What is the internal accuracy for Pi in the DM42?
What is the internal accuracy for Pi in the DM42?
There is a discussion of the correct result of SIN of exactly 3.141592654 Radians in the HP 15C Advanced Functions handbook Appendix...The Dm42 gets it correct to a full 34 significant digits, (as I verified in Mathematica...). All of the result numbers are correct...To get that result, the internal figure for Pi must be much higher than the result. Curious how Pi is represented internally in the DM42 and or what the IEEE Quad Standard calls for regarding the number Pi..
Re: What is the internal accuracy for Pi in the DM42?
I disagree. I'd guess it's well within Pi +/ 10^(34).zeno333 wrote: ↑Sat Dec 30, 2017 7:52 amThere is a discussion of the correct result of SIN of exactly 3.141592654 Radians in the HP 15C Advanced Functions handbook Appendix...The Dm42 gets it correct to a full 34 significant digits, (as I verified in Mathematica...). All of the result numbers are correct...To get that result, the internal figure for Pi must be much higher than the result.
DM42 SN: 00041  Follower of Platon.
HP35, HP45, ..., HP50, WP 34S, WP 31S, DM16L
HP35, HP45, ..., HP50, WP 34S, WP 31S, DM16L

 Posts: 503
 Joined: Tue May 02, 2017 3:48 pm
 Location: New Jersey, USA
 Contact:
Re: What is the internal accuracy for Pi in the DM42?
The PI function in Free42 Decimal, and thus the DM42, simply returns the 34digit approximation of pi.
To get the accuracy you noticed for sin(3.141592654), or, in fact, for sin(3.141592653589793238462643383279503), the floatingpoint library has to use a higherprecision approximation of pi internally.
You could find out exactly how many digits it uses by checking the Intel library's source code. The IEEE754 standard requires transcendental functions to be accurate to within one ULP; in order for the sine to achieve that on the domain (pi, pi), its approximation of pi would have to have at least twice as many digits as its highestprecision external floatingpoint format.
To get the accuracy you noticed for sin(3.141592654), or, in fact, for sin(3.141592653589793238462643383279503), the floatingpoint library has to use a higherprecision approximation of pi internally.
You could find out exactly how many digits it uses by checking the Intel library's source code. The IEEE754 standard requires transcendental functions to be accurate to within one ULP; in order for the sine to achieve that on the domain (pi, pi), its approximation of pi would have to have at least twice as many digits as its highestprecision external floatingpoint format.
Re: What is the internal accuracy for Pi in the DM42?
Section 9 of the 2008 version of IEEE 754 says that the recommended functions should be correctly rounded not 1ULP accurate. These functions are optional in the sense that you can claim conformance without them but ...language standards should define, to be implemented according to this subclause, as many of the functions of 9.2 as are appropriate to the language.
Interestingly, the specified domain is the entire real range where the function is defined. This makes the modulo reduction for trigonometric functions quite a challenge. The Intel decimal library uses the binary functions for an initial estimate, so they are possibly religating these difficulties to that library instead.
The WP 34S uses 451 digits in it's internal representation of 2π. This is sufficient to correctly range reduce for single precision (16 digit) inputs but not for double precision (34 digit) ones. To do the latter requires upwards of 8,000 digits and more RAM than the hardware has.
Pauli
Interestingly, the specified domain is the entire real range where the function is defined. This makes the modulo reduction for trigonometric functions quite a challenge. The Intel decimal library uses the binary functions for an initial estimate, so they are possibly religating these difficulties to that library instead.
The WP 34S uses 451 digits in it's internal representation of 2π. This is sufficient to correctly range reduce for single precision (16 digit) inputs but not for double precision (34 digit) ones. To do the latter requires upwards of 8,000 digits and more RAM than the hardware has.
Pauli

 Posts: 453
 Joined: Wed May 03, 2017 5:46 pm
 Location: Malone, NY USA
Re: What is the internal accuracy for Pi in the DM42?
It doesn't need to calculate anything if the value of PI is stored as a literal and the calculator uses tables to display common values (0, PI/2, PI, 3*PI/2, 2*PI) of transcendentals.zeno333 wrote: ↑Sat Dec 30, 2017 7:52 amThere is a discussion of the correct result of SIN of exactly 3.141592654 Radians in the HP 15C Advanced Functions handbook Appendix...The Dm42 gets it correct to a full 34 significant digits, (as I verified in Mathematica...). All of the result numbers are correct...To get that result, the internal figure for Pi must be much higher than the result. Curious how Pi is represented internally in the DM42 and or what the IEEE Quad Standard calls for regarding the number Pi..
Tom L
Ducator meus nihil agit sine lagunculae leynidae accedunt
DM42 SN: 00025 (Beta)
SN: 00221 (Shipping)
Ducator meus nihil agit sine lagunculae leynidae accedunt
DM42 SN: 00025 (Beta)
SN: 00221 (Shipping)

 Posts: 503
 Joined: Tue May 02, 2017 3:48 pm
 Location: New Jersey, USA
 Contact:
Re: What is the internal accuracy for Pi in the DM42?
The Intel library uses a 384bit approximation of pi, equivalent to 115 decimal digits.
Re: What is the internal accuracy for Pi in the DM42?
This might cause issues for huge arguments: e.g. sin(1.234E350).Thomas Okken wrote: ↑Sat Dec 30, 2017 2:13 pmThe Intel library uses a 384bit approximation of pi, equivalent to 115 decimal digits.
Pauli

 Posts: 503
 Joined: Tue May 02, 2017 3:48 pm
 Location: New Jersey, USA
 Contact:
Re: What is the internal accuracy for Pi in the DM42?
Oh, it causes issues much sooner. When I tried the series sin(pi * 10^n), for n = 0, 1, 2, ..., I could see the error start to creep in at around n = 17.pauli wrote: ↑Sun Dec 31, 2017 4:16 amThis might cause issues for huge arguments: e.g. sin(1.234E350).Thomas Okken wrote: ↑Sat Dec 30, 2017 2:13 pmThe Intel library uses a 384bit approximation of pi, equivalent to 115 decimal digits.

 Posts: 503
 Joined: Tue May 02, 2017 3:48 pm
 Location: New Jersey, USA
 Contact:
Re: What is the internal accuracy for Pi in the DM42?
N.B. To keep this in perspective: On the real HP42S, you can see the error start to creep in at n = 6. Given that my main objective for Free42 is that it be compatible with the 42S, to the point of being viable as a dropin replacement, the Intel library is Good Enough™.Thomas Okken wrote: ↑Sun Dec 31, 2017 4:24 amOh, it causes issues much sooner. When I tried the series sin(pi * 10^n), for n = 0, 1, 2, ..., I could see the error start to creep in at around n = 17.pauli wrote: ↑Sun Dec 31, 2017 4:16 amThis might cause issues for huge arguments: e.g. sin(1.234E350).Thomas Okken wrote: ↑Sat Dec 30, 2017 2:13 pmThe Intel library uses a 384bit approximation of pi, equivalent to 115 decimal digits.
Re: What is the internal accuracy for Pi in the DM42?
Wow, I'm surprised it is that soon. There must be something else going on for the error to appear this early  possibly the change from binary to decimal after the initial estimate.Thomas Okken wrote: ↑Sun Dec 31, 2017 4:24 amOh, it causes issues much sooner. When I tried the series sin(pi * 10^n), for n = 0, 1, 2, ..., I could see the error start to creep in at around n = 17.
As you wrote, this is better than the real 42S.
Pauli