Numeric Testing

To test numeric functions we should use cases like so:

| Fcalc(x) - Ftrue(x) | < tolerance

We can calculate Ftrue using an arbitrary precision maths package.

This is as there are inherent losses with floating point numbers so our calculated value may be off (within reason).

Tolerance

We can calculate the tolerance with the following function:

\[\text{Tolerance} \leq \text{Cn}(x) \times \text{ULP}\]

Where:

$\text{ULP}$ - Unit of least precision which is dependant on the data type used.
$\text{Cn}(x)$ - Function of the condition number where the greater the condition number, the greater the effect of loss in precision.

These are particular values which are proven mathematically such as:

We can test these using the tolerance calculation above.

This is a case where we use multiple functions in conjunction to produce an output:

\[\sin(x)^2+\cos(x)^2=1\]

Using identities may hide compensated error (where the functions are equally wrong in the opposite direction).

These are similar to identities:

\[x = (\sqrt x)^2\]

Unlike identities, they test for accuracy as in the previous example - $\sin(x)=1$ and $\cos(x)=0$, while still passing the test.

Consider that we are storing a decimal number in an 8-bit float:

sign	exponent				significand
0	0	0	0	0	0	0	0

We can calculate the error when storing 0.1:

0 1111 001

This is stored perfectly.

Or calculate the error when storing 0.25:

0 1111 011

We cannot store 25 in the 3 bit significand so the closes digit is 3. This gives an error of 0.05.

We can also use a fixed point representation like so:

$2^2$	$2^1$	$2^0$	$2^{-1}$	$2^{-2}$	$2^{-3}$	$2^{-4}$	$2^{-5}$
1	0	1	0	0	1	1	0

This can also be written as:

\[1101\ .\ 0011_2\]

We can calculate the error of storing 0.1 in 8 bits:

\[0\ .\ 0001\ 1010_2 = 0.1015625\]

We have to round to fit this in 8 bits.

\[\vert0.1 - 0.1015625\vert = 1.5625{\times}10^{-3}\]