arcsinc( 1-y ), for small y
07-07-2018, 06:04 PM (This post was last modified: 08-18-2019 01:17 PM by Albert Chan.)
Post: #6
 Albert Chan Senior Member Posts: 2,103 Joined: Jul 2018
RE: arcsinc( 1-y ), for small y
Did plots of how good e = sqrt((m + z)z), vs e = sqrt(6 m d) / 4
The new formula is 4 times more accurate !

Because the ratio is so consistent around 4, this is a fast way to estimate e:
Previous example, d = 100 ft:

e ~ (446.5492654 * 4 + 444.9719092) / 5 = 446.2337942 ft

With 6.5X warp or less (d < 5.5 *miles*), accuracy can be improved:
y = 100/5380
w = 4 - 27/56 y = 3.991038237
e ~ (446.5492654 w + 444.9719092) / (w+1) = 446.2332309 ft

This is not far from exact e of 446.233229831394 ...

We could even reverse the process, use good e to estimate arcsinc !
To simplify, let m = 1, so e = 1/2 tan( x/2 )

Example: calculate arcsinc(0.99):

sinc(x) = 0.99 = 1/(1+ d)
d = 1/99
z = 3/8 d = 1/264

e(min) = √(z) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ = 0.06154574549
e(max) = √(z + z²) = 0.06166219923

w ~ 4 - 27/56 y = 3.995178571
e ~ (w e(max) + e(min)) / (w + 1) = 0.061638886
x ~ 2 arctan( 2 e ) = 0.2453178089

For comparison, asinc(0.99) = 0.2453178088 54025 ...
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 Messages In This Thread arcsinc( 1-y ), for small y - Albert Chan - 07-05-2018, 11:43 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-06-2018, 03:22 PM RE: arcsinc( 1-y ), for small y - Thomas Klemm - 07-06-2018, 09:07 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-06-2018, 11:17 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-07-2018, 06:11 AM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-07-2018 06:04 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-08-2018, 03:28 AM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-09-2018, 01:12 AM RE: arcsinc( 1-y ), for small y - Albert Chan - 07-12-2018, 05:26 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 08-20-2018, 02:20 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 08-20-2018, 03:23 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 08-25-2018, 03:51 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 10-01-2019, 06:03 PM RE: arcsinc( 1-y ), for small y - Albert Chan - 06-18-2020, 11:54 PM

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