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Subject: Re: [boost] [fixed_point] First presentation from GSoC 2015 and more
From: John McFarlane (john_at_[hidden])
Date: 2016-10-14 13:11:56

> Date: Thu, 13 Oct 2016 20:41:27 +0000 (UTC)
> From: Christopher Kormanyos <e_float_at_[hidden]>
> > I discuss P0106R0 (a revision to N3352) in my paper, the latest draft of
> > which is always available here:
> > johnmcfarlane/fixed_point/blob/master/doc/
> > I'm honestly not sure which approach is preferable overall. The set of
> > features and qualities requested during SG14 and SG6 meetings is large
> and
> > conflicting. It includes minimal build times, minimal run-time overhead,
> > every rounding and overflow strategy listed in P0105, decimal radix and
> > support for specialized fixed-point instructions on embedded
> architectures.
> Hi John. Thanks for visiting this thread.
> My recommendation would be to limit the scope of the proposal
to a realistic level in the SGs so that we can end up with anything
at all within a reasonable time scale. We could consider back porting
some (but probably not all) of SG14's specified interface to Boost
if that helps move anything along.

A second pair of implementor eyes would be a huge help.

I should point out that although it hasn't seen much attention lately,
P0106 may still be active. (Frankly, the evolution of my design has been
largely a process of slowly copying or facilitating all the most popular
from Lawrence's proposal!)

One problem P0037 faces - even if it were to be standardized - would be
that alone, it only provides fixed-point arithmetic. It's designed to work
with types that also solve arbitrary width, rounding and overflow but those
types are not part of any proposal. In particular, without an
arbitrary-width integer type, it's very difficult to use fixed_point<>

> I was also wondering about decimal radix, but I don't know if it
might simply be better to identify those 80% of the most popular

use cases and simply specify these. At least it would get the library
moving forward and people could use fixed-point in C++. This is

basically what we tried to do with the GSoC project.

That makes sense. I at least want to have a solution which doesn't close
the door on decimal in future additions. Here's my current plan:

> >> In your work I've been experiencing some trouble with>> the division
> routine.
> > Can you elaborate? Was this recently? I've been trying out numerous
> > strategies for arithmetic operators. My current preferred approach is to
> > pad the left-hand operand before performing the division but that's only
> > been checked in for a few months.
> I'm not sure about the design, and error on my side is

a likely possibility. But division operator in the master

branch seems like it does not pad before divide.

This might give the wrong result.

I hope that's not the case but it's a tough operator to get right -
especially with 32-bit operands. If you ever find a good example of this
happening I'd be very interested in it. At least when using the elastic
alias, precision loss should be avoidable (beyond inevitable approximation
of rational numbers and x/0).

tl;dr: The details of padding in the division operator follow...

The padding is done on this line:
It's on display in this test:

In the above test, the two operands have a single fractional digits. If
operator/ did not pad the LHS, then the integer operation would be 126/8
which would lose precision. Instead, the LHS is promoted and shifted left
by 8 bits. Then, integer division occurs and the value is stored with
(hopefully!) no further shifts.

> For division, I have used the following approach.

* handle sign

* extend n bits to 2n bits
> * shift left by radix split plus one round bit

* divide 2n / n -> n bits
> * handle round and sign
> So you're right this requires 64-bit extension for a 32-bit fixed-point

unless you tackle the problem of two-component

division, which I have done using a simplified

version of Knuth long division. At the moment,

this is only in develop branch. The compiler switch


This is core to the difference in approach between P0037 and P0106. I
guarantee to use built-in integer math for any type specialized on built-in
integers. I believe that types with arbitrary width can then be built upon
this foundation using novel integer types.

Here is an example using Boost.Multiprecision to express a Googol:
fixed_point<> uses the same algorithm regardless of whether the `Rep`
integer is an uint8_t or a 400-bit custom type.

> >> I also had some problems compiling

on VC12 (but you don't target that
> >> compiler anyway). So that's no big deal.
> > You will need a C++11-compliant compiler as sg14::fixed_point<> creates
> > literal types. Lack of support for features like constexpr constructors
> is
> > the main obstacle to compiling under VC++ 2013 and earlier.
> I actually agree with that approach. Either it uses C++11 or

it does not. I don't see the need to cater to a specific older

compiler for new library work.

It's a pragmatic choice. Three reasons are:
* C++11 has everything I need to implement (with difficulty) a literal
type. In particular, I can write most of my unit tests using
`static_assert` which is the best thing ever!
* Once you start returning novel specializations from your arithmetic
operators, `auto` type deduction is essential to usability.
* Many of my target users in SG14 are stuck with compilers which are behind
the curve.

> >> In your work you are using wrapped versions of elementary transcendental
> >> functions designed for built-in float. You might appreciate our
> hand-crafted
> >> functions designed to be highly efficient in several digit ranges.
> > Yes! I would be very interested in seeing if I could adapt those
> function.
> We worked hard on these. And there are many good

algorithms available in

> Having spent a non-trivial amount of time writing the simplest sqrt I
could find, I appreciate the effort this must have involved.

> Mine are placeholders and I've deliberately postponed the non-trivial task
> > of adding cmath equivalents. It's come up - maybe a couple of times -
> > whether these functions are desired and the response has been to just
> keep
> > the proposal narrow so they are not included.
> > There are interesting questions regarding how to implement some
> functions.
> > For example, you surely don't always want the same inputs and outputs
> from
> > functions involving angles. And hopefully these functions can be
> constexpr
> > - unlike existing functions.
> I agree.
> > Also, how have you approached the trade-off between speed and accuracy? I
> > went for a very simple sqrt function which is adequate if the value is
> > calculated at compile time but is likely not desirable at run-time. This
> > isn't a problem that floating-point variants face.
> The elementary transcendental functions use many techniques

including polynomial approximation, Pade approximation, Taylor series,
> Newton iteration. For low digit counts, mostly polynomial approximation. We
> express the coefficients directly as integral values of the fixed_point
> representation and avoid as much as possible full division and multiply,
> favoring integer div and mul. These functions are quite efficient. We have
> not constexpr-ed them. But it looks like there is potential for that.

Writing constexpr functions in C++11 can be a big challenge.

> >> I have also benched our fixed_point on bare-metal embedded systems both
> >> 8-bit and 32-bit with very satisfying efficiency results. This is a key
> >> feature for our embedded users that might also be interesting for SG14.
> > My current aim is to match integer performance for common arithmetic
> > operations when using built-ins. This leads to some situations where
> > overflow is a big problem - but no worse than dealing with raw integers.
> > I'm curious how you deal with multiplying two int-sized values together
> > without resorting to wider, slower types.I was under the impression that
> > P0106 types always widen to fit their values.
> We have resorted to wider integer for internal mul and div or used higher
> level algorithms when the compiler switch BOOST_FIXED_POINT_DISABLE_WIDE_INTEGER_MATH
> is activated as mentioned above (develop branch only).
> In our approach, full multiplication requires:

* sign conversion
* left shift for round bit
> * multiply n*n --> 2n bits

* right shift and round

* sign conversion
> On a 32-bit system, a full 32*32 -> 64 mul is required.
> This full multiplication approach can not achieve your desired efficiency
> goal on 8-bit and 32-bitCPUs. Division uses similar approach.
> Multiplication and division with plain int or unsigned can achieve the
> efficiency goal --- if it were not for handling rounding and overflow.
> In my current branch, I'm actually trying to get even closer to
integer-level efficiency. No longer does 32-bit multiplication result in
64-bit results on systems with 32-bit integer. This means that overflow on
32-bit arithmetic operations is even more common. The solution is explicit
casting to 64-bit or use of an integer type which auto-widens:

This is the toughest design issue I face - especially when combined with

> >> Thank you and best regards, Chris
> > And thank you. Keep up the good work!
> > John
> Thanks again and best regards, Chris
> Cheers,

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