arcsinc( 1y ), for small y

07052018, 11:43 PM
Post: #1




arcsinc( 1y ), for small y
Come across a fun civil engineering problem from HP archive [1]
"Five minute challenge" essentially evaluate arcsinc( 1y ) sin(x) / x = ( 1y )  y = 1/5281 = rail expansion effect x = arcsinc( 1y )  then, rail warp e = 2640 tan(x/2) From Gerson W. Barbosa (msg #21), sinc(x) ~ cos(x / sqrt(3)) x = arcsinc( 1y ) ~ sqrt(3) arccos( 1y ) = 0.03370733324  using my Casio fx115ms Estimate is good, exact x = 0.033707758809879429 ... I wanted a simpler, and more accurate arcsinc( 1y ) arccos( 1y ) taylor series = sqrt(2y) (1 + y/12 + ...) [2] x = arcsinc( 1y ) ~ sqrt(6y) = 0.03370680134 For better accuracy, apply Newton's method, x0 = sqrt(6y): f(x) = sinc(x)  (1y) = y  x^2/6 + x^4/120  x^6/5040 + ... f'(x) = x/3 + x^3/30  (3/70) x^5 + ... f(x0) = y  y + 0.3 y^2  (3/70) y^3 + ... ~ 0.3 y^2 f'(x0) = x0 (1/3 + y/5  (54/35) y^2 + ...) ~ x0 / 3 x = x0  f(x0) / f'(x0) ~ x0 + (0.9 y^2 / x0) * (x0/x0) = x0 (1 + 0.9 y^2 / (6y)) = sqrt(6y) (1 + 0.15 y) x = arcsinc( 1y ) ~ sqrt(6y) (1 + 0.15 y) = 0.03370775874 with x this good, f(x) = 0.0 on my Casio :) [1] http://www.hpmuseum.org/cgisys/cgiwrap/...ead=177273 [2] https://math.stackexchange.com/questions...twoterms 

07062018, 03:22 PM
(This post was last modified: 10012019 10:54 PM by Albert Chan.)
Post: #2




RE: arcsinc( 1y ), for small y
This new formula is slightly more accurate, and just as simple:
For small y: arcsinc( 1y ) ~ √(6y) / (1  0.15 y) arcsinc(1  1/5281) ~ √(6/5281) / (1  0.15/5281) = 0.033707 75877 Example: x = 0.1 radian y = 1  sin(x) / x = 0.001665833532 Old formula, x ~ √(6y) * (1 + 0.15 y) = 0.099999 98408 New formula, x ~ √(6y) / (1  0.15 y) = 0.099999 99033 Update: newer formula from post #11 x ~\(\large \sqrt{6y ÷ \sqrt{1 + 0.6 y}} \) = 0.099999 99997 // almost roundtrip! 

07062018, 09:07 PM
Post: #3




RE: arcsinc( 1y ), for small y
(07052018 11:43 PM)Albert Chan Wrote: Come across a fun civil engineering problem from HP archive [1] Thank you for reminding me of this thread. Good memories. Quote:"Five minute challenge" essentially evaluate arcsinc( 1y ) And this was supposed to be a 5minute challenge... Well, probably not. Cheers Thomas 

07062018, 11:17 PM
(This post was last modified: 07102018 09:38 PM by Albert Chan.)
Post: #4




RE: arcsinc( 1y ), for small y
To show new formula is *really* more accurate: (below from Mathematica)
arcsinc( 1y ) / 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( 1y ) = 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( 1y ) ~ 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))) 

07072018, 06:11 AM
(This post was last modified: 08182019 01:14 PM by Albert Chan.)
Post: #5




RE: arcsinc( 1y ), for small y
I reread the Five Minute challenge posts (highly recommended)
I found a simple formula that is slightly better than e = sqrt(6 m d) / 4 Let z = gain in length of of hypotenuse Using post #30 3/4 rule, z = d/2 * 3/4 = 3/8 d e^2 = (m/2 + z)^2  (m/2)^2 = m z + z^2 e = sqrt((m + z) z) With m=5280, d=1: e = sqrt((5280 + 3/8)(3/8)) = 44.49877105 ~ 44.50 ft 3/4 rule is the upper bound for tiny angle, so z (thus e) is overestimated. Let's try with bigger sector angle, say, d = 100 ft: Old formula, e = sqrt(6 * 5280 * 100) / 4 = 444.9719092 ~ 444.97 ft New formula, e = sqrt((5280 + 300/8)(300/8)) = 446.5492694 ~ 446.55 ft We have a nice bound, correct e is between 444.97 ft to 446.55 ft For correct value of e, with big angle, do arcsinc with corrected k: y = 1  5280/5380 = 100/5380 = 0.01858736 k = 0.15 / (1  13/56 y / (1  0.19068 y / (1  0.21627 y))) = 0.1506523749 x = sqrt(6y) / (1  k y) = 0.3348901066 (about 19 degree) e = (m/2) tan( x/2 ) = 446.2332298 ~ 446.23 ft (correct) 

07072018, 06:04 PM
(This post was last modified: 08182019 01:17 PM by Albert Chan.)
Post: #6




RE: arcsinc( 1y ), 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 ... 

07082018, 03:28 AM
(This post was last modified: 07132018 09:46 PM by Albert Chan.)
Post: #7




RE: arcsinc( 1y ), for small y
Redoing the Five minutes challenge with weighted e:
Note: sqrt(6 m d) / 4 = sqrt(6 m (8/3 z) / 16) = sqrt(m z) m = 5280 z = 3/8 d = 3/8 e(min) = sqrt(m z) = 44.49719092 ~ 44.50 e(max) = sqrt((m + z) z) = 44.49877105 ~ 44.50 The nice error bound *guaranteed* 4 digits accuracy e ~ (4 e(max) + e(min)) / 5 = 44.49845502 4 weighted e reach 10 digits accuracy ! w weighted e do even better, see next post ... 

07092018, 01:12 AM
(This post was last modified: 07132018 08:03 PM by Albert Chan.)
Post: #8




RE: arcsinc( 1y ), for small y
To show weighted e method work, I use Mathematica:
To simplify, assumed m = 1, so e = 1/2 tan(x/2) sinc[x_] := Sin[x] / x; t = Series[1  sinc[Sqrt[x]], {x, 0, 5}]; t = InverseSeries[t] /. x > y; exact5 = 1/2 Tan[Sqrt[t] / 2]; w = 4  27/56 y; weighted = (w * Sqrt[z + z z] + Sqrt[z]) / (w+1); weighted = weighted /. z > 3/8d /. d > y/(1y); ratio = Series[exact5 / weighted, {y, 0, 5}] // Simplify // Normal ratio // N >> 1  9459/25088000 y^3 + 2651751/309084160000 y^4 >> 1.  3.77033e4 y^3 + 8.57938e6 y^4 weighted e estimate is simple, and very good. :) To check if above is valid, use ratio to improve Five Minute Challenge: First, calculate exact e, with 30 significant digits: FindRoot[sinc[x] == 5280/5281, {x, 0.1}, WorkingPrecision > 50] >> {x > 0.033707758809879429348218498305111 ... } e = N[5280/2 Tan[x / 2] /. %, 30] (* Reference: all good digits *) >> 44.4984550191007992545541600167 (* Normally no need for scaling, but above assumed m=1 *) e = N[5280 weighted /. y> 1/5281, 30] (* 14 digits accurate *) >> 44.4984550191009131676665673849 e = N[5280 weighted ratio /. y> 1/5281, 30] (* 22 digits accurate *) >> 44.4984550191007992545475544533 

07122018, 05:26 PM
(This post was last modified: 07162018 01:54 PM by Albert Chan.)
Post: #9




RE: arcsinc( 1y ), for small y
On the railroad warp problem, same formulas still work for pi/2 < x < pi:
In other words, circle center and warp ok on the *same* side: Prove sin(x) / x = m / (m + d): m/2 = R sin(pi  x)  chord length (m + d) / 2 = R x  arc length sin(pi  x)/x = sin(x)/x = m / (m + d)  QED Prove e = (m/2) tan( x/2 ): (m/2) / e = tan((pi  x) / 2)  triangle: chord + e (contained circle center) e = (m/2) / tan(pi/2  x/2) = (m/2) tan( x/2 )  QED My weighted e is not designed for big angle, but still work ok For x < pi/2 (thus, warp < pi/2), weighted e has max error of 0.001% For HUGE warp, weighted e formula run into problem: Example: let m=1, 6.5X warp (d=5.5), calculate e: sin(x) / x = 1 / (1+d) = 0.1538461538 x = 2.71131291  my Casio solver, x ~ 155 degrees R = (1+d) / (2x) = 1.198681269 e = 1/2 tan(x/2) = 2.288101656  almost diameter of circle ! Assumed we don't know x, use 4 weighted e method: z = 3/8 d = 2.0625 e(min) = sqrt(z) = 1.436140662 e(max) = sqrt(z + z z) = 2.513246158 e ~ (4 e(max) + e(min)) / 5 = 2.297825059  error = +0.425% With warp that high, "improved" weight does not help: y = 1  0.1538461538 = 0.8461538462 w = 4  27/56 y = 3.5920322967 e ~ (w e(max) + e(min)) / (w+1) = 2.27868654  error = 0.411% For HUGE warp, weighted e method is not suited, below fit is better: For pi/2 < x < pi, fitted e (below) has max error of 0.013% e (warp 2.4X or more) ~ (((0.690948 y + 1.95561) y  2.47546) y + 1.52907) d Redo above case (6.5X warp): e ~ 0.4160261229 * 5.5 = 2.288143676  error = +0.002% 

08202018, 02:20 PM
(This post was last modified: 08212018 10:45 PM by Albert Chan.)
Post: #10




RE: arcsinc( 1y ), for small y
Forman Acton's book Numerical Method that work (page 68) have a neat solution to the railroad problem:
sinc(x) = 5280/5281 = 1/(1+e), where e=1/5280 > 1  x^2/3! + x^4/5!  x^6/7! + ... = 1  e/(1+e) Subtract 1 from both side, factor out x^2, we get: x^2 = 6e / (1+e) / (1  x^2/20 + ...) Last denominator is about 1, we get x^2 ~ 6e / (1 + e) = 6/5281 We get x = sqrt(6/5281) = 0.0337068013 Iterate once more: x^2 = 6e / (1 + e) / (1  (6e / (1+e)) / 20) = 6e / (1 + 0.7e) = 6/5280.7 We get x = sqrt(6/5280.7) = 0.0337077588 With this x, warp = (5280/2) tan(x/2) = 44.498455 ft (matched rounded exact value) The book use this example to illustrate two points: 1. Remove large masking quantities, (in this case, the 1.0 term) 2. Do not solve quadratic equation if the quadratic term carry little weight. Amazing book ... 

08202018, 03:23 PM
(This post was last modified: 06182020 01:10 PM by Albert Chan.)
Post: #11




RE: arcsinc( 1y ), for small y
Rearange Acton fromula in terms of arcsinc(k = 1y):
x = arcsinc(k) ~ sqrt(6y / (1  0.3 y)) this is much better than my estimate of sqrt(6y) / (1  0.15y) For small y, errors cut down by almost 33% ! To push the idea a bit further, below cut down error (small y) by 97% ! x = arcsinc(k) ~ sqrt(6y / sqrt(1  0.6 y)) For y > 0.5, asymptotic formula is better: x ~ Pi / (1 + k + 1.244 k^3) For 0<y<1, above combined setup kept relative error to 0.1% or less. For more accuracy, use either form of Newton's method: f = sin(x)  kx → x = x  (sin(x)  k*x) / (cos(x)  k) f = sin(x)/x  k → x = x + x * (z  k) / (z  cos(x)), where z = sin(x)/x Update: More accurate Asinc (small y), but more complicated formula: http://www.hpmuseum.org/forum/thread291...ight=Asinc Update2: instead of asymptotic formula, correction also work: For y >= 0.4, asinc(1y) ~ sqrt(6y / sqrt(1  0.6 y)) * (1.001 + y^6/53) 

08252018, 03:51 PM
(This post was last modified: 10022019 12:01 PM by Albert Chan.)
Post: #12




RE: arcsinc( 1y ), for small y
The book Numerical Method that work showed Acton's favorite way of doing interpolation (no difference table)
A variation of Aitken scheme of intepolation (p 94), using post #8 as an example Solve x, such that sinc(x) = k = 1/6.5 ~ 0.153846154 Using my asympotic formula, x ~ Pi/(1 + k + 1.244 k^3) ~ 2.712066499 sinc(2.712) = 0.1535768820, error ~ 0.00027, thus x too high (sinc(x) is decreasing for x = [0, 4.4934]) sinc(2.711) = 0.1539688052, error ~ +0.00013, thus x too low sinc(2.7115) = 0.1537728267, error ~ 0.00007, used for 3 point quadratic fit. Acton variation is to sort the values, interpolated estimate up on top. This ensures top numbers likely the best estimate, with other entries very "tight" Mistakes during manual calculations are easier to spot. All entries below are linear interpolation (against top entry), for k = 1/6.5 PHP Code: sinc(x) x Quadratic fit might not be needed, if interpolated estimate used as 3rd point PHP Code: sinc(x) x For Casio FX115MS, Interpolation (A,B), (C,D) Formula: With CALC: 0A + B + (CA)^{1}(DB)(XA) Or, SOLVE: (XA)(B0CD) + (YB)(CA) update: instead of manually getting the points to fit, we can use iterations: sinc(pi/2) = 1/(pi/2) = 2/pi ≈ 0.6366197724 If k ≥ 2/pi: x = sin(x)/k If k < 2/pi: x = pi  asin(kx) For above example, k < 2/pi, we get 2.712 → 2.7111966 → 2.711332599 ... Using Aitken Δ² method, above 3 numbers extrapolated to 2.711312910 

10012019, 06:03 PM
(This post was last modified: 10032019 12:26 PM by Albert Chan.)
Post: #13




RE: arcsinc( 1y ), for small y
Inspired by Happy Coincidence for the approximate solution of x tan(x) = k.
I tried the same trick for sin(x)/x = k, using Mathematica. > <<Calculus`Pade` > sinc[x_] := Sin[x] / x > Pade[sinc[x], {x,0,4,4}] // Simplify ⇒ (166320  22260 x^2 + 551 x^4) / (166320 + 5460 x^2 + 75 x^4) > Solve[166320  22260 x^2 + 551 x^4 == 0, x] // N ⇒ {{x → 3.14572}, {x → 3.14572}, {x → 5.52302}, {x → 5.52302}} Pade approximant numerator had 2 roots very close to ±Pi, so solve this instead: \(\Large { x^2\pi^2 \over sinc(x) } = { x^2\pi^2 \over k}\) We don't want expression too complicated, and limit it to solving Quadratics: > lhs = Pade[(x^2Pi^2) / sinc[x], {x,0,4,2}]; > soln = Solve[ Numerator[lhs] * k  Denominator[lhs] * (x^2Pi^2) == 0, x] Although it look like a quartic, it is really a quadratic, with variables x^2. Instead of solving asinc(k) for x, I choose solving asinc(1y) for x. This is a rule I learned from Acton's book, "Numerical Methods that works", p71 Quote:Store the small quantites; Compute the larger ones; Never subtract nearly equal quantities. \(\large asinc(1y) ≈ \Large \sqrt{{12y \over 10.20409y\;+ \sqrt{1  0.792y  0.0318y^2}}} \) The estimate formula is good for full y range. y = 1.00 → worst overestimated: abs error = 0.00023, rel error = 0.0075% y = 0.85 → worst underestimated: abs error = 0.00021, rel error = 0.0075% Previous post example is close to worst case, y = 11/6.5 ≈ 0.846154 Using above formula, we get asinc(1/6.5) ≈ 2.71111, underestimated 0.00020 

06182020, 11:54 PM
Post: #14




RE: arcsinc( 1y ), for small y
Rephrase previous post asinc(1y) in terms of k, with shorter constants:
\(\large asinc(k) ≈ \Large \sqrt{{12(1k) \over 10.2041(1k)\;+ \sqrt{(27.11k)(0.0065+0.0318k)}}} \) Free42 code, Newton's method using above guess: Code: 00 { 85Byte Prgm } 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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