(12C Platinum) Normal Distribution

12012018, 07:53 AM
(This post was last modified: 12022018 01:26 AM by Gamo.)
Post: #1




(12C Platinum) Normal Distribution
This Normal Distribution program is adapted from TI SR56 Applications Library.
Instruction is attached here and procedure is mostly the same. Above Example: FIX 6 2.02 [R/S] display 0.051863574 // Z(x) [R/S] display 0.021691583 // Q(x) Remark: Answer is a bit less accurate due to the Pi precisions Program: Normal Distribution (ALG Mode) 94 Program Lines Code:
Check answer with your calculator from this Normal Distribution Calculator website. https://keisan.casio.com/exec/system/1180573188 Percentile X = your desire input Mean u = 0 Standard Deviation q = 1 Gamo 

12012018, 10:59 AM
(This post was last modified: 12012018 12:04 PM by Dieter.)
Post: #2




RE: (12C Platinum) Normal Distribution
(12012018 07:53 AM)Gamo Wrote: Above Example: FIX 9 See below. ;) (12012018 07:53 AM)Gamo Wrote: Remark: Answer is a bit less accurate due to the Pi precisions Yes, but why don't you simply use the exact value? Gamo, you are wasting no less than 11 steps to get an approximate √(2pi) value with only seven (!) valid digits: Code: ÷ Why don't you simply use the exact value directly (even with less steps): Code: x BTW, on a ten digit calculator multiplying by 1/√2pi is slightly more accurate than dividing by √(2pi) because the value rounds much better to ten digits. Edit: But let's not bee too picky here. For large x the accuracy for Z(x) will lose up to three digits anyway. I also wonder if the 12C (Platinum) will accept the final digit 4 in the constant. Considering its internal 12digit precision this might be the best way: Code: ÷ This should give 11 digits of √(2pi): 2,50662827460 instead of ...63. Requires the same 11 steps as before, but has much better accuracy. (12012018 07:53 AM)Gamo Wrote: Program: Normal Distribution (ALG Mode) 94 Program Lines First of all: while Z(x) works for positive and negative x, the program calculates Q(x) only for x≥0. It does not handle negative x, for instance the program will return the same value for x=1 and –1. Here the user has to apply the formula Q(–x) = 1–Q(x) manually. Of course you can also make the program do so. The cumulative distribution function Q(x) – the upper tail Normal integral – is approximated by a polynomial, the wellknown Hastings approximation which for instance is listed in Abramovitz & Stegun, formula 26.2.17. But... the constants are rounded to six digits (probably due to the SR56's limited memory), so the accuracy of the original formula – a largest absolute error of 7,5 E–8 – is not met. That's why FIX 9 makes no sense here. For x≤2 the relative error is within 5 E–6, so you get about 5 valid significant digits. For larger x, similar to the original approximation, the number of valid digits decreases. For x=5 only three significant digits are left, and finally only two or one are correct. The accuracy can be improved if the original untruncated coefficients are used: p = 0,2316419 b1 = 0,31938153 b2 = 0,356563782 b3 = 1,781477937 b4 = 1,821255978 b5 = 1,330274429 This way you can set FIX 7, and what you see is correct within ±1 ULP. If the 12C Platinum does it like the original 12C, it will eventually switch to scientific notation. At this point you know you shouldn't trust more than two and finally only one significant digit: Try x=8, the exact Qvalue here is 6,221 E–16 while the Hastings approximation with the above coefficients returns 6,285 E–16. That's why I'd prefer a different way of calculating the Normal integral. For instance with a customized rational function. Same effort, better accuracy. ;) Here is such an approximation (for x≥0) that I set up some time ago: Q(x) ≈ e^{–x²/2} · (1 + a1·x + a2·x² + a3·x³) / (2 + b1·x + b2·x² + b3·x³) where a1 = 0,433563 a2 = 0,084144052 a3 = 9,6367 E5 b1 = 2,46295834 b2 = 1,13288719 b3 = 0,20590615 Again there are six constants, but with less digits. For x≤5 the largest relative error is less than 1,2 E–6, and the absolute error is < 1 unit in the 6th significant digit. For x>5 this approach is still better than Hastings'. For instance it returns Q(8) as 6,220 E–16. Near the end of the working range, at Q(21) = 3,28 E–98 the above method returns 3,26 E–98 while the Hastings approximation yields 3,56 E–98. All in all even at the underflow limit the relative error is less than 0,007. But for large x you can apply a continued fraction method that keeps the error on the 1 E–6 level also for x>5. Actually the accuracy increases as x gets larger. I have also designed such a CF method that complements the above rational approximation and requires just three terms. But I think I'm getting lost here... ;) Dieter 

12012018, 05:37 PM
Post: #3




RE: (12C Platinum) Normal Distribution
(12012018 10:59 AM)Dieter Wrote: Here is such an approximation (for x≥0) that I set up some time ago: Using x = 0 to 5 step 0.01, confirmed your claim of 1.2e6 max rel. error ... Nice! Hasting approximation (even with best constants) is 1000X worse, max rel. error = 1.6e3 

12012018, 07:44 PM
(This post was last modified: 12012018 07:50 PM by Dieter.)
Post: #4




RE: (12C Platinum) Normal Distribution
(12012018 05:37 PM)Albert Chan Wrote: Using x = 0 to 5 step 0.01, confirmed your claim of 1.2e6 max rel. error ... With exact coefficents it's even a tad less, about 1,10 E–6. I like that it's "exact by design" for x=0. ;) (12012018 05:37 PM)Albert Chan Wrote: Nice! It gets even nicer: take a look at this thread in the HP67 software forum. This implements a (4;5) rational approximation. Finally, here is the CF expansion that complements the above rational approximation for x>5: Q(x) ≈ Z(x) / (x+1/(x+2/(x+3/(x+0,66)))) The two largest errors are at x=5 (–1,04 E–6) and x=6,13 (+8,9 E–7). For larger x the error decreases continuously towards zero. At the underflow limit (x=21,16517934) it is merely 1,4 E–9. If exactly evaluated, that is. Which is not possible with the standard 12C implementation of Z(x). The linked HP67/97 program evaluates the PDF differently to achieve better accuracy. Dieter 

12012018, 11:16 PM
(This post was last modified: 12012018 11:44 PM by Albert Chan.)
Post: #5




RE: (12C Platinum) Normal Distribution
(12012018 07:44 PM)Dieter Wrote:(12012018 05:37 PM)Albert Chan Wrote: Using x = 0 to 5 step 0.01, confirmed your claim of 1.2e6 max rel. error ... I see it now. Q(x = 0) = 1/2 * exp(0) = 1/2 (exactly) Can you post the exact coefficients too ? Edit: playing with the coefficients, I managed to lower max rel. errors to 1.12e6 a1 = 0.433563 a2 = 0.084144052 a3 = 9.63659e6 b1 = 2.4629583 b2 = 1.1328872 b3 = 0.2059062 

12022018, 09:11 AM
(This post was last modified: 12022018 09:42 AM by Dieter.)
Post: #6




RE: (12C Platinum) Normal Distribution
(12012018 11:16 PM)Albert Chan Wrote: Can you post the exact coefficients too ? According to my results the following set has a largest error of 1,0064 E6: a1 = 0,4335629429 a2 = 0,0841440309 a3 = 9,636444 E5 b1 = 2,462958275 b2 = 1,132887108 b4 = 0,20590614476 (12012018 11:16 PM)Albert Chan Wrote: Edit: playing with the coefficients, I managed to lower max rel. errors to 1.12e6 Fine, even with less digits. But it can still be tweaked a bit: a1 = 0,433563 a2 = 0,084144052 a3 = 9,6366 E5 b1 = 2,4629583 b2 = 1,1328872 b3 = 0,2059062 This way the error seems to stay a tiny bit below 1,12E6. And for calculators every digit counts. For such applications I would now recommend this set of coefficients. Maybe someone with access to Mathematica, Maple or similar software can post the exact largest errors. Dieter 

12022018, 06:59 PM
(This post was last modified: 12022018 07:38 PM by Dieter.)
Post: #7




RE: (12C Platinum) Normal Distribution
(12022018 09:11 AM)Dieter Wrote: Fine, even with less digits. But it can still be tweaked a bit: There now is a new approximation with focus on the absolute error. But in a somewhat different way: With the following coefficients the error is less than 6,5 units in the 7th significant digit. a1 = 0,4367117 a2 = 0,0851264 a3 = 9,34575 E5 b1 = 2,46927 b2 = 1,1397766 b3 = 0,2085059 The valid domain is even a bit wider than before: the above error threshold is maintained up to x=5,1993..., which is the point where the CDF reaches 1 E–7. Which is easy to memorize: any result down to 1 E–7 has an error less than 7 units in its 7th significant digit. ;) This rational approximation can be complemented with the above CF expansion. Replace the constant 0,66 with 0,6444 and switch between both methods at x=5. This way the error stays below 7 units in the 7th significant digit for all x≥0. Dieter 

12022018, 08:14 PM
(This post was last modified: 12022018 08:16 PM by Albert Chan.)
Post: #8




RE: (12C Platinum) Normal Distribution
(12022018 06:59 PM)Dieter Wrote: There now is a new approximation with focus on the absolute error. But in a somewhat different way: With the following coefficients the error is less than 6,5 units in the 7th significant digit ... For x = 0 to 5, confirmed on the 6.5e7 maximum absolute error ... However, the old constants did even better, max abs. error = 5.1e7 

12022018, 09:43 PM
(This post was last modified: 12022018 09:46 PM by Dieter.)
Post: #9




RE: (12C Platinum) Normal Distribution
(12022018 08:14 PM)Albert Chan Wrote:(12022018 06:59 PM)Dieter Wrote: There now is a new approximation with focus on the absolute error. But in a somewhat different way: With the following coefficients the error is less than 6,5 units in the 7th significant digit ... It's 6,5 units in the 7th significant (!) digit. Not an absolute error of 6,5 E7. The previous approximation had a largest error of about 1 unit in the 6th significant digit, now this is down to 65% of that. Compare the results for x=3,1: Result for the coefficients in post #6: 9,676022 E04 Compare this to the true result 9,676032 E04 The error is about 1 unit in the 6th significant digit. Result for the coefficients in post #7: 9,676026 E04 Compare this to the true result 9,676032 E04 The error is about 6 units in the 7th significant digit. The coefficients in post #7 limit this kind of error to less than 6,5 units in the 7th digit. These are simply two different targets both approximations are designed for. Dieter 

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