slow. And there is something to be said for this. M. Holman also said that bc is non-portable, which I find astounding, since it has been in Unix since 1974, but sadly plausible.

So supposing that you take this complaint seriously, what can you do? Are you just doomed? No, I found a solution to the problem that solves all the problems. It is portable, efficient, and correct. It is also slightly insane.

Portable fractions in bash

	$ x=3.5; echo $((x + 1))
	bash: 3.5: syntax error: invalid arithmetic operator (error token is ".5")

	$ x_n=7; x_d=2; echo $((x_n + x_d))/$((x_d))
        9/2

        # given an input number which might be a decimal, convert it to
        # a rational number; set n and d to its numerator and
        # denominator.  For example, 3.3 becomes n=33 and d=10;
        # 17 becomes n=17 and d=1.
        to_rational() {
          # Crapulent bash can't handle decimal numbers, so we will convert
          # the input number to a rational
          if [[ $1 =~ (.*)\.(.*) ]] ; then
              i_part=${BASH_REMATCH[1]}
              f_part=${BASH_REMATCH[2]}
              n="$i_part$f_part";
              d=$(( 10 ** ${#f_part} ))
          else
              n=$1
              d=1
          fi
        }

Now we can rewrite all the shell arithmetic in terms of rational numbers. If a_n and a_d are the numerator and denominator of a, and b_n and b_d are the numerator and denominator of b, then addition, subtraction, multiplication, and even division of a and b are fast, easy, and even portable:

        # a + b
        sum_n = $((a_n * b_d + a_d * b_n))
        sum_d = $((a_d * b_d))

        # a - b
        diff_n = $((a_n * b_d - a_d * b_n))
        diff_d = $((a_d * b_d))

        # a * b
        prod_n = $((a_n * b_n))
        prod_d = $((a_d * b_d))

        # a / b
        quot_n = $((a_n * b_d))
        quot_d = $((a_d * b_n))

        greatest_int = $((a_n / a_d))

        nearest_int = $(( (a_n * 2 + a_d) / (a_d * 2) ))

For complicated calculations, you can work the thing out as several steps, or you can solve it on paper and then just embed a big rational expression. For example, suppose you want to calculate ((x-min)·number_of_tiers)/range, where number_of_tiers is known to be an integer. You could do each operation in a separate step, or you could use instead:

  tick_index_n=$(( ( x_n * min_d - min_n * x_d ) * number_of_tiers * range_d ))
  tick_index_d=$(( range_n * x_d * min_d ))

	function to_dec {
	  n=$1
	  d=$2
          maxit=$(( 1 + ${3:-10} ))
	  while [ $n != 0 -a $maxit -gt -1 ]; do
	    next=$((n/d))
	    if [ "$r" = "" ]; then r="$next."; else r="$r$next"; fi
	    n=$(( (n - d * next) * 10 ))
	    maxit=$(( maxit - 1 ))
	  done
	  r=${r:-'0.'}
	}

Coming next month, libraries in bash for computing with continued fractions using Gosper's algorithms. Ha ha, just kidding. The obvious next step is to implement base-10 floating-point numbers in bash like this:

  prod_mantissa=$((a_mantissa * b_mantissa))
  prod_exponent=$((a_exponent + b_exponent))

[ Addendum 20180101: Shane Hansen did something similar to calculate Euler's number (2.71818…) in Bash a while back. It might be fun to compare our implementations. ]

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