arcsinc( 1-y ), for small y
07-06-2018, 11:17 PM (This post was last modified: 07-10-2018 09:38 PM by Albert Chan.)
Post: #4
 Albert Chan Senior Member Posts: 1,241 Joined: Jul 2018
RE: arcsinc( 1-y ), for small y
To show new formula is *really* more accurate: (below from Mathematica)

arcsinc( 1-y ) / sqrt(6y)
= 1 + 3/20 y + 321/5600 y^2 + 3197/112000 y^3 + 445617/27596800 y^4 + ...
~ 1 + 0.15 y + 0.0573214 y^2 + 0.0285446 y^3 + 0.0161474 y^4 + ...
~ 1 + 0.15 y

sqrt(6y) / arcsinc( 1-y )
= 1 - 3/20 y - 39/1120 y^2 - 1649/112000 y^3 - 5285631/689920000 y^4 - ...
~ 1 - 0.15 y - 0.0348214 y^2 - 0.0147232 y^3 - 0.00766122 y^4 - ...
~ 1 - 0.15 y

Looking at the size of dropped terms, new formula is more accurate:

arcsinc( 1-y ) ~ sqrt(6y) / (1 - k y), where k = 0.15

For more accurate arcsinc, correct for k using above divide trick.

k = 0.15 / (1 - 13/56 y / (1 - 0.19068 y / (1 - 0.21627 y)))
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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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