Re: Exact arithmetic with quadratic irrationals
On Thu, Apr 20, 2017 at 09:11:56PM +0200, Timon Gehr via Digitalmars-d wrote: > On 20.04.2017 20:29, H. S. Teoh via Digitalmars-d wrote: [...] > > Having said that, I haven't scrutinized the performance > > characteristics of QRat too carefully just yet -- there is probably > > room for optimization. > > > > Gcd is the problem. [...] [...] > If I change the gcd computation in QRat (line 233) from > > auto g = gcd(abs(a), abs(b), c); > > to > > auto g = gcd(abs(a), c, abs(b)); > > I get performance that is a lot closer to the matrix version and also beats > the linear computation. (This is because if one of the operands is 1, gcd is > cheap to compute.) Fixed, thanks! [...] > > The +1 is for the denominator, assuming integer coefficients. Since > > having 2^n rational coefficients is equivalent to having 2^n integer > > coefficients (which are half the size of rational coefficients, > > computer-representation-wise) + 1 denominator. ... > > Ah, I see. Personally, I'm more in the one denominator per coefficient > camp. :) I think having a designated ℚ type is cleaner, and it might > even be more performant. It's hard to say without profiling it, I think. Having one denominator per coefficient does need more storage space, and you do have to separately reduce each rational coefficient to lowest terms per operation. I think some profiling is needed to know for sure. Besides, Phobos is badly in need of a Rational type that's compatible with both native int types and BigInt. Maybe I should adapt some of the code in QRat for that purpose. :-D (Since rationals, after all, are a subset of QRats.) On Thu, Apr 20, 2017 at 09:25:19PM +0200, Timon Gehr via Digitalmars-d wrote: > On 20.04.2017 21:18, Timon Gehr wrote: > > On 20.04.2017 21:11, Timon Gehr wrote: > > > > Update: QRat now supports ^^. :-) Integral exponents only, of > > > > course. I also implemented negative exponents, because QRat > > > > supports division and the same algorithm can be easily reused > > > > for that purpose. ... > > > > > > Nice! :) > > > > It does not work with BigInt-based QRats (T(1) does not work, as 1 > > does not implicitly convert to BigInt.) > > I guess the best fix is to templatize the QRat constructor such that > it accepts all argument types that can be used to construct the > coefficients. Fixed, thanks! T -- If the comments and the code disagree, it's likely that *both* are wrong. -- Christopher
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 21:18, Timon Gehr wrote: On 20.04.2017 21:11, Timon Gehr wrote: Update: QRat now supports ^^. :-) Integral exponents only, of course. I also implemented negative exponents, because QRat supports division and the same algorithm can be easily reused for that purpose. ... Nice! :) It does not work with BigInt-based QRats (T(1) does not work, as 1 does not implicitly convert to BigInt.) I guess the best fix is to templatize the QRat constructor such that it accepts all argument types that can be used to construct the coefficients.
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 21:11, Timon Gehr wrote: Update: QRat now supports ^^. :-) Integral exponents only, of course. I also implemented negative exponents, because QRat supports division and the same algorithm can be easily reused for that purpose. ... Nice! :) It does not work with BigInt-based QRats (T(1) does not work, as 1 does not implicitly convert to BigInt.)
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 20:29, H. S. Teoh via Digitalmars-d wrote:
On Thu, Apr 20, 2017 at 02:51:12PM +0200, Timon Gehr via Digitalmars-d wrote:
On 20.04.2017 03:00, H. S. Teoh via Digitalmars-d wrote:
On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmars-d wrote:
[...]
Yes, there is in fact a beautifully simple way to do better. :)
...
Ahh, so *that's* what it's all about. I figured that's what I was
missing. :-D Thanks, I'll include this in QRat soon.
Update: QRat now supports ^^. :-) Integral exponents only, of course. I
also implemented negative exponents, because QRat supports division and
the same algorithm can be easily reused for that purpose.
...
Nice! :)
...
[...]
BTW, you are right that with std.bigint, computation using a linear
number of additions is actually faster for my example (10th
Fibonacci number). The asymptotic running time of the version with
pow on QRats is lower though, so there ought to be a crossover point.
(It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement
anything that is asymptotically faster than Karatsuba.)
Yeah, probably there is a crossover point. But it might be quite large.
I suppose one could make a graph of the running times for increasing n,
and either find the crossover point that way or extrapolate using the
known curve shapes.
Having said that, I haven't scrutinized the performance characteristics
of QRat too carefully just yet -- there is probably room for
optimization.
Gcd is the problem. The following code which implements a strategy based
on matrix multiplication instead of QRat multiplication is significantly
faster than naive linear computation:
BigInt fib(long n){
BigInt[2]
a=[BigInt(0),BigInt(1)],b=[BigInt(1),BigInt(2)],c=[BigInt(1),BigInt(-1)];
for(;n;n>>=1){
foreach(i;1-n&1..2){
auto d=a[i]*a[1];
a[i]=a[i]*b[1]+c[i]*a[1];
b[i]=b[i]*b[1]-d;
c[i]=c[i]*c[1]-d;
}
}
return a[0];
}
If I change the gcd computation in QRat (line 233) from
auto g = gcd(abs(a), abs(b), c);
to
auto g = gcd(abs(a), c, abs(b));
I get performance that is a lot closer to the matrix version and also
beats the linear computation. (This is because if one of the operands is
1, gcd is cheap to compute.)
...
Yes, at most, except you don't need "+1". (For each radical ri, you
will at most need to pick a power between 0 to deg(ri)-1 to index into
the coefficients.)
[...]
The +1 is for the denominator, assuming integer coefficients. Since
having 2^n rational coefficients is equivalent to having 2^n integer
coefficients (which are half the size of rational coefficients,
computer-representation-wise) + 1 denominator.
...
Ah, I see. Personally, I'm more in the one denominator per coefficient
camp. :) I think having a designated ℚ type is cleaner, and it might
even be more performant.
Re: Exact arithmetic with quadratic irrationals
On Thu, Apr 20, 2017 at 02:51:12PM +0200, Timon Gehr via Digitalmars-d wrote: > On 20.04.2017 03:00, H. S. Teoh via Digitalmars-d wrote: > > On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmars-d > > wrote: > > [...] > > > Yes, there is in fact a beautifully simple way to do better. :) > > > ... > > > > Ahh, so *that's* what it's all about. I figured that's what I was > > missing. :-D Thanks, I'll include this in QRat soon. Update: QRat now supports ^^. :-) Integral exponents only, of course. I also implemented negative exponents, because QRat supports division and the same algorithm can be easily reused for that purpose. Interestingly enough, std.math has an overload of pow() that pretty much does exactly the same thing, except that its sig constraints require a built-in floating-point type. I'm half-tempted to submit a Phobos PR to relax the sig constraints so that we could actually use it for QRat without essentially duplicating the code. [...] > BTW, you are right that with std.bigint, computation using a linear > number of additions is actually faster for my example (10th > Fibonacci number). The asymptotic running time of the version with > pow on QRats is lower though, so there ought to be a crossover point. > (It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement > anything that is asymptotically faster than Karatsuba.) Yeah, probably there is a crossover point. But it might be quite large. I suppose one could make a graph of the running times for increasing n, and either find the crossover point that way or extrapolate using the known curve shapes. Having said that, I haven't scrutinized the performance characteristics of QRat too carefully just yet -- there is probably room for optimization. [...] > > > That would certainly be nice. Note that QRat is basically already > > > there for different quadratic radicals, the only reason it does > > > not work already is that we cannot use a QRat as the base field > > > instead of ℚ (because ℚ is hardcoded). > > > > Oh? I didn't try it myself, but if QRat itself passes > > isArithmeticType, I'd venture to say QRat!(n, QRat!m) ought to > > work... There are some hidden assumptions about properties of the > > rationals, though, but I surmise none that couldn't be replaced by > > prerequisites about the relative linear dependence of the mixed > > radicals over Q. ... > > The issue is that gcd does not work on QRats. If QRat had two > coefficients from an arbitrary (possibly ordered) field instead of > encoding rationals explicitly, I think it would work. You're right, without gcd it won't work. The current implementation is a bit overzealous on using gcd (cf. the division algorithm), mainly because I'm concerned with integer overflow on native types. Probably some of these uses can be dispensed with, where BigInt is involved. But normalize() still needs gcd, otherwise sgn() and opCmp() may produce wrong results. One thought is that if there is a QInt base type (i.e., implementing numbers of the form a+b√r, without the denominator), then we could implement a gcd algorithm for it, and we'd be able to instantiate QRat!(r, QInt). > > What I had in mind, though, was a more direct approach that perhaps > > may reduce the total number of operations, since if the code is > > aware that multiple radicals are involved, it could potentially > > factor out some commonalities to minimize recomputations. ... > > This is probably the case. Upon reviewing the algorithms I've come up with in the past, it appears that QRat!(r, QInt) may in fact produce essentially the same code. > > Also, the current implementation of QRat fixes the radical at > > compile-time; I wanted to see if I could dynamically handle > > arbitrary radicals at runtime. It would have to be restricted by > > only allowing operations between two QRats of the same extension, of > > course, but if the code could handle arbitrary extensions > > dynamically, then that restriction could be lifted and we could > > (potentially, anyway) support arbitrary combinations of expressions > > involving radicals. That would be far more useful than QRat, for > > some of the things I'd like to use it for. ... > > What applications do you have in mind? Computational geometry? Yes. In particular, manipulating the coordinates of certain kinds of polytopes. I currently do have code that can do this with floating-point, but I'd like to be able to deal with exact coordinates rather than floating-point approximations. [...] > > > All your conjectures are true, except the last one. (Galois theory > > > is not an obstacle, because here, we only need to consider > > > splitting fields of particularly simple polynomials that are > > > solvable in radicals.) > > > > That's nice to know. But I suppose Galois theory *would* become an > > obstacle if I wanted to implement, for example, Q(x) for an > > arbitrary algebraic x? ... > > All that the result about the quintic
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 03:00, H. S. Teoh via Digitalmars-d wrote: On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmars-d wrote: [...] Yes, there is in fact a beautifully simple way to do better. :) ... Ahh, so *that's* what it's all about. I figured that's what I was missing. :-D Thanks, I'll include this in QRat soon. In particular, this leads to multiple ways to compute the n-th Fibonacci number using O(log n) basic operations. (One way is to use your QRat type, but we can also use the matrix (1 1; 1 0).) True, though I'm still jealous that with transcendental functions like with floating-point, one could ostensibly compute that in O(1). ... BTW, you are right that with std.bigint, computation using a linear number of additions is actually faster for my example (10th Fibonacci number). The asymptotic running time of the version with pow on QRats is lower though, so there ought to be a crossover point. (It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement anything that is asymptotically faster than Karatsuba.) For computations over field extensions of (small) finite fields, pow is a lot faster though. Another module I'm thinking about is an extension of QRat that allows you to mix different radicals in the same expression. For example, (√3+√5)/√7 and so forth. ... That would certainly be nice. Note that QRat is basically already there for different quadratic radicals, the only reason it does not work already is that we cannot use a QRat as the base field instead of ℚ (because ℚ is hardcoded). Oh? I didn't try it myself, but if QRat itself passes isArithmeticType, I'd venture to say QRat!(n, QRat!m) ought to work... There are some hidden assumptions about properties of the rationals, though, but I surmise none that couldn't be replaced by prerequisites about the relative linear dependence of the mixed radicals over Q. ... The issue is that gcd does not work on QRats. If QRat had two coefficients from an arbitrary (possibly ordered) field instead of encoding rationals explicitly, I think it would work. What I had in mind, though, was a more direct approach that perhaps may reduce the total number of operations, since if the code is aware that multiple radicals are involved, it could potentially factor out some commonalities to minimize recomputations. ... This is probably the case. Also, the current implementation of QRat fixes the radical at compile-time; I wanted to see if I could dynamically handle arbitrary radicals at runtime. It would have to be restricted by only allowing operations between two QRats of the same extension, of course, but if the code could handle arbitrary extensions dynamically, then that restriction could be lifted and we could (potentially, anyway) support arbitrary combinations of expressions involving radicals. That would be far more useful than QRat, for some of the things I'd like to use it for. ... What applications do you have in mind? Computational geometry? This is the relevant concept from algebra: https://en.wikipedia.org/wiki/Splitting_field All your conjectures are true, except the last one. (Galois theory is not an obstacle, because here, we only need to consider splitting fields of particularly simple polynomials that are solvable in radicals.) That's nice to know. But I suppose Galois theory *would* become an obstacle if I wanted to implement, for example, Q(x) for an arbitrary algebraic x? ... All that the result about the quintic really says is that you will not, in general, be able to express x using field operations on radicals. It is still possible to compute the roots to arbitrary precision. Computing the field operations in ℚ(x) will actually still be quite straightforward but you'd have to think about what to do with toString and opCmp. (Or more generally, you'd have to think about how to pick one of the roots of a given polynomial.) You can even mix radicals of different degrees. Yes, I've thought about that before. So it should be possible to represent Q(r1,r2,...rn) using deg(r1)*deg(r2)*...*deg(rn)+1 coefficients? ... Yes, at most, except you don't need "+1". (For each radical ri, you will at most need to pick a power between 0 to deg(ri)-1 to index into the coefficients.)
Re: Exact arithmetic with quadratic irrationals
On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmars-d wrote: [...] > Yes, there is in fact a beautifully simple way to do better. :) > > Assume we want to compute some power of x. With a single > multiplication, we obtain x². Multiplying x² by itself, we obtain x⁴. > Repeating this a few times, we get: > > x, x², x⁴, x⁸, x¹⁶, x³², etc. > > In general, we only need n operations to compute x^(2ⁿ). > > To compute xʸ, it basically suffices to express y as a sum of powers > of two (i.e., we write it in binary). > > For example, 22 = 16 + 4 + 2, and x²² = x¹⁶·x⁴·x². > > My last post includes an implementation of this algorithm. ;) Ahh, so *that's* what it's all about. I figured that's what I was missing. :-D Thanks, I'll include this in QRat soon. > In particular, this leads to multiple ways to compute the n-th > Fibonacci number using O(log n) basic operations. (One way is to use > your QRat type, but we can also use the matrix (1 1; 1 0).) True, though I'm still jealous that with transcendental functions like with floating-point, one could ostensibly compute that in O(1). > > Another module I'm thinking about is an extension of QRat that > > allows you to mix different radicals in the same expression. For > > example, (√3+√5)/√7 and so forth. ... > > > > That would certainly be nice. Note that QRat is basically already > there for different quadratic radicals, the only reason it does not > work already is that we cannot use a QRat as the base field instead of > ℚ (because ℚ is hardcoded). Oh? I didn't try it myself, but if QRat itself passes isArithmeticType, I'd venture to say QRat!(n, QRat!m) ought to work... There are some hidden assumptions about properties of the rationals, though, but I surmise none that couldn't be replaced by prerequisites about the relative linear dependence of the mixed radicals over Q. What I had in mind, though, was a more direct approach that perhaps may reduce the total number of operations, since if the code is aware that multiple radicals are involved, it could potentially factor out some commonalities to minimize recomputations. Also, the current implementation of QRat fixes the radical at compile-time; I wanted to see if I could dynamically handle arbitrary radicals at runtime. It would have to be restricted by only allowing operations between two QRats of the same extension, of course, but if the code could handle arbitrary extensions dynamically, then that restriction could be lifted and we could (potentially, anyway) support arbitrary combinations of expressions involving radicals. That would be far more useful than QRat, for some of the things I'd like to use it for. > This is the relevant concept from algebra: > https://en.wikipedia.org/wiki/Splitting_field > > > All your conjectures are true, except the last one. (Galois theory is > not an obstacle, because here, we only need to consider splitting > fields of particularly simple polynomials that are solvable in > radicals.) That's nice to know. But I suppose Galois theory *would* become an obstacle if I wanted to implement, for example, Q(x) for an arbitrary algebraic x? > You can even mix radicals of different degrees. Yes, I've thought about that before. So it should be possible to represent Q(r1,r2,...rn) using deg(r1)*deg(r2)*...*deg(rn)+1 coefficients? > To get the formula for multiplicative inverses, one possible algorithm is: > https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm [...] Thanks, will look into this at some point. :-) T -- Some ideas are so stupid that only intellectuals could believe them. -- George Orwell
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 02:01, Timon Gehr wrote:
My last post includes an implementation of this algorithm. ;)
But in that implementation I used the parameter 'a' instead of the
variable 'x' as a result of being tired, which makes it slightly more
confusing than necessary even though it is correct. More readable version:
auto pow(T,S)(T a,S n){
T r=T(ℕ(1),ℕ(0));
for(auto x=a;n;n>>=1,x*=x)
if(n&1) r*=x;
return r;
}
Re: Exact arithmetic with quadratic irrationals
On 20.04.2017 02:01, Timon Gehr wrote: To get the formula for multiplicative inverses, one possible algorithm is: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm Better reference: https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Arithmetic_of_algebraic_extensions
Re: Exact arithmetic with quadratic irrationals
On 19.04.2017 23:39, H. S. Teoh via Digitalmars-d wrote: On Wed, Apr 19, 2017 at 10:47:04PM +0200, Timon Gehr via Digitalmars-d wrote: On 19.04.2017 21:32, H. S. Teoh via Digitalmars-d wrote: I alluded to this in D.learn some time ago, and finally decided to take the dip and actually write the code. So here it is: exact arithmetic with numbers of the form (a+b√r)/c where a, b, c are integers, c!=0, and r is a (fixed) square-free integer. Code: https://github.com/quickfur/qrat ... Nice. :) Some suggestions: - You might want to support ^^ (it is useful for examples like the one below). I would, except that I doubt it would perform any better than an actual recursive or iterative algorithm for computing Fibonacci sequences, because I don't know of any simple way to exponentiate a quadratic rational using only integer arithmetic other than repeated multiplication. (For all I know, it may perform even worse, because multiplying n quadratic rationals involves a lot more than just summing n+1 integers as in an iterative implementation of Fibonacci.) Hmm, come to think of it, I *could* expand the numerator using the binomial theorem, treating (a+b√r) as a binomial (a+bx) where x=√r, and folding even powers into the integral part (since x^2 = r, so x^(2k) = r^k). The denominator could be exponentiated using plain ole integer exponentiation. Then it's just a matter of summing coefficients. But it still seems to be about the same amount of work as (or more than) summing n+1 integers in an iterative implementation of Fibonacci. Am I missing something? ... Yes, there is in fact a beautifully simple way to do better. :) Assume we want to compute some power of x. With a single multiplication, we obtain x². Multiplying x² by itself, we obtain x⁴. Repeating this a few times, we get: x, x², x⁴, x⁸, x¹⁶, x³², etc. In general, we only need n operations to compute x^(2ⁿ). To compute xʸ, it basically suffices to express y as a sum of powers of two (i.e., we write it in binary). For example, 22 = 16 + 4 + 2, and x²² = x¹⁶·x⁴·x². My last post includes an implementation of this algorithm. ;) In particular, this leads to multiple ways to compute the n-th Fibonacci number using O(log n) basic operations. (One way is to use your QRat type, but we can also use the matrix (1 1; 1 0).) ... Another module I'm thinking about is an extension of QRat that allows you to mix different radicals in the same expression. For example, (√3+√5)/√7 and so forth. ... That would certainly be nice. Note that QRat is basically already there for different quadratic radicals, the only reason it does not work already is that we cannot use a QRat as the base field instead of ℚ (because ℚ is hardcoded). This is the relevant concept from algebra: https://en.wikipedia.org/wiki/Splitting_field All your conjectures are true, except the last one. (Galois theory is not an obstacle, because here, we only need to consider splitting fields of particularly simple polynomials that are solvable in radicals.) You can even mix radicals of different degrees. To get the formula for multiplicative inverses, one possible algorithm is: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm
Re: Exact arithmetic with quadratic irrationals
On Wed, Apr 19, 2017 at 10:47:04PM +0200, Timon Gehr via Digitalmars-d wrote: > On 19.04.2017 21:32, H. S. Teoh via Digitalmars-d wrote: > > I alluded to this in D.learn some time ago, and finally decided to > > take the dip and actually write the code. So here it is: exact > > arithmetic with numbers of the form (a+b√r)/c where a, b, c are > > integers, c!=0, and r is a (fixed) square-free integer. > > > > Code: https://github.com/quickfur/qrat > > > > ... > > Nice. :) > > Some suggestions: > > - You might want to support ^^ (it is useful for examples like the one > below). I would, except that I doubt it would perform any better than an actual recursive or iterative algorithm for computing Fibonacci sequences, because I don't know of any simple way to exponentiate a quadratic rational using only integer arithmetic other than repeated multiplication. (For all I know, it may perform even worse, because multiplying n quadratic rationals involves a lot more than just summing n+1 integers as in an iterative implementation of Fibonacci.) Hmm, come to think of it, I *could* expand the numerator using the binomial theorem, treating (a+b√r) as a binomial (a+bx) where x=√r, and folding even powers into the integral part (since x^2 = r, so x^(2k) = r^k). The denominator could be exponentiated using plain ole integer exponentiation. Then it's just a matter of summing coefficients. But it still seems to be about the same amount of work as (or more than) summing n+1 integers in an iterative implementation of Fibonacci. Am I missing something? > - constructor parameter _b should have a default value of 0. Good point, done. > - formatting should special case b==-1 like it special cases b==1. Done, good catch! > (also: as you are using Unicode anyway, you could also use · instead > of *. Another cute thing to do is to add a vinculum.) Well, I would, but it gets a bit too fancy for my tastes and may not render well on all displays. But I'll put it on my list of things to consider. Another module I'm thinking about is an extension of QRat that allows you to mix different radicals in the same expression. For example, (√3+√5)/√7 and so forth. I have discovered algorithms that, given n distinct radicals, allow a closed-form expression with 2^n coefficients (+1 denominator), closed under field operations. The only trouble in this case is that reciprocating such things will be very slow, as will comparisons, and both have a high chance of overflow (so BigInt is probably a necessity). And 2^n+1 coefficients for n radicals quickly gets expensive space-wise as n increases. Yesterday I also discovered an algorithm for expressing the reciprocal of numbers of the form: (a + b∛r + c∛r^2)/d in the same form. I.e., for rewriting: d/(a + b∛r + c∛r^2) in the first form. Which means it's possible to implement a QRat-like representation for cubic rationals. (The actual computation is rather expensive, as it involves quite a lot of multiplications, squaring, and cubing. But it's *possible*.) I'm still trying to verify the correctness of the formula I obtained, since while checking a concrete example last night I discovered a possible error. If this works out, I might consider 4th roots as well -- though I'm expecting that might be near the limit of the usefulness of these representations, since the complexity becomes so great that symbolic manipulation like in Mathematica may turn out to be more feasible after all. But it may be of some theoretical interest whether such representations are possible, even if they are ultimately impractical. A particularly interesting question is whether such representations exist for *all* algebraic numbers (of bounded degree). Currently I have a conjecture that given a rational extension of n radicals of degree k, field closure can be achieved with a representation of k^n+1 coefficients. But it's still too early to say whether algorithms exist for inverting radicals of degree k for large k -- I have a creeping suspicion that perhaps somewhere around k=5 or k=6 the unsolvability of the general quintic may raise its ugly head and prevent further progress. T -- INTEL = Only half of "intelligence".
Re: Exact arithmetic with quadratic irrationals
On 19.04.2017 21:32, H. S. Teoh via Digitalmars-d wrote:
I alluded to this in D.learn some time ago, and finally decided to take
the dip and actually write the code. So here it is: exact arithmetic
with numbers of the form (a+b√r)/c where a, b, c are integers, c!=0, and
r is a (fixed) square-free integer.
Code: https://github.com/quickfur/qrat
...
Nice. :)
Some suggestions:
- You might want to support ^^ (it is useful for examples like the one
below).
- constructor parameter _b should have a default value of 0.
- formatting should special case b==-1 like it special cases b==1.
(also: as you are using Unicode anyway, you could also use · instead
of *. Another cute thing to do is to add a vinculum.)
Example application: Computing large Fibonacci numbers efficiently:
import qrat;
import std.bigint;
alias ℕ=BigInt;
enum φ=(1+surd!(5,ℕ))/2,ψ=(1-surd!(5,ℕ))/2;
auto pow(T,S)(T a,S n){
T r=T(ℕ(1),ℕ(0));
for(auto x=a;n;n>>=1,a*=a)
if(n&1) r*=a;
return r;
}
auto fib(long n){
return (pow(φ,n)-pow(ψ,n))/surd!(5,ℕ);
}
void main(){
import std.stdio;
foreach(i;0..40) writeln(fib(i));
writeln(fib(10));
}
Re: Exact arithmetic with quadratic irrationals
On Wed, Apr 19, 2017 at 07:54:02PM +, Stanislav Blinov via Digitalmars-d
wrote:
> Awesome! Congrats and thanks for sharing.
>
> On Wednesday, 19 April 2017 at 19:32:14 UTC, H. S. Teoh wrote:
>
> > Haha, it seems that the only roadblocks were related to the
> > implementation quality of std.numeric.gcd... nothing that a few
> > relatively-simple PRs couldn't fix. So overall, D is still awesome.
>
> There's another one, which is more about dmd: dmd does not inline gcd,
> which, when arguments are const, turns gcd into a double function call
> :D
If I weren't such a sucker for the bleeDing edge with dmd (I actually
compile even my serious projects with git HEAD dmd, except when
performance matters), I'd be standardizing on ldc or gdc, which have far
superior optimizers.
I consistently find that my CPU-intensive projects perform at least
20-30% worse on dmd than gdc (and I presume ldc), sometimes even as bad
as 40-50%, due to dmd's inliner giving up far too easily. I don't know
if this has been fixed yet, but the last time I checked, if you wrote
this:
int func(int x, int y) {
if (x<0)
return y;
return x;
}
instead of:
int func(int x, int y) {
if (x<0)
return y;
else
return x;
}
then the dmd inliner would not inline the function.
Because of sensitivities like this, the inliner gives up far too early
in the process, thus the optimizer misses out on further optimization
opportunities that would have opened up, had the function been inlined.
The last time I checked, I also found that dmd was rather weak at loop
optimizations (and loops are very important in performance as we all
know) compared to gdc. Again, the same domino effect (or rather, the
missing thereof) applies: by failing to, for example, hoist a constant
expression out of the loop, further optimization opportunities are
missed, whereas gdc, after hoisting the expression out, would discover
that the loop can be reduced further, perhaps via a strength reduction,
and then unrolled, and then inlined inside the caller, then vectorized,
etc.. This chain of optimizations were missed because of one missed
optimization early in the process. Hence the suboptimal resulting code.
T
--
If blunt statements had a point, they wouldn't be blunt...
Re: Exact arithmetic with quadratic irrationals
Awesome! Congrats and thanks for sharing. On Wednesday, 19 April 2017 at 19:32:14 UTC, H. S. Teoh wrote: Haha, it seems that the only roadblocks were related to the implementation quality of std.numeric.gcd... nothing that a few relatively-simple PRs couldn't fix. So overall, D is still awesome. There's another one, which is more about dmd: dmd does not inline gcd, which, when arguments are const, turns gcd into a double function call :D
