[racket] how to get exact results from get-text-extent

>
> Here's what I'd do: decide on a smallest fractional point and a longest
> length, and determine how many bits that requires. If 52 bits is enough,
> use flonums. If it's not, write a prototype using bigfloats. If that's too
> slow, try double-doubles or fixed-point numbers.


What's the best way to impose systematic number typing on existing untyped
code, for instance, forcing everything to be a flonum? Does it require
switching to Typed Racket? Or does mean making explicit calls to
`exact->inexact` and using the flonum math operators? Or some other
technique?


Indexing two structs to get operands to add together takes more time than
> actually adding them;


What are the preferred data structures for optimizing math performance, if
structs are too slow? Vectors?



On Tue, Dec 2, 2014 at 7:17 PM, Neil Toronto <neil.toronto at gmail.com> wrote:
>
> Here are some options for representing lengths.
>
>  * Fixed-point numbers; i.e. a fixnum n represents n/2^k (where k = 16 for
> TeX). These are cheap and easy until you want to multiply or divide them.
> Then they get more expensive and bit-shifty. You can forget about doing
> anything else quickly. On the plus side, if you allow arbitrary integers,
> lengths that fit in a fixnum are still reasonably fast to operate on, and
> there's no limit on maximum length.
>
>  * Flonums. With these, you can exactly represent lengths up to 2^36 =
> 68719476736 points at TeX's 1/2^16-point precision. In Racket, they're just
> about as fast as fixnums, unless a result is put in a struct or is
> heap-allocated. Those results are 2x-3x slower.
>
>  * Exact rationals. Exact for arithmetic, slow, and generally grow without
> bound as the number of operations that produces them increases.
> Non-arithmetic forces deciding on a precision, but can be done with just a
> little pain by converting to bigfloats and back.
>
>  * Double-doubles; i.e. two flonums that represent one value, for about
> 106 bits precision. (So at TeX's precision, they exactly represent lengths
> up to 2^90 = 1237940039285380274899124224 points.) 10x slower than flonums,
> and 4x more annoying to work with.
>
>  * Bigfloats. About 100x slower than flonums. The default precision of 128
> bits is likely way more than you'll ever need for layout.
>
> Here's the thing: layout code will do a lot more than compute arithmetic.
> Indexing two structs to get operands to add together takes more time than
> actually adding them; sometimes a lot more. IMO, this makes the speed of
> fixnums vs. flonums, or even fixnums vs. integers (usually), a non-issue.
> You may even be able to get away with using bigfloats if you need them.
>
> Here's what I'd do: decide on a smallest fractional point and a longest
> length, and determine how many bits that requires. If 52 bits is enough,
> use flonums. If it's not, write a prototype using bigfloats. If that's too
> slow, try double-doubles or fixed-point numbers.
>
> (If you use fixed-point numbers, double the number of fractional bits to
> make area calculations exact and square roots decently precise. The latter
> can be computed using `integer-sqrt` without too much trouble.)
>
> Neil ⊥
>
> On 12/01/2014 07:07 PM, Matthew Butterick wrote:
>
>> Right, I meant "exact" in the Racket sense of "exact rational."
>>
>> The broader issue I'm thinking about is what kind of units to use in a
>> typesetting system in order to get the best balance of precision and
>> speed.
>>
>> For instance, the flexibility of a 64-bit flonum doesn't necessarily buy
>> you anything over a 64-bit fixnum, since typesetting systems have a
>> practical limit on both precision (in the subpixel direction) and scale
>> (in the megapixel direction).
>>
>> TeX, for instance, is based on a "scaled point" which represents
>> 1/65536th of a point, with a max dimension of 2^30 scaled points, or
>> about 19 feet. One could imagine a 64-bit version of this concept that
>> extends both the scale and precision (to ludicrous degrees) while
>> remaining a fixnum (which I gather from the docs are typically cheapest).
>>
>>
>>
>>
>>
>> On Mon, Dec 1, 2014 at 2:01 PM, Matthew Flatt <mflatt at cs.utah.edu
>> <mailto:mflatt at cs.utah.edu>> wrote:
>>
>>     We should probably improve the contracts on `racket/draw` to promise
>>     flonum results for text metrics. The intent is to make metric-derived
>>     calculations have a predictable cost, instead of potentially
>> triggering
>>     expensive exact arithmetic.
>>
>>     When you say that Pango produces "exact" results, do you mean
>> "integer"
>>     or "exact rational"? The latter is certainly true: Pango's raw API
>>     produces integers to be divided by 1024 to convert to drawing units.
>>     Taking that conversion into account, Pango doesn't always produce
>>     integer drawing units (at least on some platforms; I'm not sure about
>>     all). Even so, the intent is that representing an integer divided by
>>     1024 as a flonum will not normally lose any prevision (i.e., for
>> normal
>>     drawing sizes), and so `inexact->exact` on the immediate result from
>>     `racket/draw` recovers the exact Pango result when exact arithmetic is
>>     specifically wanted.
>>
>>     At Mon, 1 Dec 2014 11:56:12 -0800, Matthew Butterick wrote:
>>      > The `get-text-extent` method in racket/draw does not
>>     contractually guarantee
>>      > either exact or inexact numbers, though in practice I find it
>>     produces inexact.
>>      >
>>      > This function, however, calls into the Pango text-layout system.
>>     I find that
>>      > when I invoke Pango's text measuring directly through the FFI, it
>>     produces
>>      > exact results.
>>      >
>>      > Is this difference in behavior deliberate, or does
>>     `get-text-extent` preserve
>>      > exactness under certain circumstances?
>>      >
>>      >
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