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afp.cpp
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afp.cpp
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// AFP
// Floating Point Precision Number class (Binary Coded Decimal)
// An Arbitrary float always has the format [sign][digit][.[digit]*][E[sign][digits]+] where sign is either '+' or '-'
// And is always stored in normalized mode after an operation or conversion
// The length or the representation is always >= 2
// A null string is considered as an error and an exception is thrown
// Floating Point Numbers is stored in BASE 10 or BASE 256 depends of the F_RADIX setting
//
// Improvement of the float_precision library of Henrik Vestermark (hve@hvks.com). See www.hvks.com
//
#include "Stdafx.h"
#include "afp.h"
#include <math.h>
// Core Supporting functions. Works directly on string class
// All forward declarations to statics in this implementation file
static int _afp_normalize( std::tstring * );
static int _afp_rounding(std::tstring* m, int sign, unsigned int precision, enum round_mode mode );
static void _afp_strip_leading_zeros( std::tstring * );
static void _afp_strip_trailing_zeros( std::tstring * );
static void _afp_right_shift( std::tstring *, int );
static void _afp_left_shift( std::tstring *, int );
static int _afp_compare( std::tstring *, std::tstring * );
static void _int_real_fourier( double data[], unsigned int n, int isign );
static void _int_fourier( std::complex<double> data[], unsigned int n, int isign );
static void _int_reverse_binary( std::complex<double> data[], unsigned int n );
static std::tstring _afp_uadd_short( std::tstring *, unsigned int );
static std::tstring _afp_uadd( std::tstring *, std::tstring * );
static std::tstring _afp_usub_short( int *, std::tstring *, unsigned int );
static std::tstring _afp_usub( int *, std::tstring *, std::tstring * );
static std::tstring _afp_umul_short( std::tstring *, unsigned int );
static std::tstring _afp_umul( std::tstring *, std::tstring * );
static std::tstring _afp_umul_fourier( std::tstring *, std::tstring * );
static std::tstring _afp_udiv_short( unsigned int *, std::tstring *, unsigned int );
// Inline functions for the internals of the AFP
inline _TUCHAR FDIGIT(_TUCHAR x)
{
#if (F_RADIX == BASE_10)
return (_TUCHAR) (x - '0');
#else
return (_TUCHAR) x;
#endif
}
inline _TUCHAR FCHARACTER(_TUCHAR x )
{
#if (F_RADIX == BASE_10)
return (_TUCHAR)(x + '0');
#else
return (_TUCHAR) x;
#endif
}
inline _TUCHAR FCHARACTER10(_TUCHAR x)
{
return (_TUCHAR)( x + '0');
}
inline int FCARRY( unsigned int x ) { return x / F_RADIX; }
inline int FSINGLE( unsigned int x ) { return x % F_RADIX; }
inline std::tstring SIGN_STRING( int x ) { return x >=0 ? _T("+") : _T("-") ; }
inline int CHAR_SIGN(TCHAR x ) { return x == '-' ? -1 : 1; }
//////////////////////////////////////////////////////////////////////////////////////
///
/// CONSTRUCTORS
///
//////////////////////////////////////////////////////////////////////////////////////
// Description:
// Constructor through a char number
// Validate and initialize with a character
// Input Always in BASE_10
//
afp::afp(TCHAR p_number, unsigned int p_precision, enum round_mode p_mode)
{
int i;
if( p_number < '0' || p_number > '9' )
{
throw bad_int_syntax();
}
else
{
i = p_number - '0'; // Convert to integer
m_number = _T("+");
m_number += FCHARACTER( FSINGLE( i ) );
m_expo = 0;
m_rmode = p_mode;
m_prec = p_precision;
}
}
// Description:
// Constructor through an integer
// Just convert integer to string representation in BASE RADIX
// The input integer is always BASE_10
// Only use core base functions to create multi precision numbers
//
afp::afp(int p_number, unsigned int p_precision, enum round_mode p_mode)
{
int sign = 1;
std::tstring number;
m_rmode = p_mode;
m_prec = p_precision;
m_expo = 0;
// Optimize null
if(p_number == 0)
{
m_number = _T("+");
m_number.append( 1, FCHARACTER( 0 ) );
return;
}
// Take care of sign
if(p_number < 0 )
{
p_number = -p_number;
sign = -1;
}
if(F_RADIX == BASE_256) // Fast BASE_256 conversion
{
int j;
_TUCHAR* p = (_TUCHAR *)&p_number;
for( j = sizeof( int ); j > 0; j-- )
{
if( p[j-1] != 0 ) // Strip leading zeros
{
break;
}
}
for( ; j > 0; j-- )
{
number.append( 1, p[j-1] );
}
}
else
{
// All other BASE
for( ; p_number != 0; p_number /= F_RADIX )
{
number.insert((std::tstring::size_type)0, 1, FCHARACTER(p_number % F_RADIX));
}
}
_afp_strip_leading_zeros( &number ); // First strip for leading zeros
m_expo = (int)number.length() -1; // Always one digit before the dot
_afp_strip_trailing_zeros( &number ); // Get rid of trailing non-significant zeros
m_expo += _afp_rounding( &number, sign, m_prec, m_rmode ); // Perform rounding
m_number = SIGN_STRING( sign ) + number; // Build number
}
// Description:
// Constructor trough a character string
// Validate input and convert to internal representation
// Always add sign if not specified
// Only use core base functions to create multi precision numbers
// The float can be any integer or decimal float representation
//
afp::afp(LPCTSTR p_number, unsigned int p_precision, enum round_mode p_mode)
{
if(p_number == NULL || *p_number== '\0' )
{
throw bad_int_syntax();
}
m_rmode = p_mode;
m_prec = p_precision;
*this = _afp_atof(p_number,p_precision,p_mode);
}
// Description:
// Constructor from a double
// Validate input and convert to internal representation
// Always add sign if not specified
// Only use core base functions to create multi precision numbers
// The float can be any integer or decimal float representation
//
afp::afp(double p_number, unsigned int p_precision, enum round_mode p_mode)
{
m_rmode = p_mode;
m_prec = p_precision;
*this = _afp_dtof(p_number,p_precision,p_mode);
}
// When initialized through another afp
afp::afp(const afp& p_number)
:m_number(p_number.m_number)
,m_rmode (p_number.m_rmode)
,m_prec (p_number.m_prec)
,m_expo (p_number.m_expo)
{
// Simply copy constructor. Nothing more to do
}
//////////////////////////////////////////////////////////////////////////////////////
///
/// END AFP CONSTRUCTORS
///
//////////////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
//
// AFP PUBLIC FUNCTIONS
//
//////////////////////////////////////////////////////////////////////////
unsigned
afp::precision(unsigned int p_precision)
{
int sign;
std::tstring m;
sign = CHAR_SIGN(m_number[0]);
m_prec = p_precision > 0 ? p_precision : PRECISION;
// [HVKS] Adjust rounding only works in base_10 mode of the class
if(F_RADIX == BASE_10)
{
m = m_number.substr(1); // Bypass sign
m_expo += _afp_rounding(&m, sign, m_prec, m_rmode);
m_number = SIGN_STRING(sign) + m;
}
return m_prec;
}
// Assign another AFP to this AFP
// Works like the copy constructor, but can always be called as in
// afp number;
// afp another;
// number.assign(another)
afp
afp::assign(afp& p_another)
{
m_rmode = p_another.m_rmode;
m_prec = p_another.m_prec;
m_expo = p_another.m_expo;
m_number = p_another.m_number;
return *this;
}
// Change the sign of the AFP
// and returns the new sign
int
afp::change_sign()
{
// Change and return sign
if(m_number.length() != 2 || FDIGIT(m_number[1]) != 0 ) // Don't change sign for +0!
{
if(m_number[0] == '+')
{
m_number[0] = '-';
}
else
{
m_number[0] = '+';
}
}
return CHAR_SIGN(m_number[0]);
}
// Return the sign of the number
int
afp::sign() const
{
return CHAR_SIGN( m_number[0] );
}
// Cast to an int
afp::operator int()
{
// Conversion to int
std::tstring s = _afp_ftoa(*this);
return _ttoi( s.c_str() );
}
// Cast to a double
afp::operator double()
{
// Conversion to double
std::tstring s = _afp_ftoa(*this);
return _ttof( s.c_str() );
}
//////////////////////////////////////////////////////////////////////////////////////
///
/// AFP OPERATORS
///
//////////////////////////////////////////////////////////////////////////////////////
// Description:
// Assign operator
// Round it to precision and mode of the left hand side
// Only the exponent and mantissa is assigned
// Mode and precision is not affected by the assignment
//
afp&
afp::operator=(const afp& p_other)
{
int sign;
m_expo = p_other.m_expo;
sign = p_other.sign();
m_number = p_other.m_number.substr(1);
// [HVKS] In 10 base, see to rounding of
if(F_RADIX == BASE_10)
{
if(_afp_rounding(&m_number,sign,m_prec,m_rmode) != 0 )
{
// Round back to left hand side precision
m_expo++;
}
}
m_number.insert((std::tstring::size_type)0,SIGN_STRING(sign));
return *this;
}
// Description:
// The essential += operator
// 1) Align to same exponent
// 2) Align to same precision
// 3) Add Mantissa
// 4) Add carry to exponent
// 4) Normalize
// 5) Rounding to precision
// Early out algorithm. i.e.
// - x+=0 return x
// - x+=a where x is 0 return a
//
// @todo Still missing code for x += a where add make sense. fx. if a is so small it does
// not affect the result within the given precision is should ignored. same is true
// if x is insignificant compare to a the just assign a to x
//
afp&
afp::operator+=(const afp& p_other)
{
int sign, sign1, sign2, wrap;
int expo_max, digits_max;
unsigned int precision_max;
std::tstring s, s1, s2;
if(p_other.m_number.length() == 2 && FDIGIT(p_other.m_number[1]) == 0) // Add zero
{
return *this;
}
if(m_number.length() == 2 && FDIGIT(m_number[1]) == 0) // Add a (not zero) to *this (is zero) Same as *this = a;
{
return *this = p_other;
}
// extract sign and unsigned portion of number
sign1 = p_other.sign();
s1 = p_other.m_number.substr(1); // Extract Mantissa
sign2 = CHAR_SIGN(m_number[0]);
s2 = m_number.substr( 1 ); // Extract Mantissa
expo_max = MAX(m_expo, p_other.m_expo);
precision_max = MAX(m_prec,p_other.precision());
// Check if add makes sense. Still missing
// Right shift (padding leading zeros) to the smallest number
if(p_other.m_expo != expo_max )
{
_afp_right_shift(&s1, expo_max - p_other.m_expo);
}
if(m_expo != expo_max)
{
_afp_right_shift(&s2,expo_max - m_expo);
}
// Round to same precision
if( _afp_rounding(&s1,sign1,precision_max,p_other.mode()) != 0) // If carry when rounding up then one right shift
{
_afp_right_shift( &s1, 1 );
}
if( _afp_rounding(&s2,sign2,precision_max,m_rmode) != 0) // If carry when rounding up then one right shift
{
_afp_right_shift( &s2, 1 );
}
// Alignment to same number of digits, so add can be performed as integer add
digits_max = (int) MAX( s1.length(), s2.length() );
if( s1.length() != digits_max )
{
_afp_left_shift( &s1, digits_max - (int)s1.length());
}
if( s2.length() != digits_max )
{
_afp_left_shift( &s2, digits_max - (int)s2.length() );
}
// Now s1 and s2 is aligned to the same exponent. The biggest of the two
if( sign1 == sign2 )
{
s = _afp_uadd( &s1, &s2 );
if( s.length() > s1.length() ) // One more digit
{
expo_max++;
}
sign = sign1;
}
else
{
int cmp = _afp_compare( &s1, &s2 );
if( cmp > 0 ) // Since we subtract less the wrap indicator need not to be checked
{
s = _afp_usub( &wrap, &s1, &s2 );
sign = sign1;
}
else
{
if( cmp < 0 )
{
s = _afp_usub( &wrap, &s2, &s1 );
sign = sign2;
}
else
{ // Result zero
sign = 1;
s = _T("0");
s[0] = FCHARACTER( 0 );
expo_max = 0;
}
}
}
expo_max += _afp_normalize( &s ); // Normalize the number
if( _afp_rounding( &s, sign, m_prec, m_rmode ) != 0 ) // Round back left hand side precision
{
expo_max++;
}
m_number = SIGN_STRING( sign ) + s;
m_expo = expo_max;
return *this;
}
// Description:
// The essential -= operator
// n = n - a is the same as n = n + (-a). so change sign and use the += operator instead
//
afp&
afp::operator-=(const afp& p_other)
{
afp b;
b.precision(p_other.precision());
b.mode(p_other.mode());
b = p_other;
b.change_sign();
*this += b;
return *this;
}
// Description:
// The essential *= operator
// 1) Multiply mantissa
// 2) Add exponent
// 3) Normalize
// 4) Rounding to precision
//
afp&
afp::operator*=(const afp& p_other)
{
int expo_res;
int sign, sign1, sign2;
std::tstring s, s1, s2;
// extract sign and unsigned portion of number
sign1 = p_other.sign();
s1 = p_other.m_number.substr( 1 );
sign2 = CHAR_SIGN( m_number[0] );
s2 = m_number.substr( 1 );
sign = sign1 * sign2;
s = _afp_umul_fourier( &s1, &s2 );
expo_res = m_expo + p_other.m_expo;
if( s.length() - 1 > s1.length() + s2.length() - 2 ) // A carry
{
expo_res++;
}
expo_res += _afp_normalize( &s ); // Normalize the number
if( _afp_rounding( &s, sign, m_prec, m_rmode ) != 0 ) // Round back left hand side precision
{
expo_res++;
}
if( sign == -1 && s.length() == 1 && FDIGIT( s[0] ) == 0 ) // Avoid -0 as result +0 is right
{
sign = 1; // Change sign
}
if( s.length() == 1 && FDIGIT( s[0] ) == 0 ) // Result 0 clear exponent
{
expo_res = 0;
}
m_expo = expo_res;
m_number = SIGN_STRING( sign ) + s;
return *this;
}
// Description:
// The essential /= operator
// We do a /= b as a *= (1/b)
// Bug
// 1/27/2006 Inverse was always done with the precision of a instead of the Max precision of both this & a
//
afp&
afp::operator/=(const afp& p_other)
{
if(m_number.length() == 2 && FDIGIT(m_number[1]) == 0) // If divisor is zero the result is zero
{
return *this;
}
afp c;
c.precision(p_other.precision());
if(p_other.precision() < m_prec)
{
c.precision(m_prec);
}
c = p_other;
afp inv = _afp_inverse(c);
afp test = p_other * inv;
*this *= inv;
// *this *= _afp_inverse(c);
return *this;
}
// Description:
// Binary add two afp numbers
// Implementing using the essential += operator
//
afp
operator+(const afp& p_left, const afp& p_right)
{
afp result;
if(p_left.precision() < p_right.precision())
{
result.precision(p_right.precision());
}
else
{
result.precision(p_left.precision());
}
result = p_left;
result += p_right;
return result;
}
// Description:
// Unary add. Do nothing and return a
//
afp
operator+(const afp& p_right)
{
// Otherwise do nothing
return p_right;
}
// Description:
// Binary subtract two afp numbers
// Implementing using the essential -= operator
//
afp
operator-(const afp& p_left,const afp& p_right)
{
unsigned int precision;
afp result;
precision = p_left.precision();
if( precision < p_right.precision() )
{
precision = p_right.precision();
}
result.precision(precision);
result = p_left;
result -= p_right;
return result;
}
// Description:
// Unary hypen: Just change sign
//
afp
operator-(const afp& p_right)
{
afp result;
result.precision(p_right.precision());
result = p_right;
result.change_sign();
return result;
}
// Description:
// Binary multiplying two afp numbers
// Implenting using the essential *= operator
//
afp
operator*(const afp& p_left,const afp& p_right)
{
unsigned int precision;
afp result;
precision = p_left.precision();
if( precision < p_right.precision() )
{
precision = p_right.precision();
}
result.precision(precision);
result = p_left;
result *= p_right;
return result;
}
// Description:
// Binary divide two afp numbers
// Implementing using the essential /= operator
//
afp
operator/(const afp& p_left,const afp& p_right)
{
unsigned int precision;
afp result;
precision = p_left.precision();
if(precision < p_right.precision())
{
precision = p_right.precision();
}
result.precision(precision + 1);
result = p_left;
result /= p_right;
return result;
}
// Description:
// If both operands has the same mantissa length and same exponent
// and if the mantissa is identical then it's the same.
// However a special test of +0 == -0 is done
// Precision and rounding mode does not affect the comparison
//
bool
operator==(const afp& p_left,const afp& p_right)
{
if(const_cast<afp&>(p_left).ref_mantissa()->length() != const_cast<afp&>(p_right).ref_mantissa()->length() ||
p_left.exponent() != p_right.exponent() ) // Different therefore false
{
return false;
}
else
{
if((const_cast<afp&>(p_left).ref_mantissa())->compare(*const_cast<afp&>(p_right).ref_mantissa()) == 0) // Same return true
{
return true;
}
else
{
if( const_cast<afp&>(p_left) .ref_mantissa()->length() == 2 &&
(*const_cast<afp&>(p_left) .ref_mantissa())[1] == FDIGIT(0) &&
(*const_cast<afp&>(p_right).ref_mantissa())[1] == FDIGIT(0) )
{
// This conditions is only true if +0 is compare with -0 and therefore true
return true;
}
}
}
return false;
}
// Description:
// 1) Test for both operand is zero and return false if condition is meet
// 2) If signs differs then return the boolean result based on that
// 3) Now if same sign and one operand is zero then return the boolean result
// 4) If same sign and not zero check the exponent
// 5) If same sign and same exponent then check the mantissa for boolean result
// Precision and rounding mode does not affect the comparison
//
bool
operator<(const afp& p_left,const afp& p_right)
{
int sign1, sign2, cmp;
bool zero1, zero2;
sign1 = p_left.sign();
sign2 = p_right.sign();
zero1 = const_cast<afp&>(p_left ).ref_mantissa()->length() == 2 && FDIGIT( ( *const_cast<afp&>(p_left ).ref_mantissa())[1] ) == 0 ? true : false;
zero2 = const_cast<afp&>(p_right).ref_mantissa()->length() == 2 && FDIGIT( ( *const_cast<afp&>(p_right).ref_mantissa())[1] ) == 0 ? true : false;
if( zero1 == true && zero2 == true ) // Both zero
{
return false;
}
// Different signs
if( sign1 < sign2 )
{
return true;
}
if( sign1 > sign2 )
{
return false;
}
// Now a & b has the same sign
if( zero1 == true ) // If a is zero and a & b has the same sign and b is not zero then a < b
{
return true;
}
if( zero2 == true ) // If b is zero and a & b has the same sign and a is not zero then a > b
{
return false;
}
// Same sign and not zero . Check exponent
if(p_left.exponent() < p_right.exponent() )
{
return sign1 > 0 ? true : false;
}
if( p_left.exponent() > p_right.exponent() )
{
return sign1 > 0 ? false: true;
}
// Same sign & same exponent. Check mantissa
cmp = (const_cast<afp&>(p_left).ref_mantissa())->compare( *const_cast<afp&>(p_right).ref_mantissa());
if( cmp < 0 && sign1 == 1 )
{
return true;
}
else
{
if( cmp > 0 && sign1 == -1 )
{
return true;
}
}
return false;
}
// Description:
// implemented negating the == comparison
//
bool
operator!=(const afp& p_left,const afp& p_right)
{
return p_left == p_right ? false : true;
}
// Description:
// Implemented using the equality a>b => b<a
//
bool
operator>(const afp& p_left,const afp& p_right)
{
return p_right < p_left ? true : false;
}
// Description:
// Implemented using the equality a<=b => not b<a
//
bool
operator<=(const afp& p_left,const afp& p_right)
{
return p_right < p_left ? false : true;
}
// Description:
// Implemented using the equality a>=b => not a<b
//
bool
operator>=(const afp& p_left,const afp& p_right)
{
return p_left < p_right ? false: true;
}
// The stream << operator
std::ostream&
operator<<( std::ostream& strm, const afp& d )
{
return strm << _afp_ftoa( const_cast<afp &>(d) ).c_str();
}
std::tistream&
operator>>( std::tistream& strm, afp& d )
{
TCHAR ch; std::tstring s; int cnt, exp_cnt;
strm >> std::ws >> ch;
if( ch == '+' || ch == '-' )
{
s += ch; strm >> ch;
}
else
{
s += '+'; // Parse sign
}
for( cnt = 0; ch >= '0' && ch <= '9'; cnt++, strm >> ch )
{
s += ch; // Parse integer part
}
if( ch == '.' ) // Parse fraction
{
for( s += '.', strm >> ch; ch >= '0' && ch <= '9'; cnt++, strm >> ch )
{
s += ch; // Parse fraction part
}
}
if( ch == 'e' || ch == 'E' )
{
s += 'e'; strm >> ch; if( ch == '+' || ch == '-' )
{
s += ch; strm >> ch;
}
else
{
s += '+'; // Parse Expo sign
}
for( exp_cnt =0; ch >= '0' && ch <= '9'; exp_cnt++, strm >> ch ) s += ch; // Parse expo number
}
std::cin.putback((char) ch ); // ch contains the first character not part of the number, so put it back
if( !strm.fail() && ( cnt > 0 || exp_cnt > 0 ) ) // Valid number
{
d = afp( const_cast<PTCHAR>( s.c_str() ), PRECISION, ROUND_NEAR );
}
return strm;
}
//////////////////////////////////////////////////////////////////////////////////////
///
/// END AFP OPERATORS
///
//////////////////////////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
//
// PUBLIC AFP SUPPORT FUNCTIONS
//
//////////////////////////////////////////////////////////////////////////
// Description:
// Reciprocal/Inverse of V
// Using a Newton iterations Un = U(2-UV)
// Always return the result with 2 digits higher precision that argument
// _afp_inverse() return a interim result for a basic operation like /
//
afp
_afp_inverse(const afp& p_number)
{
unsigned int precision,i;
int expo;
double fv, fu;
afp r, u, v, c2;
std::tstring::reverse_iterator rpos;
std::tstring::iterator pos;
std::tstring *p;
precision = p_number.precision();
v.precision(precision + 2);
v = p_number;
p= v.ref_mantissa();
if( p->length() == 2 && FDIGIT( (*p)[1] ) == 0 )
{
throw afp::divide_by_zero();
return p_number;
}
expo = v.exponent();
v.exponent( 0 );
r.precision( precision + 2 ); // Do iteration using 2 digits higher precision
u.precision( precision + 2 );
c2.precision(precision + 2 );
c2 = afp( 2, precision + 2 );
// Get a initial guess using ordinary floating point
rpos = v.ref_mantissa()->rbegin();
fv = FDIGIT( (_TUCHAR)*rpos );
for( rpos++; rpos+1 != v.ref_mantissa()->rend(); rpos++ )
{
fv *= (double)1/(double)F_RADIX;
fv += FDIGIT( *rpos );
}
if( v.sign() < 0 )
{
fv = -fv;
}
fu = 1 / fv;
u = afp( fu );
// Now iterate using Newton Un=U(2-UV)
for(;;)
{
r = u * v; // UV
r = c2-r; // 2-UV
u *= r; // Un=U(2-UV)
for( pos = r.ref_mantissa()->begin(), pos+=2, i = 0; pos != r.ref_mantissa()->end(); i++, pos++ )
{
if( FDIGIT( *pos ) )
{
break;
}
}
if( pos == r.ref_mantissa()->end() || i >= precision )
{
break;
}
}
u.exponent( u.exponent() - expo );
u.mode(p_number.mode());
return u;
}
// Description:
// sqrt(V)
// Equivalent with the same standard C function call
// Seperate exponent. e.g. sqrt(V*10^x)=10^x/2*sqrt(V)
// Un=0.5U(3-VU^2)
// Then Un == 1/Sqrt(V). and sqrt(V) = VUn
//
afp
sqrt(const afp p_number)
{
unsigned int precision,i;
int expo, expo_sq;
double fv, fu;
afp r, u, v;
afp c3(3);
afp c05(0.5);
std::tstring::reverse_iterator rpos;
std::tstring::iterator pos;
std::tstring *p;
precision = p_number.precision();
v.precision( precision + 2 );
v = p_number;
if( v.sign() < 0 )
{
throw afp::domain_error();
return p_number;
}
p= v.ref_mantissa();
if( p->length() == 2 && FDIGIT( (*p)[1] ) == 0 ) // Sqrt(0) is zero
{
return afp( 0 );
}
expo = v.exponent();
expo_sq = expo / 2;
v.exponent( expo - 2 * expo_sq );
r.precision( precision + 2 ); // Do iteration using 2 digits higher precision
u.precision( precision + 2 );
// Get a initial guess using ordinary floating point
rpos = v.ref_mantissa()->rbegin();
fv = FDIGIT( (_TUCHAR)*rpos );
for( rpos++; rpos+1 != v.ref_mantissa()->rend(); rpos++ )
{
fv *= (double)1/(double)F_RADIX;
fv += FDIGIT( *rpos );
}
if( expo - 2 * expo_sq > 0 )
{
fv *= (double)F_RADIX;
}
else
{
if( expo - 2 * expo_sq < 0 )
{
fv /= (double)F_RADIX;
}
}
fu = 1 / sqrt( fv );
u = afp( fu );
// Now iterate using Newton Un=0.5U(3-VU^2)
for(;;)
{
r = v * u * u; // VU^2
r = c3-r; // 3-VU^2
r *= c05; // (3-VU^2)/2
u *= r; // U=U(3-VU^2)/2
for( pos = r.ref_mantissa()->begin(), pos+=2, i = 0; pos != r.ref_mantissa()->end(); i++, pos++ )
{
if( FDIGIT( *pos ) )
{
break;