1 Representing Numbers… in General

In any base (greater than 2), the digits $b_i$ (i^th mantissa digit/bit) and $s_i$ (i^th exponent digit/bit) will fall in the range $[0, \beta-1]$. It is useful to use ordered set notation:

$b_i, s_i \in <0, 1, 2, \ldots, \beta-1>$

Consider base 2 and base 10:

• base 2 - $<0, 1>$
• base 10 - $<0, 1, 2, 3, 4, 5, 6, 7, 8, \beta-1>$ where $\beta-1 = 9$.

1.1 Smallest Mantissa

We are interested in the smallest possible mantissa for any base, $\beta \ge 2$. Naively, one might write $0.0$. However, we are interested in the smallest non-zero mantissa. This leads to

$\frac{1}{\beta} = 0.10000000000\ldots$

The normalization constraint requires that the first digit (i.e., $b_{-1}$) after the decimal place be non-zero. For all numeric bases two or greater, the smallest digit is $1$.

1.2 Largest Mantissa

Think about the largest number in…

• Base 2 - $0.1111\ldots$
• Base 8 - $0.7777\ldots$
• Base 10 - $0.9999\ldots$
• Base 16 - $0.FFFF\ldots$

Do you notice a pattern? The largest digit in any base is always the base minus 1 (i.e., $\beta - 1$).

1.3 Representing Real Numbers

Let $x^{*}$ (read as “x-star” or “x-chop”) represent a number that is subject to finite precision (i.e., must be stored using a finite number of digits/bits).

$x^{*} = \pm \left( \sum\limits_{i=1}^t b_{-i} \beta^{-i} \right) * \beta^{e^{*}}$ where $e^{*} = \pm \sum\limits_{i=1}^s s_{i} \beta^{i}$

Let us break this notation down…

$$\sum\limits_{i=1}^t b_{-i} \beta^{-i}$$

represents the mantissa (i.e., the digits to the right of the decimal place). The number $t$ represents the number of mantissa digits.

$$\pm \sum\limits_{i=1}^s s_{i} \beta^{i} $$

represents the exponent digits. The number $s$ represents the number of exponent digits. Keep in mind that similar to traditional base-10 scientific notation the sign (positive or negative) determines whether a left or right shift should occur.

1.4 Convenient Notation

We can use a convenient notation to capture $x$ and $x^{*}$.

$$ x^{*} \in \mathbb{R}(t,s) $$

$x^{*}$ is a real number that can be represented by $t$ mantissa digits and $s$ exponent digits. $x$ is more interesting…

$$ x \in \mathbb{R}(\infty,\infty) $$

$x$ is a real number for which we have access to infinite precision (i.e., infinite digits).

2 Smallest Number and Largest Number

We know that $x^{*}$ is defined as

$x^{*} = \pm \left( \sum\limits_{i=1}^t b_{-i} \beta^{-i} \right) * \beta^{e^{*}}$ where $e^{*} = \pm \sum\limits_{i=1}^s s_{i} \beta^{i}$

Our goal is to determine the lower limit (smallest number) and upper limit (largest number) that can be represented. Let us find four (4) values:

• smallest possible mantissa
• largest possible mantissa
• largest positive exponent, $|e|$
• “smallest” possible exponent, $-|e|$

2.1 The Mantissa Pieces

The smallest possible non-zero mantissa is

$$ \beta^{-i} = \frac{1}{\beta} $$

by the normalization constraint. That is the first piece of the puzzle. The largest possible mantissa is next. Let us start with…

$$ \sum\limits_{i=1}^t b_{-i} \beta^{-i} $$

To obtain the largest number… we need the largest possible digit in every position. That leads to $b_{-i} = (\beta - 1) \phantom{2} \forall i$… which leads to…

$$(\beta - 1 ) \left(\sum\limits_{i=1}^t \beta^{-i}\right)$$

The sum can be evaluated using the geometric series formula.

$$ \frac{\alpha(1-r^k)}{(1-r)} $$

where

• $k = t$
• $r = \beta^{-1}$
• $\alpha = \beta^{-1}$

Applying the geometric series formula results in…

$$ \begin{eqnarray} (\beta - 1 ) \left(\sum\limits_{i=1}^t \beta^{-i}\right) &=& \frac{\beta^{-1}(1- \beta^{-t})(\beta - 1 )}{1 - \beta^{-1}}\\ &=& \frac{\beta^{-1}(\beta - 1)(1- \beta^{-t})}{1 - \beta^{-1}}\\ &=& \frac{(1 - \beta^{-1})(1- \beta^{-t})}{1 - \beta^{-1}}\\ &=& (1- \beta^{-t}) \end{eqnarray} $$

2.2 Mantissa Bounds

The mantissa ($f$) bounds can be written succinctly.

$$ \beta^{-1} \le f \le 1 - \beta^{-t} $$

2.3 The Exponent Pieces

The exponent portion of a real number is defined as…

$\beta^{e^{*}}$ where $e^{*} = \pm \sum\limits_{i=1}^s s_{i} \beta^{i} $

We are interested in $max(|e^{*}|)$. Let us set $s_i = \beta - 1$ for all $i$ (or using math notation… $\forall_i$ let $s_i = \beta - 1$).

$$ \begin{eqnarray} |e^{*}| &=& |\pm \sum\limits_{i=1}^s s_{i} \beta^{i}| \\ |e^{*}| &=& \sum\limits_{i=1}^s s_{i} \beta^{i} \\ |e^{*}| &\le& \sum\limits_{i=1}^s (\beta - 1) \beta^{i} \\ |e^{*}| &\le& \frac{(\beta - 1)(1 - \beta^{s})}{1 - \beta} \\ |e^{*}| &\le& \frac{(\beta - 1)(1 - \beta^{s})}{-(\beta - 1)} \\ |e^{*}| &\le& -(1 - \beta^{s})\\ |e^{*}| &\le& \beta^{s} - 1 \\ \end{eqnarray} $$

Note how the geometric series formula was applied:

• $k = s$
• $\alpha = -1$
• $\beta - 1 = -(1 - \beta)$

Note the result…

$$ max(|e|) = \beta^{s} - 1 $$

This is the largest magnitude for an exponent. This leads to…

$$ min(\beta^{e^*}) = \beta^{-(\beta^{s} - 1)} $$

and

$$ max(\beta^{e^*}) = \beta^{\beta^{s} - 1} $$

as the largest and smallest possible $\beta^{e^*}$ terms, respectively

2.4 Exponent Bounds

The exponent ($\beta^{e^*})$ bounds can be written succinctly.

$$ \beta^{-(\beta^{s} - 1)} \le \beta^e \le \beta^{\beta^{s} - 1} $$

2.5 Putting the Pieces Together

Combining the minimum and maximum terms from the mantissa analysis and exponent analysis leads to…

$$ \beta^{-1}\beta^{-(\beta^{s} - 1)} \le x^{*} \le (1 - \beta^{-t})\beta^{\beta^{s} - 1} $$

If we let $\beta = 2$… the result matches our previous base-2 analysis.

$$ 2^{-1}2^{-(2^{s} - 1)} \le x^{*} \le (1 - 2^{-t})2^{2^{s} - 1} $$

3 Bounding Relative Error

Now… it is time to derive an upper bound for the error between $x$ and $x^*$. We know that (by definition):

$$ x^{*} = \pm \left( \sum\limits_{i=1}^t b_{-i} \beta^{-i} \right) * \beta^{e^{*}} $$

where

$$ e^{*} = \pm \sum\limits_{i=1}^s s_{i} \beta^{i} $$

and

$$ x = \pm \left( \sum\limits_{i=1}^{\infty} b_{-i} \beta^{-i} \right) * \beta^{e^{*}} $$

where

$$e = \pm \sum\limits_{i=1}^{\infty} s_{i} \beta^{i} $$

The difference comes down to finite (i.e., limited) precision, i.e., $x^* \in \mathbb{R}(t, s)$ vs $x \in \mathbb{R}(\infty, \infty)$.

3.1 Absolute Error

Relative error is defined as $\frac{|x-x^*|}{|x|}$. Let us start with the numerator… which happens to be absolute error!

$$ \left| x - x^{*} \right| = \left| \sum\limits_{i=0}^{\infty} b_{-i}\beta^{-i} * \beta^{e} - \sum\limits_{i=0}^{k} b_{-i}\beta^{-i} * \beta^{e^{*}} \right| $$

Let…

• $b_{-i} = b_{-i}^{*}$ for $i \le k$ (truncation)
• $e = e^{*}$ (same magnitude)

These two “small” observations lead to…

$$ \begin{eqnarray} \left| x - x^{*} \right| &=& \left| \sum\limits_{i=0}^{k} b_{-i}\beta^{-i} * \beta^{e} + \sum\limits_{i=k+1}^{\infty} b_{-i}\beta^{-i} * \beta^{e} - \sum\limits_{i=0}^{k} b_{-i}\beta^{-i}* \beta^{e} \right| \\ &=& \left| \sum\limits_{i=0}^{k} b_{-i}\beta^{-i} + \sum\limits_{i=k+1}^{\infty} b_{-i}\beta^{-i} - \sum\limits_{i=0}^{k} b_{-i}\beta^{-i} \right| \beta^{e} \\ &=& \left| \sum\limits_{i=k+1}^{\infty} b_{-i}\beta^{-i} \right| \beta^{e} \\ \end{eqnarray} $$

Now… we need to bound the error by using the largest possible mantissa. Letting $b_{-i} = (\beta - 1)$ for all $i$ leads to

$$ \begin{eqnarray} \left| x - x^{*} \right| &=& \left| \sum\limits_{i=k+1}^{\infty} b_{-i}\beta^{-i} \right| \beta^{e} \\ &\le& \left| \sum\limits_{i=k+1}^{\infty} (\beta - 1) \beta^{-i} \right| \beta^{e} \\ &\le& \left| (\beta - 1) \sum\limits_{i=k+1}^{\infty}\beta^{-i} \right| \beta^{e} \\ &\le& \left| (\beta - 1) \beta^{-k} \sum\limits_{i=1}^{\infty}\beta^{-i} \right| \beta^{e} \\ &\le& \left| (\beta - 1) \beta^{-k - 1} \sum\limits_{i=0}^{\infty}\beta^{-i} \right| \beta^{e} \\ \end{eqnarray} $$

Now we need to tackle the sum

$$ \sum\limits_{i=0}^{\infty}\beta^{-i} $$

We can use the geometric series formula for a convergent infinite series..

$S_{\infty} = \frac{1}{1-r}$ iff $|r| < 1$

In this case… $r = \beta^{-1}$.

$$ \begin{eqnarray} S_{\infty} &=& \frac{1}{1-r} \\ &=& \frac{1}{1 - \frac{1}{\beta}} \frac{\beta}{\beta} &=& \frac{\beta}{\beta - 1} \end{eqnarray} $$

Using this result leads to…

$$ \begin{eqnarray} \left| x - x^{*} \right| &\le& \left| (\beta - 1) \beta^{-k - 1} \left(\frac{\beta}{\beta - 1}\right) \right| \beta^{e} \\ &\le& \left| \beta^{-k - 1} \beta \right| \beta^{e} \\ &\le& \left| \beta^{-k} \right| \beta^{e} \\ &\le& \beta^{-k}\beta^{e} \\ \end{eqnarray} $$

The worst case error (i.e., upper bound for error) can be written as…

$$ \left| x - x^{*} \right| \le \beta^{-k}\beta^{e} $$

3.2 Relative Error

We know that relative error is defined as

$$ \frac{|x - x^*|}{|x|} $$

From the absolute error derivation, we know that

$$ \left| x - x^{*} \right| \le \beta^{-k}\beta^{e} $$

That leads to…

$$ \frac{|x - x^*|}{|x|} \le \frac{\left|\beta^{-k}\beta^{e}\right|}{min(|x|) * \beta^{e}} $$

Notice how $|x|$ became $min(|x|)$. We are bounding the bound. As |x| gets smaller, the error gets larger. The smallest legal non-zero mantissa is $\beta^{-1}$.

$$ \begin{eqnarray} \frac{|x - x^*|}{|x|} &\le& \frac{\left|\beta^{-k}\beta^{e}\right|}{min(|x|) * \beta^e} \\ &\le& \frac{\left|\beta^{-k}\beta^{e}\right|}{\beta^{-1}\beta^e} \\ &\le& \frac{\left|\beta^{-k}\right|}{\beta^{-1}} \\ &\le& \beta\left|\beta^{-k}\right| \\ &\le& \beta\beta^{-k} \\ &\le& \beta^{-k + 1} \\ \end{eqnarray} $$