arcsinc( 1-y ), for small y
06-18-2020, 11:54 PM
Post: #14
 Albert Chan Senior Member Posts: 1,549 Joined: Jul 2018
RE: arcsinc( 1-y ), for small y
Re-phrase previous post asinc(1-y) in terms of k, with shorter constants:

$$\large asinc(k) ≈ \Large \sqrt{{12(1-k) \over 1-0.2041(1-k)\;+ \sqrt{(27.11-k)(0.0065+0.0318k)}}}$$

Free-42 code, Newton's method using above guess:

Code:
00 { 85-Byte Prgm } 01▸LBL "ASINC" 02 27.11 03 X<>Y 04 - 05 LASTX 06 STO ST Z     ; k   27.11-k   k 07 0.0318 08 × 09 65ᴇ-4 10 + 11 × 12 SQRT 13 1 14 STO- ST Z 15 +            ; 1+sqrt1   k-1 k 16 0.2041 17 RCL× ST Z 18 + 19 ÷ 20 -12 21 × 22 SQRT 23▸LBL 01       ; newton, f = sinx - kx 24 RCL× ST Z    ; kx    .   k   . 25 LASTX 26 STO ST Z     ; x     kx  x   k 27 SIN 28 - 29 RCL ST Y     ; x    -f   x   k 30 COS 31 RCL- ST T    ; f'   -f   x   k 32 ÷            ; e    x    k   k 33 STO+ ST Y    ; e    x'   k   k 34 X↑2 35 X<>Y 36 STO+ ST Y    ; x' x'+e^2 k   k 37 X≠Y? 38 GTO 01 39 END

Redo previous posts examples:

asinc(5280/5281) → 0.03370775880987942934821849830508089
asinc(0.99) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿→ 0.2453178088540252967075843126095618
asinc(5280/5380) → 0.3348901065998630063407861032211992
asinc(1/6.5) ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿→ 2.711312910356983336479873410532187
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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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