Threaded Mode | Linear Mode

Albert Chan · (This post was last modified: 03-30-2022 01:20 AM by Albert Chan.)

If arugment for asin is big, taylor series take a long time to converge.

Ratio of coefficient = (2k+1)^2 / ((2k+2)*(2k+3)), approach 1 when k is huge.
For asin(1), this implied dropped term is almost same size as the last kept term.

We can use the "half-angle" formula for asin()

(05-31-2021 07:30 PM)Albert Chan Wrote: \(\displaystyle\arcsin(x) = 2\arcsin\left( {x \over \sqrt{2\sqrt{1-x^2}+2}} \right)\)

Code:

function asinq(x) -- = asin(sqrt(x))

    if x > 4e-4 then return 2 * asinq(0.5*x/(sqrt(1-x)+1)) end

    return sqrt(x) * (1+x/6*(1+x*9/20/(1-x*25/42)))

end

lua> asinq(1/4), pi/6
0.5235987755982989      0.5235987755982988
lua> asinq(2/4), pi/4
0.7853981633974483      0.7853981633974483
lua> asinq(3/4), pi/3
1.0471975511965979      1.0471975511965976
lua> asinq(4/4), pi/2
1.5707963267948966      1.5707963267948966

Another way is to use Carlson Elliptic Integrals: RC(1-x,1) = asin(√x)/(√x)

Code:

function RC(x,y, verbose) -- RC(1-x,1) = asin(sqrt(x))/sqrt(x)

    local k = y + 2*sqrt(x*y)

    if k==x+2*y then return sqrt(3/k) end

    if verbose then print(sqrt(3/k)) end

    return RC((x+k)*0.25, (y+k)*0.25, verbose)

end

This is a simplified version of RF, RC(x,y) = RF(x,y,y)
see https://www.hpmuseum.org/forum/thread-17...#pid148498

It has no problem getting asin(1)/1 = pi/2, 1 sqrt per iteration

lua> RC(1-1, 1, true)
1.7320508075688772
1.5764775210064272
1.5711174700143078
1.5708159330462599
1.5707975451380392
1.570796402832061
1.5707963315455151
1.5707963270917837
1.5707963268134517
1.5707963267960565
1.5707963267949692
1.5707963267949012
1.570796326794897
1.5707963267948968
1.5707963267948968

Another example, from area of inscribed and circumsribed square, for pi

lua> A, B = 2, 4
lua> r = (B-A)/B -- = 1/2
lua> c = RC(1-r, 1, true) -- = asin(sqrt(1/2)) / sqrt(1/2) = pi/4 * sqrt(2)
1.1147379454918027
1.1109478170877694
1.1107345982528842
1.1107215960382895
1.1107207883059862
1.110720737898786
1.1107207347495225
1.110720734552712
1.1107207345404118
1.1107207345396428
1.1107207345395949
1.1107207345395917
1.1107207345395915

lua> A * c/sqrt(1-r)
3.141592653589793

lua> A * asinq(r)/sqrt(r-r*r)
3.141592653589793