smallest |cos(x)| ?

[Albert Chan]

(10-11-2021 01:16 PM)Werner Wrote:  For the following values of k , (k+0.5)*pi is not the same as k*180+90 ->RAD, and the latter is correct:
k=0, 2, 8, 15, 18, 25, ..
consequently, (k+0.5)*pi does not yield the lowest |cos(x)|, but the ->RAD version does.

Assuming we already tried k=0,1,2 (i.e, IP(x) of single digit), difference does not matter.

We are now searching for exact(x) *close* to round(x,34).
For this, only error of concern is inaccuracy of PI, (rel_err < 0.37 ULP)
(Here, ULP ≡ 1E-34)

If |cos(x)| is minimized, both ways of calculating x will match.
If they are different, it is guaranteed |cos(x)| is not minimized.

Example, for IP(x) of 2 digits:

c:\> spigot -d3 -C "abs(pi/2*1e32 rem 1)"
0/1
1/6
1/7
15/104

7 2 / PI *       → 10.99557428756427633461925184147826
7 90 * ->RAD → 10.99557428756427633461925184147826
COS               → -9.469009289781287037341230607307734e-35

---

From Free42 core_helpers.cc, deg_to_rad(x) = x / (180 / PI)

180/pi = 57.29577951308232087679815481410517 0332 ...

Rounded to 34 digits, this has relerr ≈ 0.0332 / 0.573 ≈ 0.058 ULP

This is much more accurate than stored PI (relerr ≈ 0.369 ULP)
If it had used flipped constant:

pi/180 = 0.01745329251994329576923690768488612 7134 ...

relerr ≈ (1-0.7134) / 0.1745 ≈ 1.64 ULP ... very bad

Comment:
Here, relerr calculations ignored sign, but sign may be important.
I use this convention for sign of error: exact = approx + error

PI is over-estimated, relerr(PI) = −0.369 ULP
180/PI under-estimated, relerr(180/PI) = +0.058 ULP

But, ->RAD() use division of constant, we subtract relerr instead of add.
Or, we can think of multiply by inverse of constant, with relerr of −0.058 ULP

Since both formula have relerr of same sign, ->RAD() always better.