Page 282 - ARM 64 Bit Assembly Language

Non-integral mathematics 271

Table 8.2: Reciprocals for the terms two
through nine of the Taylor series.

Term Reciprocal Reciprocal
Format Value (Hex)
1
− S(−2,32) AAAAAAAA
3!
1 S(−6,32) 44444445
5!
1
− S(−12,32) 97F97F97
7!
1 S(−18,32) 5C778E96
9!
1
− S(−25,32) 9466EA60
11!
1 S(−32,32) 5849184F
13!
1
− S(−40,32) 94603063
15!
1 S(−48,32) 654B1DC1
17!
In Hexadecimal, the number to be multiplied is 44444445 16 . Note that since 1 is a positive
5!
number, the reciprocal was incremented by one to avoid round-off errors. We can apply the
same procedure to the remaining terms. Table 8.2 shows the reciprocals and their formats for
terms two through nine. The first term is just one, so it is not included in the table.

8.6.3 Putting it all together

We want to keep as much precision as is reasonably possible for our intermediate calculations.
However, we also want to provide the best performance possible. Using 64 bits of precision
for all intermediate calculations will give a good trade-off between performance and preci-
sion. Keeping more than 64 bits would require more time to perform the multiplication and
addition steps, and would not result in any appreciable difference in our final 32-bit result.
The integer portion should never require more than two bits, so we choose an S(2,61) as our
intermediate representation. If we combine Table 8.1 and Table 8.2, we can determine what
the format of each complete term will be. This is shown in Table 8.3.

Note that the formats were truncated to fit in a 64 bit result. We can now see that the formats
for the first nine terms of the Taylor series are reasonably similar. They all require exactly 64
bits, and the radix points can be shifted so that they are aligned for addition. In order to make
the shifting and adding process easier, we will pre-compute the shift amounts and store them
in a look-up table. In fact, we will create a look-up table to hold the reciprocal values and
the shift amounts together, so that our Taylor series code is greatly simplified and extremely
efficient. The procedure will involve calculating the next power of x, multiplying it by the