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Subject: Re: [boost] Is there interest in Computable Calculus?
From: Damian Vicino (damian.vicino_at_[hidden])
Date: 2016-01-15 06:03:09

> Considering, for example, the Delaunay condition:
> bool delaunay(T ax, T ay, T bx, T by, T cx, T cy, T dx, T dy)
> {
> T cos_abc = (ax-bx)*(cx-bx) + (ay-by)*(cy-by);
> T cos_cda = (cx-dx)*(ax-dx) + (cy-dy)*(ay-dy);
> if (cos_abc >= 0 && cos_cda >= 0) return false;
> if (cos_abc < 0 && cos_cda < 0) return true;
> T sin_abc = (ax-bx)*(cy-by) - (cx-bx)*(ay-by);
> T sin_cda = (cx-dx)*(ay-dy) - (ax-dx)*(cy-dy);
> T sin_sum = sin_abc * cos_cda + cos_abc * sin_cda;
> return sin_sum < 0;
> }
> I'd be interested to know how this approach of effectively "lazy
> evaluation" compares to using eagerly-evaluated arbitrary-precision
> types (e.g. Boost.Multiprecision's floating point types). I.e. in
> terms of performance and any complexity introduced to the code.
> For what sorts of problems is it better? How much work is actually
> saved by laziness in practice?

Short answer: The complexity depends in the definition of ax, ay, bx, by, cx, cy, dx, dy.
Longer one below (with a lot of oversimplification of the algorithms)

Suppose the function pi(int digits) providing individual digits of pi. pi(0) = 3, pi(1)=1, pi(2) = 4 … and e(digits) providing digits of e, e(0)=2, e(1)=7…, and for simplicity in the example let assume all numbers have the decimal comma in position 1, and we do not need to check it.
Then, we have that the number pi is the sum(pi(digit)*10^(-digit)), and e is sum(e(digit)*10^(-digit)).
Rational numbers, can be easily expanded having the numerator and denominator.

Doing the sum from 0 to n, has an error of “10^(-n)”, one unit in the last computed digit.

Now, we want to evaluate the expression: pi < 3/1
Iteration 0:
Pi(0) = 3
3/1(0) = 3
3 - 3 = 0 ?, yes, then we need to expand another.
Iteration 1:
Pi(1) = 1
3/1(1) = 0
1 - 0 > 0, then we can answer false, and stop expanding digits.

Arithmetic operations are thought as interval operations defined to produce digits based in the parameters.
For example:
(e + e) (0) = 5 is computed using first 2 digits of e, and not just the first one.
Iteration 0:
e(0) = 2 : meaning e is in [2, 3)
[2, 3) + [2, 3) = [4, 6), then the result could be 4, or 5, we require a second iteration
Iteration 1:
e(1) = 7 : meaning e is in [2.7, 2.8)
[2.7, 2.8) + [2.7, 2.8) = [3.4, 3.6), then we are certain about first digit (not second yet).

Spacial complexity:
The complexity of addition is just keeping pointer to the functions received in the parameters.
The complexity of comparing, is the complexity of expanding the operands until they can be decided, if an operand is a sum, complexity is linear to the size of the sum.

Time complexity:
The complexity of addition is assignation of 2 pointers, one per operand.
The complexity of comparison is the complexity of expanding enough digits to be able to decide.

Disclaimer 1: in practice it is not necessary to compute every digit of the numbers, numbers does not change, large database of interesting numbers with millions of digits exist, and in the case they are not available, computing once is enough if using some kind of cache or internal database.

Going back to comparing with multi precision.
Multi precision will be definitely faster, assuming you know how many digits you need. However, failing the estimation has a cost. If overestimated, you paying on memory and time. If underestimated, you are getting the wrong results. Usually, that is what happens when using floating-point, you have 23 bits, then pi is 23 bits, but if the values you evaluating differs in the bit 50 (not represented) you believe them equal when they are not.
About space complexity, one depends quantity of variables, the other in how they are combined by operators.

Disclaimer 2: Using arbitrary numbers (something you really want to avoid), it is proven that equality comparison is undecidable, then pi_version_1 == pi_version_2 will never end execution.
For this reason, it is common to set a max_expansion constant, for most jobs a few thousands will do the job (which is orders of magnitude away from what you can usually get in other approaches). If an operation expanded to this limit and did not decide an operation, most certain using other approaches produced an inexact result.

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