Page 286 - ARM 64 Bit Assembly Language

Non-integral mathematics 275

Listing 8.7 ARM assembly implementation of sinx and cosx using fixed point
calculations.
1 //*************************************************************
2 // Name: sincos.S
3 // Author: Larry Pyeatt
4 // Date: 2/22/2018
5 //*************************************************************
6 /*
7 This is a version of the sin/cos functions that uses symmetry
8 to enhance precision. The actual sin and cos routines convert
9 the input to lie in the range 0 to pi/2, then pass it to the
10 worker routine that computes the result. The result is then
11 converted back to correspond with the original input.
12
13 We calculate sin(x) using the first seven terms of the Taylor
14 Series: sin(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...
15 and we calculate cos(x) using the relationship:
16 cos(x) = sin(pi/2-x)
17
18 We start by defining a helper function, which we call sinq.
19 The sinq function calculates sin(x) for 0<=x<=pi/2. The
20 input, x, must be an S(1,30) number. The factors of x that
21 sinq will use are: x, x^3, x^5, x^7, x^9, x^11, and x^13.
22
23 Dividing by (2n+1)! is changed to a multiply by a coefficient
24 as we compute each term, we will add it to the sum, stored as
25 an S(2,61). Therefore, we want the product of each power of x
26 and its coefficient to be converted to an S(2,61) for the add.
27 It turns out that this just requires a small shift.
28
29 We build a table to decide how much to shift each product
30 before adding it to the total. x^2 will be stored as an
31 S(2,29), and x is given as an S(1,30). After multiplying x by
32 x^2, we will shift left one bit, so the procedure is:
33 x will be an S(1,30) - multiply by x^2 and shift left
34 x^3 will be an S(3,28) - multiply by x^2 and shift left
35 x^5 will be an S(5,26) - multiply by x^2 and shift left
36 x^7 will be an S(7,24) - multiply by x^2 and shift left
37 x^9 will be an S(9,22) - multiply by x^2 and shift left
38 x^11 will be an S(11,20)- multiply by x^2 and shift left
39 x^13 will be an S(13,18)- multiply by x^2 and shift left
40 x^15 will be an S(15,16)- multiply by x^2 and shift left
41 x^17 will be an S(17,14)
42 */
43 /*
44 The following table shows the constant coefficients
45 needed for calculating each term.
46