Benchmark test e^x^3

[Beginner]

Some time ago I came across a youtube video in which as a benchmark test is suggested to calculate the intervall of e^x^3 between 0 and 6. The result is 5,96 E91. It takes about 1 second for the HP Prime, it takes more than 1 min for the HP 50G.

https://www.youtube.com/watch?v=DHRsvSTGiBc

I used this test for several other calculators (it takes about 4 seconds for the new Casio CG50 to accomplish this task, several smaller scientific calculators need more than 1 minute) and tried some variants for larger calculators:

It is possible to calculate the intervall of e^x^3 between 0 and 10,5 with the HP 50G (which takes several minutes but finally succeedes as expected, because the resultat is under E499 (2,22 E498).

I was astonished, that the HP Prime could calculate the intervall of e^x^3 only between 0 and 8,9 - the result being 6,14 E303. It should be able to do the same as the HP 50G because the limit is also E499. Why does it fail?

Does the second generation of HP Prime succeede?

The TI NSpire can calculate the intervall of e^x^3 between 0 and 13 (result: 2,75 E951) in about 2 sec. It's limit is E999.

[pier4r]

This http://www.hpmuseum.org/forum/thread-111...#pid110819 is related

[John Keith]

(02-11-2019 03:28 PM)Beginner Wrote:  I was astonished, that the HP Prime could calculate the intervall of e^x^3 only between 0 and 8,9 - the result being 6,14 E303. It should be able to do the same as the HP 50G because the limit is also E499. Why does it fail?

Does the second generation of HP Prime succeede?

The reason for the narrower range on the Prime is that the Prime's CAS uses 64-bit binary numbers which have a range of about 10^-308 ~ 10^308.

[Beginner]

It is possible to work with numbers up to E499 on the HP Prime and the HP 50G. There ist apparently no difference.

I assume, that the problems I encountert on the HP Prime must be due to the specific task (calculating this specific intervall).

Can there be a bug that did not exist on the HP 50G?

[Albert Chan]

Using XCas 1.4.9-57 (win32), try to see the closed form integral

y := integrate(exp(x^3), x, 0, n)
==> No check were made for singular points of antiderivative ...
==> - igamma(1/3, -n^3) / 3

Ignoring the warning, and no clue what igamma does:

subst(y, n=6.0) => 5.96393809189e+91 // ok

subst(y, n=7.0) => 6.25946729139e+146 // close
integrate(exp(x^3), x, 0, 7.0) => 6.25946729142e+146

subst(y, n=8.0) => 7.91127730599e+154 // bad
integrate(exp(x^3), x, 0, 8.0) => 1.19135302958e+220

How to check if the result is right ? (last one were off by a lot)
What does igamma function do ?

[rprosperi]

(02-11-2019 09:43 PM)Beginner Wrote:  I assume, that the problems I encountert on the HP Prime must be due to the specific task (calculating this specific intervall).

As John pointed out just above your post, the range for CAS MODE on the Prime is not the same as it is for Home mode; CAS numeric range is limited to 10^-308 ~ 10^308

[Beginner]

If I try to calculate the intervall by using the CAS I do not recieve any numeric output at all but a long, long formula with several transformations of the gamma-function.

If I calculate the itervall by using the home mode I recieve a numeric output with the limitations that are here attributed to the CAS. How can that be? Normally I can operate with numbers up to E499 in the home mode.

[Albert Chan]

(02-12-2019 09:47 AM)Beginner Wrote:  If I try to calculate the intervall by using the CAS I do not recieve any numeric output at all ...

Try the formula with interval of float values. Instead of 0 to 6, do 0.0 to 6.0

[Beginner]

(02-12-2019 12:18 PM)Albert Chan Wrote:  

(02-12-2019 09:47 AM)Beginner Wrote:  If I try to calculate the intervall by using the CAS I do not recieve any numeric output at all ...

Try the formula with interval of float values. Instead of 0 to 6, do 0.0 to 6.0

Thank your for your advice. This procedure indeed enables a numerical output in the CAS.

But my problem in the Home environment remains: If I calculate the intervall in the Home environment - why do the limitations of the CAS environment apply there as well? I do not understand it.

I assumed so far, that Home and Cas are two differend parts of the calculator. Am I mistaken in this assumption? Does the Calculator automatically shift to the settings of the CAS environment if I try to do an integration in the Home environment?

[rprosperi]

(02-12-2019 06:21 PM)Beginner Wrote:  But my problem in the Home environment remains: If I calculate the intervall in the Home environment - why do the limitations of the CAS environment apply there as well? I do not understand it.

I assumed so far, that Home and Cas are two differend parts of the calculator. Am I mistaken in this assumption? Does the Calculator automatically shift to the settings of the CAS environment if I try to do an integration in the Home environment?

It will be easier to both understand what you are seeing and offer help if you post the exact equation, values and keystrokes with lots of detail (e.g. "starting in Home mode...") you are using, so that it may be exactly replicated by others. Don't assume the reader knows anything, except how to use a Prime.

Also, as you consistently write "intervall", I'd just like to make sure you are referring to an Integral, right? Maybe intervall is a term used these days for integral and I'm simply not familiar, but it's best to be clear.

[Beginner]

O.K., I´ll try it:

1. I press the home button.
2. I press the 3rd button in the first row and choose the mask of the integral function.
3. I fill in the lower (0) and the upper (6) border.
4. I fill in the function e^x^3.
5. I press "enter".
6. I recieve the value 5,96... E91.

I repeat the procedure and choose as upper border 7.
I revieve the value 6,259.... E146.
I repeat the procedure anc choose as upper border 8.
I recieve the value 1,19135... E220.
I repeat the procedure and choose as upper border 9.
I have to wait 3 sec an then recieve "Fehler: Ungültige Eingabe"
I repeat the procedure and choose as upper border 8,9.
I recieve the value 6,1459... E303.

[DA74254]

Seems that the highest number you can do on the Prime (G2) is 8,92 which gives ~7,168E305
Running "time" function, the 0-8,92 reveals that the HP Prime G2 does this (in CAS) in 0,041s.

[ijabbott]

(02-12-2019 06:21 PM)Beginner Wrote:  I assumed so far, that Home and Cas are two differend parts of the calculator. Am I mistaken in this assumption? Does the Calculator automatically shift to the settings of the CAS environment if I try to do an integration in the Home environment?

It might be borrowing the CAS code to do numeric integration, or perhaps it is not using the CAS code but is using IEEE-754 double precision floating point operations for speed or convenience.

[Beginner]

Interesting. But this should not be the case, if the HP Prime is able to deal with numbers up to E499 in the Home environment.

The new hardware of the Second Generation of the HP Prime is revolutionary, but the software does not seem to use all these wonderful resources. Will there be an update in the not so distant future that makes use of this wonderful hardware?

SwissMicros edited the DM 42 (an "update" oft the hp 42s) which is able to handle this stuff (but it is slower than the HP Prime):

>>The DM42 runs Free42, which uses the Intel Decimal Floating-Point Math Library; it uses IEEE 754-2008 quadruple precision decimal floating-point, which consumes 16 bytes per number, and gives 34 decimal digits of precision, with exponents ranging from -6143 to +6144.<<

https://www.swissmicros.com/dm42.php

I wonder if I should buy one or wait for improvements of the Second Generation of the HP Prime?

[Beginner]

The DM42 by SwissMicros is performing this benchmark test not so fast as the HP Prime but has a far wider range of data. I quote from the forum of SwissMicros:
_________

On battery (Accuracy factor = 0.01):

0..6 = 5.9640E91 (<2 s)
0..13 = 2.7549E951 (<4 s)
0..22 = 1.6059E4621 (<9 s)
0..24 = 2.8145E6000 (<9 s)
0..25 = Out of range

On USB (Accuracy factor = 0.01):

0..6 = 5.9640E91 (<1 s)
0..13 = 2.7549E951 (<2 s)
0..22 = 1.6059E4621 (<3 s)
0..24 = 2.8145E6000 (<3 s)
0..25 = Out of range

Accuracy factor (On USB):

0..24 (acc=1) = 3.638951888E5803 (<1 s)
0..24 (acc=0.1) = 2.809107013E6000 (<2 s)
0..24 (acc=0.01) = 2.814494035E6000 (<3 s)
0..24 (acc=0.001) = 2.814459733E6000 (<6 s)
0..24 (acc=0.0001) = 2.814459733E6000 (<6 s)
0..24 (acc=0.00001) = 2.814459726E6000 (<12 s)
0..24 (acc=0.000001) = 2.814459726E6000 (<12 s)
0..24 (acc=1E-10) = 2.814459726E6000 (<23 s) More digits: 2.814459725921252477432075662808156
0..24 (acc=1E-20) = (Stuck in a loop) Edit: Actually I did not had the patience to wait enough, it is not looping, it takes around 6 minutes. Final result: 2.814459725921296162203929219415341E6000

In comparison with Wolfram result:
2.8144597259212961622039292556760324066E6000
2.814459725921296162203929219415341E6000

[pier4r]

This may also be of interest.

http://www.hpmuseum.org/forum/thread-9750.html

[rprosperi]

(02-12-2019 08:32 PM)Beginner Wrote:  O.K., I´ll try it....

Thanks, this made it easy to follow and reproduce, I get the exact same results as you did ("Error: Invalid input" for range 0..9). I tend to agree there is either a bug in Prime Home mode, or solving this integral is being handed to the CAS for some non-obvious reason.

According to the manual, "In the CAS, MAXREAL=1.79769313486*10^308"

Here are some interesting results that may help someone (Tim, Joe, anybody...) better understand what's going on:

0..9 Error: Invalid input
0..8.9 6.1469095538E303
0..8.92 7.13760485169E305
0..8.92025 7.60793580705E305

etc. tweaking the exponent until overflow...
0..8.920259874454 => 7.62587315929E305
0..8.920259874455 => +Inf

then looking for the next cliff...
0..8.921 +Inf
0..8.924 +Inf
0..8.925 +Inf
0..8.93 +Inf
0..8.935 +Inf
0..8.94 Error: Invalid input
0..8.95 Error: Invalid input

Definitely looks like the CAS is somehow being used here...

[Albert Chan]

With a bit of work, you can extend the range quite a bit.
Example, this is how I do it on my Casio FX-115MS, for x = 0 .. 24

For x = 24, decimal digits ~ 24³/ln(10) ~ 6003.7
For x = 23.99, decimal digits ~ 23.99³/ln(10) ~ 5996.2, a difference of 7.5 digits
Since Casio integrate function only gives 6 digits precision, that is enough.

A = integ(exp(x³), 0, 24)
≈ exp(24³) * integ(exp(x³-24³, 23.99, 24)
≈ 1e6000 * exp(24³ - 6000*ln(10)) * 5.78732e-4
≈ 1e6000 * exp(8.4894421) * 5.78732e-4
≈ 2.81446e6000

[Tim Wessman]

Integration has always been handled by the CAS and this will not change.

[rprosperi]

(02-13-2019 02:16 AM)Tim Wessman Wrote:  Integration has always been handled by the CAS and this will not change.

Thanks Tim, mystery solved. Though Integration in Home is handled by CAS is not a bad thing, is it documented anywhere (other than you post above)?