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Speed up GcdInt, PValuation, SmallestRootInt, IsPrimePowerInt #2086

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Jan 15, 2018
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Speed up GcdInt, PValuation, SmallestRootInt, IsPrimePowerInt #2086

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6a635c9
kernel: optimize GcdInt for small inputs
fingolfin Jan 9, 2018
8bf97a8
kernel: optimize PVALUATION_INT for small inputs
fingolfin Jan 9, 2018
3563e60
Optimize SmallestRootInt
fingolfin Jan 9, 2018
9021830
Optimize IsPrimePowerInt (speed up 10x)
fingolfin Jan 9, 2018
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70 changes: 59 additions & 11 deletions hpcgap/lib/integer.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -923,8 +923,51 @@ InstallGlobalFunction( IsOddInt, n -> n mod 2 = 1 );
##
#F IsPrimePowerInt( <n> ) . . . . . . . . . . . test for a power of a prime
##
InstallGlobalFunction( IsPrimePowerInt,
n -> IsPrimeInt( SmallestRootInt( n ) ) );
InstallGlobalFunction( IsPrimePowerInt, function(n)
local k, r, s, p, l, q, i;

# check the argument
if n > 1 then k := 2; s := 1;
elif n < -1 then k := 3; s := -1; n := -n;
else return false;
fi;

# exclude small divisors, and thereby large exponents
for p in Primes do
if p*p > n then return true; fi; # n is prime
r := PVALUATION_INT(n, p);
if r > 0 then
if s = -1 and IsEvenInt(r) then return false; fi;
return n = p^r;
fi;
od;
l := LogInt( n, p );

# loop over the possible prime divisors of exponents
# use Fermat's little theorem to cast out impossible ones:
# for suppose we had r such that n = r^k. Then by Fermat,
# n^((q-1)/k) = r^(q-1) is congruent 0 or 1 mod q
i := Position(Primes, k);
while k <= l do
q := 2*k+1; while not IsPrimeInt(q) do q := q+2*k; od;
if PowerModInt( n, (q-1)/k, q ) <= 1 then
r := RootInt( n, k );
if r ^ k = n then
n := r;
l := QuoInt( l, k );
continue;
fi;
fi;
if i <> fail and i < Length(Primes) then
i := i + 1;
k := Primes[i];
else
k := NextPrimeInt( k );
fi;
od;

return IsPrimeInt(n);
end);


#############################################################################
Expand Down Expand Up @@ -1118,7 +1161,7 @@ InstallGlobalFunction( SignInt, SIGN_RAT ); # support rationals for backwards co
#F SmallestRootInt( <n> ) . . . . . . . . . . . smallest root of an integer
##
InstallGlobalFunction(SmallestRootInt,function ( n )
local k, r, s, p, l, q;
local k, r, s, p, l, q, i;

# check the argument
if n > 0 then k := 2; s := 1;
Expand All @@ -1127,25 +1170,30 @@ InstallGlobalFunction(SmallestRootInt,function ( n )
fi;

# exclude small divisors, and thereby large exponents
if n mod 2 = 0 then
p := 2;
else
p := 3; while p < 100 and n mod p <> 0 do p := p+2; od;
fi;
for p in Primes do
if p*p > n then return s * n; fi;
if n mod p = 0 then break; fi;
od;
l := LogInt( n, p );

# loop over the possible prime divisors of exponents
# use Euler's criterion to cast out impossible ones
# use Fermat's little theorem to cast out impossible ones:
# for suppose we had r such that n = r^k. Then by Fermat,
# n^((q-1)/k) = r^(q-1) is congruent 0 or 1 mod q
i := Position(Primes, k);
while k <= l do
q := 2*k+1; while not IsPrimeInt(q) do q := q+2*k; od;
if PowerModInt( n, (q-1)/k, q ) <= 1 then
r := RootInt( n, k );
if r ^ k = n then
n := r;
l := QuoInt( l, k );
else
k := NextPrimeInt( k );
continue;
fi;
fi;
if i <> fail and i < Length(Primes) then
i := i + 1;
k := Primes[i];
else
k := NextPrimeInt( k );
fi;
Expand Down
70 changes: 59 additions & 11 deletions lib/integer.gi
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Expand Up @@ -910,8 +910,51 @@ InstallGlobalFunction( IsOddInt, n -> n mod 2 = 1 );
##
#F IsPrimePowerInt( <n> ) . . . . . . . . . . . test for a power of a prime
##
InstallGlobalFunction( IsPrimePowerInt,
n -> IsPrimeInt( SmallestRootInt( n ) ) );
InstallGlobalFunction( IsPrimePowerInt, function(n)
local k, r, s, p, l, q, i;

# check the argument
if n > 1 then k := 2; s := 1;
elif n < -1 then k := 3; s := -1; n := -n;
else return false;
fi;

# exclude small divisors, and thereby large exponents
for p in Primes do
if p*p > n then return true; fi; # n is prime
r := PVALUATION_INT(n, p);
if r > 0 then
if s = -1 and IsEvenInt(r) then return false; fi;
return n = p^r;
fi;
od;
l := LogInt( n, p );

# loop over the possible prime divisors of exponents
# use Fermat's little theorem to cast out impossible ones:
# for suppose we had r such that n = r^k. Then by Fermat,
# n^((q-1)/k) = r^(q-1) is congruent 0 or 1 mod q
i := Position(Primes, k);
while k <= l do
q := 2*k+1; while not IsPrimeInt(q) do q := q+2*k; od;
if PowerModInt( n, (q-1)/k, q ) <= 1 then
r := RootInt( n, k );
if r ^ k = n then
n := r;
l := QuoInt( l, k );
continue;
fi;
fi;
if i <> fail and i < Length(Primes) then
i := i + 1;
k := Primes[i];
else
k := NextPrimeInt( k );
fi;
od;

return IsPrimeInt(n);
end);


#############################################################################
Expand Down Expand Up @@ -1105,7 +1148,7 @@ InstallGlobalFunction( SignInt, SIGN_RAT ); # support rationals for backwards co
#F SmallestRootInt( <n> ) . . . . . . . . . . . smallest root of an integer
##
InstallGlobalFunction(SmallestRootInt,function ( n )
local k, r, s, p, l, q;
local k, r, s, p, l, q, i;

# check the argument
if n > 0 then k := 2; s := 1;
Expand All @@ -1114,25 +1157,30 @@ InstallGlobalFunction(SmallestRootInt,function ( n )
fi;

# exclude small divisors, and thereby large exponents
if n mod 2 = 0 then
p := 2;
else
p := 3; while p < 100 and n mod p <> 0 do p := p+2; od;
fi;
for p in Primes do
if p*p > n then return s * n; fi;
if n mod p = 0 then break; fi;
od;
l := LogInt( n, p );

# loop over the possible prime divisors of exponents
# use Euler's criterion to cast out impossible ones
# use Fermat's little theorem to cast out impossible ones:
# for suppose we had r such that n = r^k. Then by Fermat,
# n^((q-1)/k) = r^(q-1) is congruent 0 or 1 mod q
i := Position(Primes, k);
while k <= l do
q := 2*k+1; while not IsPrimeInt(q) do q := q+2*k; od;
if PowerModInt( n, (q-1)/k, q ) <= 1 then
r := RootInt( n, k );
if r ^ k = n then
n := r;
l := QuoInt( l, k );
else
k := NextPrimeInt( k );
continue;
fi;
fi;
if i <> fail and i < Length(Primes) then
i := i + 1;
k := Primes[i];
else
k := NextPrimeInt( k );
fi;
Expand Down
38 changes: 38 additions & 0 deletions src/integer.c
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Expand Up @@ -690,6 +690,20 @@ UInt8 UInt8_ObjInt(Obj i)
#endif
}

// This function returns an immediate integer, or
// an integer object with exactly one limb, and returns
// its absolute value as an unsigned integer.
static inline UInt AbsOfSmallInt(Obj x)
{
if (!IS_INTOBJ(x)) {
GAP_ASSERT(SIZE_INT(x) == 1);
return VAL_LIMB0(x);
}
Int val = INT_INTOBJ(x);
return val > 0 ? val : -val;
}


/****************************************************************************
**
*F PrintInt( <gmp> ) . . . . . . . . . . . . . . . . print a GMP constant
Expand Down Expand Up @@ -2163,6 +2177,18 @@ Obj GcdInt ( Obj opL, Obj opR )
sizeL = SIZE_INT_OR_INTOBJ(opL);
sizeR = SIZE_INT_OR_INTOBJ(opR);

// for small inputs, run Euclid directly
if (sizeL == 1 || sizeR == 1) {
if (sizeR != 1) {
SWAP(Obj, opL, opR);
}
UInt r = AbsOfSmallInt(opR);
FAKEMPZ_GMPorINTOBJ(mpzL, opL);
r = mpz_gcd_ui(0, MPZ_FAKEMPZ(mpzL), r);
CHECK_FAKEMPZ(mpzL);
return ObjInt_UInt(r);
}

NEW_FAKEMPZ( mpzResult, sizeL < sizeR ? sizeL : sizeR );
FAKEMPZ_GMPorINTOBJ( mpzL, opL );
FAKEMPZ_GMPorINTOBJ( mpzR, opR );
Expand Down Expand Up @@ -2396,6 +2422,18 @@ Obj FuncPVALUATION_INT(Obj self, Obj n, Obj p)
if ( p == INTOBJ_INT(0) )
ErrorMayQuit( "PValuation: <p> must be nonzero", 0L, 0L );

if (SIZE_INT_OR_INTOBJ(n) == 1 && SIZE_INT_OR_INTOBJ(p) == 1) {
UInt N = AbsOfSmallInt(n);
UInt P = AbsOfSmallInt(p);
if (N == 0 || P == 1) return INTOBJ_INT(0);
k = 0;
while (N % P == 0) {
N /= P;
k++;
}
return INTOBJ_INT(k);
}

/* For certain values of p, mpz_remove replaces its "dest" argument
and tries to deallocate the original mpz_t in it. This means
we cannot use a fake_mpz_t for it. However, we are not really
Expand Down
71 changes: 69 additions & 2 deletions tst/testinstall/intarith.tst
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Expand Up @@ -1332,13 +1332,41 @@ gap> for m in [1..100] do
#
gap> checkPValuationInt:=function(n,p)
> local k, m;
> if n = 0 then return true; fi;
> k:=PVALUATION_INT(n,p);
> if n = 0 or p = 1 or p = -1 then return k = 0; fi;
> m:=n/p^k;
> return IsInt(m) and (m mod p) <> 0;
> end;;
gap> ForAll([-10000 .. 10000], n-> ForAll([2,3,5,7,251], p -> checkPValuationInt(n,p)));
gap> ps := [-2^60-1,-2^60,-2^28-1,-2^28];;
gap> Append(ps, [-6..-1]);
gap> Append(ps, [1..20]);
gap> Append(ps, [250..260]);
gap> Append(ps, [2^28-1,2^28,2^60-1,2^60]);;
gap> SetX([-1000 .. 1000], ps, checkPValuationInt);
[ true ]
gap> SetX([-1000 .. 1000], ps, {n,p}->checkPValuationInt(n+2^100,p));
[ true ]
gap> SetX([-1000 .. 1000], ps, {n,p}->checkPValuationInt(n-2^100,p));
[ true ]

#
gap> p:=2^255-19;; # big prime
gap> ForAll([1..30], k-> PVALUATION_INT((p+1)^k,2)=k);
true
gap> ForAll([1..30], k-> PVALUATION_INT((p+1)^k,p)=0);
true
gap> ForAll([1..30], k-> PVALUATION_INT(p^k+1,2)=1);
true
gap> ForAll([1..30], k-> PVALUATION_INT(p^k+1,p)=0);
true

#
gap> SetX(data, dataNonZero, checkPValuationInt);
[ true ]
gap> List([2..6], p->List(data, n->PVALUATION_INT(n,p)));
[ [ 0, 20, 4, 0, 0, 0, 4, 20, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 10, 2, 0, 0, 0, 2, 10, 0 ], [ 0, 20, 4, 0, 0, 0, 4, 20, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
gap> PVALUATION_INT(10,0);
Error, PValuation: <p> must be nonzero
gap> PVALUATION_INT(0,0);
Expand Down Expand Up @@ -1366,6 +1394,45 @@ Error, IsProbablyPrimeInt: <reps> must be a small positive integer
gap> IS_PROBAB_PRIME_INT(1, 0);
Error, IsProbablyPrimeInt: <reps> must be a small positive integer

#
# test SmallestRootInt
#
gap> SetX(Primes, [1..30], {p,k}->SmallestRootInt(p^k)=p);
[ true ]
gap> SetX(Primes, [1..30], {p,k}->SmallestRootInt(p^k*1009)=p^k*1009);
[ true ]
gap> SetX(Primes, [1..30], {p,k}->SmallestRootInt((p*1009)^k)=p*1009);
[ true ]
gap> p:=2^255-19;; # big prime
gap> ForAll([1..30], k-> SmallestRootInt(p^k) = p);
true
gap> List([-10..10], SmallestRootInt);
[ -10, -9, -2, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 2, 5, 6, 7, 2, 3, 10 ]
gap> List(data, SmallestRootInt);
[ -100000000000000000001, -10000, -10000, -1, 0, 1, 10, 10,
100000000000000000001 ]
gap> List([-2^101,-2^100,2^100,2^101], SmallestRootInt);
[ -2, -16, 2, 2 ]

#
# test IsPrimePowerInt
#
gap> P:=2^255-19;; # big prime
gap> Filtered([-10..10], IsPrimePowerInt);
[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ]
gap> SetX(Primes, [1..30], {p,k}->IsPrimePowerInt(p^k));
[ true ]
gap> SetX(Primes, [1..10], {p,k}->IsPrimePowerInt(P*p^k));
[ false ]
gap> SetX(Primes, [1..10], {p,k}->IsPrimePowerInt((P*p)^k));
[ false ]
gap> ForAll([1..30], k->IsPrimePowerInt(P^k));
true
gap> ForAll([1..10], k->not IsPrimePowerInt(1009*P^k));
true
gap> ForAll([1..30], k->not IsPrimePowerInt((1009*P)^k));
true

#
gap> STOP_TEST( "intarith.tst", 1);

Expand Down