[racket-dev] Sublinear functions of superfloat numbers

From: Neil Toronto (neil.toronto at gmail.com)
Date: Sun Jul 1 20:00:31 EDT 2012

How about more words and examples?

"Argument reduction" is using function properties to reduce the 
magnitude of arguments to make computation more tractable or more accurate.

I'll bet `log' uses this property:

     (log (sqrt x)) = (log (expt x 1/2)) = (* 1/2 (log x))

This form is nice for doing algebraic manipulations; not so much for 
computation. So multiply the outer sides by 2:

     (* 2 (log (sqrt x))) = (log x)

You could define your own log function like this:

     (define (my-log x)
       (* 2 (log (sqrt x))))

If `sqrt' could compute square roots of rational numbers larger than 
+max.0 (about 2^1024), then `my-log' could compute logs for those as 
well. But it can't.

Here's a version of log that does reduces its argument with 
`integer-sqrt', which computes truncated square roots of bigints:

(require racket/flonum
          (only-in unstable/flonum +max.0))

(define (real-log x)
   (cond [(x . = . +inf.0)   +inf.0]
         [(x . <= . +max.0)  (fllog (real->double-flonum x))]
         [(x . > . +max.0)
          (let loop ([x  (exact-round x)])
            (cond [(x . > . +max.0)  (* 2.0 (loop (integer-sqrt x)))]
                  [else  (fllog (real->double-flonum x))]))]
         [else  +nan.0]))

Computing (real-log #e1e800242) takes exactly the same amount of time as 
(log #e1e800242), and gives the same answer as well. (And I just 
discovered that that's how it's implemented in number.c.)

Square root (for large real numbers) is much simpler. Floating-point 
numbers above 2^52 are all integers, so bigints above 2^1024 can be 
thought of as floating-point numbers with at least 1024 - 52 = 972 bits 
of precision. That means `integer-sqrt' will do the job perfectly, 
despite the fact that it always returns integers.

(define (real-sqrt x)
         [(x . <= . +max.0)  (flsqrt (real->double-flonum x))]
         [(x . > . +max.0)   (real->double-flonum
                              (integer-sqrt (exact-round x)))]
         [else  +nan.0]))

But sine is much harder because it's periodic with an irrational period. 
It would look something like this, but with a rational approximation of 
pi whose precision depends on the magnitude of the argument:

(define (real-sin x)
   (cond ...
         [(x . > . +max.0)
          (let ([x  (- x (truncate (/ x (* 2 pi))))])
            (flsin (real->double-flonum x)))]

On 07/01/2012 04:04 PM, Robby Findler wrote:
> 3. Can you explain the issue again, using smaller words? (I think I
> understand the first example, but then I'm lost.)
> Robby
> On Sun, Jul 1, 2012 at 5:02 PM, Matthias Felleisen <matthias at ccs.neu.edu> wrote:
>> 1. What's the computational cost of such changes?

The additional cost when applying `sqrt' and `sin' to numbers in typical 
ranges would be small: the cost of wrapping a kernel function and 
checking the size of arguments.

>> 2. What is the impact on TR?

None that I can tell. But TR would remove the additional cost when it 
could prove the arguments to `sqrt' and `sin' are Float.


I think I've been trying to come up with a general rule for when 
argument reduction is necessary. But I can't, because there isn't one. 
For example, this is infinite:

     (log (make-rectangular #e1e400 1))

even though the actual answer is representable as a Float-Complex. 
Apparently, argument reduction only happens to reals, and only in a few 
functions. (But I'm glad it does, because the plot library uses `log' to 
format huge numbers.)

Neil ⊥

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