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changed: algorithms in Big::Sqrt() and ttmath::Root(x ; n)

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they were not too much accurate for some integers
         e.g. Root(16;4) returned a value very closed to 2 (not exactly 2)



git-svn-id: svn://ttmath.org/publicrep/ttmath/trunk@231 e52654a7-88a9-db11-a3e9-0013d4bc506e
This commit is contained in:
Tomasz Sowa
2009-11-01 20:26:01 +00:00
parent 4f1763d773
commit 4b4b30392a
3 changed files with 207 additions and 136 deletions
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304
ttmath/ttmath.h
View File

@@ -1856,146 +1856,189 @@ namespace ttmath
namespace auxiliaryfunctions
{
template<class ValueType>
bool RootCheckIndexSign(ValueType & x, const ValueType & index, ErrorCode * err)
template<class ValueType>
bool RootCheckIndexSign(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( index.IsSign() )
{
if( index.IsSign() )
{
// index cannot be negative
if( err )
*err = err_improper_argument;
// index cannot be negative
if( err )
*err = err_improper_argument;
x.SetNan();
x.SetNan();
return true;
}
return false;
return true;
}
template<class ValueType>
bool RootCheckIndexZero(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( index.IsZero() )
{
if( x.IsZero() )
{
// there isn't root(0;0) - we assume it's not defined
if( err )
*err = err_improper_argument;
x.SetNan();
return true;
}
// root(x;0) is 1 (if x!=0)
x.SetOne();
if( err )
*err = err_ok;
return true;
}
return false;
}
return false;
}
template<class ValueType>
bool RootCheckIndexOne(const ValueType & index, ErrorCode * err)
{
ValueType one;
one.SetOne();
if( index == one )
{
//root(x;1) is x
// we do it because if we used the PowFrac function
// we would lose the precision
if( err )
*err = err_ok;
return true;
}
return false;
}
template<class ValueType>
bool RootCheckIndexFrac(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( !index.IsInteger() )
{
// index must be integer
if( err )
*err = err_improper_argument;
x.SetNan();
return true;
}
return false;
}
template<class ValueType>
bool RootCheckXZero(ValueType & x, ErrorCode * err)
template<class ValueType>
bool RootCheckIndexZero(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( index.IsZero() )
{
if( x.IsZero() )
{
// root(0;index) is zero (if index!=0)
// RootCheckIndexZero() must be called beforehand
x.SetZero();
// there isn't root(0;0) - we assume it's not defined
if( err )
*err = err_ok;
*err = err_improper_argument;
x.SetNan();
return true;
}
// root(x;0) is 1 (if x!=0)
x.SetOne();
return false;
}
template<class ValueType>
bool RootCheckIndex(ValueType & x, const ValueType & index, ErrorCode * err, bool * change_sign)
{
*change_sign = false;
if( index.Mod2() )
{
// index is odd (1,3,5...)
if( x.IsSign() )
{
*change_sign = true;
x.Abs();
}
}
else
{
// index is even
// x cannot be negative
if( x.IsSign() )
{
if( err )
*err = err_improper_argument;
x.SetNan();
return true;
}
}
return false;
if( err )
*err = err_ok;
return true;
}
return false;
}
template<class ValueType>
bool RootCheckIndexOne(const ValueType & index, ErrorCode * err)
{
ValueType one;
one.SetOne();
if( index == one )
{
//root(x;1) is x
// we do it because if we used the PowFrac function
// we would lose the precision
if( err )
*err = err_ok;
return true;
}
return false;
}
template<class ValueType>
bool RootCheckIndexTwo(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( index == 2 )
{
x = Sqrt(x, err);
return true;
}
return false;
}
template<class ValueType>
bool RootCheckIndexFrac(ValueType & x, const ValueType & index, ErrorCode * err)
{
if( !index.IsInteger() )
{
// index must be integer
if( err )
*err = err_improper_argument;
x.SetNan();
return true;
}
return false;
}
template<class ValueType>
bool RootCheckXZero(ValueType & x, ErrorCode * err)
{
if( x.IsZero() )
{
// root(0;index) is zero (if index!=0)
// RootCheckIndexZero() must be called beforehand
x.SetZero();
if( err )
*err = err_ok;
return true;
}
return false;
}
template<class ValueType>
bool RootCheckIndex(ValueType & x, const ValueType & index, ErrorCode * err, bool * change_sign)
{
*change_sign = false;
if( index.Mod2() )
{
// index is odd (1,3,5...)
if( x.IsSign() )
{
*change_sign = true;
x.Abs();
}
}
else
{
// index is even
// x cannot be negative
if( x.IsSign() )
{
if( err )
*err = err_improper_argument;
x.SetNan();
return true;
}
}
return false;
}
template<class ValueType>
uint RootCorrectInteger(ValueType & old_x, ValueType & x, const ValueType & index)
{
if( !old_x.IsInteger() || x.IsInteger() || !index.exponent.IsSign() )
return 0;
// old_x is integer,
// x is not integer,
// index is relatively small (index.exponent<0 or index.exponent<=0)
// (because we're using a special powering algorithm Big::PowUInt())
uint c = 0;
ValueType temp(x);
c += temp.Round();
ValueType temp_round(temp);
c += temp.PowUInt(index);
if( temp == old_x )
x = temp_round;
return (c==0)? 0 : 1;
}
} // namespace auxiliaryfunctions
/*!
indexth Root of x
index must be integer and not negative <0;1;2;3....)
@@ -2026,30 +2069,33 @@ namespace ttmath
if( RootCheckIndexSign(x, index, err) ) return x;
if( RootCheckIndexZero(x, index, err) ) return x;
if( RootCheckIndexOne ( index, err) ) return x;
if( RootCheckIndexTwo (x, index, err) ) return x;
if( RootCheckIndexFrac(x, index, err) ) return x;
if( RootCheckXZero (x, err) ) return x;
// index integer and index!=0
// x!=0
uint c = 0;
ValueType old_x(x);
bool change_sign;
if( RootCheckIndex(x, index, err, &change_sign ) ) return x;
ValueType newindex;
newindex.SetOne();
c += newindex.Div(index);
c += x.PowFrac(newindex); // here can only be a carry
ValueType temp;
uint c = 0;
// we're using the formula: root(x ; n) = exp( ln(x) / n )
c += temp.Ln(x);
c += temp.Div(index);
c += x.Exp(temp);
if( change_sign )
{
// the value of x should be different from zero
// (x is actually tested by RootCheckXZero)
TTMATH_ASSERT( x.IsZero() == false )
// x is different from zero
x.SetSign();
}
c += RootCorrectInteger(old_x, x, index);
if( err )
*err = c ? err_overflow : err_ok;

33
ttmath/ttmathbig.h
View File

@@ -1511,8 +1511,7 @@ public:
/*!
this function calculates the square root
Sqrt(9) = 3
e.g. let this=9 then this.Sqrt() gives 3
return: 0 - ok
1 - carry
@@ -1529,11 +1528,33 @@ public:
if( IsZero() )
return 0;
Big<exp, man> pow;
pow.Set05();
Big<exp, man> old(*this);
Big<exp, man> ln;
uint c = 0;
// PowFrac can return only a carry because x is greater than zero
return PowFrac(pow);
// we're using the formula: sqrt(x) = e ^ (ln(x) / 2)
c += ln.Ln(*this);
c += ln.exponent.SubOne(); // ln = ln / 2
c += Exp(ln);
// above formula doesn't give accurate results for some integers
// e.g. Sqrt(81) would not be 9 but a value very closed to 9
// we're rounding the result, calculating result*result and comparing
// with the old value, if they are equal then the result is an integer too
if( !c && old.IsInteger() && !IsInteger() )
{
Big<exp, man> temp(*this);
c += temp.Round();
Big<exp, man> temp2(temp);
c += temp.Mul(temp2);
if( temp == old )
*this = temp2;
}
return CheckCarry(c);
}