/***************************************************************************/
/* */
/* ftcalc.c */
/* */
/* Arithmetic computations (body). */
/* */
/* Copyright 1996-2000 by */
/* David Turner, Robert Wilhelm, and Werner Lemberg. */
/* */
/* This file is part of the FreeType project, and may only be used, */
/* modified, and distributed under the terms of the FreeType project */
/* license, LICENSE.TXT. By continuing to use, modify, or distribute */
/* this file you indicate that you have read the license and */
/* understand and accept it fully. */
/* */
/***************************************************************************/
/*************************************************************************/
/* */
/* Support for 1-complement arithmetic has been totally dropped in this */
/* release. You can still write your own code if you need it. */
/* */
/*************************************************************************/
/*************************************************************************/
/* */
/* Implementing basic computation routines. */
/* */
/* FT_MulDiv(), FT_MulFix(), and FT_DivFix() are declared in freetype.h. */
/* */
/*************************************************************************/
#include <freetype/internal/ftcalc.h>
#include <freetype/internal/ftdebug.h>
#include <freetype/internal/ftobjs.h> /* for ABS() */
/*************************************************************************/
/* */
/* The macro FT_COMPONENT is used in trace mode. It is an implicit */
/* parameter of the FT_TRACE() and FT_ERROR() macros, used to print/log */
/* messages during execution. */
/* */
#undef FT_COMPONENT
#define FT_COMPONENT trace_calc
#ifdef FT_CONFIG_OPTION_OLD_CALCS
static const FT_Long ft_square_roots[63] =
{
1L, 1L, 2L, 3L, 4L, 5L, 8L, 11L,
16L, 22L, 32L, 45L, 64L, 90L, 128L, 181L,
256L, 362L, 512L, 724L, 1024L, 1448L, 2048L, 2896L,
4096L, 5892L, 8192L, 11585L, 16384L, 23170L, 32768L, 46340L,
65536L, 92681L, 131072L, 185363L, 262144L, 370727L,
524288L, 741455L, 1048576L, 1482910L, 2097152L, 2965820L,
4194304L, 5931641L, 8388608L, 11863283L, 16777216L, 23726566L,
33554432L, 47453132L, 67108864L, 94906265L,
134217728L, 189812531L, 268435456L, 379625062L,
536870912L, 759250125L, 1073741824L, 1518500250L,
2147483647L
};
#else
/*************************************************************************/
/* */
/* <Function> */
/* FT_Sqrt32 */
/* */
/* <Description> */
/* Computes the square root of an Int32 integer (which will be */
/* handled as an unsigned long value). */
/* */
/* <Input> */
/* x :: The value to compute the root for. */
/* */
/* <Return> */
/* The result of `sqrt(x)'. */
/* */
FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt32( FT_Int32 x )
{
FT_ULong val, root, newroot, mask;
root = 0;
mask = 0x40000000L;
val = (FT_ULong)x;
do
{
newroot = root + mask;
if ( newroot <= val )
{
val -= newroot;
root = newroot + mask;
}
root >>= 1;
mask >>= 2;
} while ( mask != 0 );
return root;
}
#endif /* FT_CONFIG_OPTION_OLD_CALCS */
#ifdef FT_LONG64
/*************************************************************************/
/* */
/* <Function> */
/* FT_MulDiv */
/* */
/* <Description> */
/* A very simple function used to perform the computation `(a*b)/c' */
/* with maximal accuracy (it uses a 64-bit intermediate integer */
/* whenever necessary). */
/* */
/* This function isn't necessarily as fast as some processor specific */
/* operations, but is at least completely portable. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. */
/* c :: The divisor. */
/* */
/* <Return> */
/* The result of `(a*b)/c'. This function never traps when trying to */
/* divide by zero; it simply returns `MaxInt' or `MinInt' depending */
/* on the signs of `a' and `b'. */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_MulDiv( FT_Long a,
FT_Long b,
FT_Long c )
{
FT_Int s;
s = 1;
if ( a < 0 ) { a = -a; s = -s; }
if ( b < 0 ) { b = -b; s = -s; }
if ( c < 0 ) { c = -c; s = -s; }
return s * ( c > 0 ? ( (FT_Int64)a * b + ( c >> 1 ) ) / c
: 0x7FFFFFFFL );
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_MulFix */
/* */
/* <Description> */
/* A very simple function used to perform the computation */
/* `(a*b)/0x10000' with maximal accuracy. Most of the time this is */
/* used to multiply a given value by a 16.16 fixed float factor. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. Use a 16.16 factor here whenever */
/* possible (see note below). */
/* */
/* <Return> */
/* The result of `(a*b)/0x10000'. */
/* */
/* <Note> */
/* This function has been optimized for the case where the absolute */
/* value of `a' is less than 2048, and `b' is a 16.16 scaling factor. */
/* As this happens mainly when scaling from notional units to */
/* fractional pixels in FreeType, it resulted in noticeable speed */
/* improvements between versions 2.x and 1.x. */
/* */
/* As a conclusion, always try to place a 16.16 factor as the */
/* _second_ argument of this function; this can make a great */
/* difference. */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_MulFix( FT_Long a,
FT_Long b )
{
FT_Int s;
s = 1;
if ( a < 0 ) { a = -a; s = -s; }
if ( b < 0 ) { b = -b; s = -s; }
return s * (FT_Long)( ( (FT_Int64)a * b + 0x8000 ) >> 16 );
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_DivFix */
/* */
/* <Description> */
/* A very simple function used to perform the computation */
/* `(a*0x10000)/b' with maximal accuracy. Most of the time, this is */
/* used to divide a given value by a 16.16 fixed float factor. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. Use a 16.16 factor here whenever */
/* possible (see note below). */
/* */
/* <Return> */
/* The result of `(a*0x10000)/b'. */
/* */
/* <Note> */
/* The optimization for FT_DivFix() is simple: If (a << 16) fits in */
/* 32 bits, then the division is computed directly. Otherwise, we */
/* use a specialized version of the old FT_MulDiv64(). */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_DivFix( FT_Long a,
FT_Long b )
{
FT_Int32 s;
FT_UInt32 q;
s = a; a = ABS(a);
s ^= b; b = ABS(b);
if ( b == 0 )
/* check for division by 0 */
q = 0x7FFFFFFFL;
else
/* compute result directly */
q = ( (FT_Int64)a << 16 ) / b;
return (FT_Int32)( s < 0 ? -q : q );
}
#ifdef FT_CONFIG_OPTION_OLD_CALCS
/* a helper function for FT_Sqrt64() */
static
int ft_order64( FT_Int64 z )
{
int j = 0;
while ( z )
{
z = (unsigned FT_INT64)z >> 1;
j++;
}
return j - 1;
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_Sqrt64 */
/* */
/* <Description> */
/* Computes the square root of a 64-bit value. That sounds stupid, */
/* but it is needed to obtain maximal accuracy in the TrueType */
/* bytecode interpreter. */
/* */
/* <Input> */
/* l :: A 64-bit integer. */
/* */
/* <Return> */
/* The 32-bit square-root. */
/* */
FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt64( FT_Int64 l )
{
FT_Int64 r, s;
if ( l <= 0 ) return 0;
if ( l == 1 ) return 1;
r = ft_square_roots[ft_order64( l )];
do
{
s = r;
r = ( r + l / r ) >> 1;
} while ( r > s || r * r > l );
return r;
}
FT_EXPORT_DEF( FT_Int32 ) FT_SqrtFixed( FT_Int32 x )
{
FT_Int64 z;
z = (FT_Int64)(x) << 16;
return FT_Sqrt64( z );
}
#endif /* FT_CONFIG_OPTION_OLD_CALCS */
#else /* FT_LONG64 */
/*************************************************************************/
/* */
/* <Function> */
/* FT_MulDiv */
/* */
/* <Description> */
/* A very simple function used to perform the computation `(a*b)/c' */
/* with maximal accuracy (it uses a 64-bit intermediate integer */
/* whenever necessary). */
/* */
/* This function isn't necessarily as fast as some processor specific */
/* operations, but is at least completely portable. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. */
/* c :: The divisor. */
/* */
/* <Return> */
/* The result of `(a*b)/c'. This function never traps when trying to */
/* divide by zero; it simply returns `MaxInt' or `MinInt' depending */
/* on the signs of `a' and `b'. */
/* */
/* <Note> */
/* The FT_MulDiv() function has been optimized thanks to ideas from */
/* Graham Asher. The trick is to optimize computation if everything */
/* fits within 32 bits (a rather common case). */
/* */
/* We compute `a*b+c/2', then divide it by `c' (positive values). */
/* */
/* 46340 is FLOOR(SQRT(2^31-1)). */
/* */
/* if ( a <= 46340 && b <= 46340 ) then ( a*b <= 0x7FFEA810 ) */
/* */
/* 0x7FFFFFFF - 0x7FFEA810 = 0x157F0 */
/* */
/* if ( c < 0x157F0*2 ) then ( a*b+c/2 <= 0x7FFFFFFF ) */
/* */
/* and 2*0x157F0 = 176096. */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_MulDiv( FT_Long a,
FT_Long b,
FT_Long c )
{
long s;
if ( a == 0 || b == c )
return a;
s = a; a = ABS( a );
s ^= b; b = ABS( b );
s ^= c; c = ABS( c );
if ( a <= 46340 && b <= 46340 && c <= 176095L && c > 0 )
{
a = ( a * b + ( c >> 1 ) ) / c;
}
else if ( c > 0 )
{
FT_Int64 temp, temp2;
FT_MulTo64( a, b, &temp );
temp2.hi = (FT_Int32)( c >> 31 );
temp2.lo = (FT_UInt32)( c / 2 );
FT_Add64( &temp, &temp2, &temp );
a = FT_Div64by32( &temp, c );
}
else
a = 0x7FFFFFFFL;
return ( s < 0 ? -a : a );
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_MulFix */
/* */
/* <Description> */
/* A very simple function used to perform the computation */
/* `(a*b)/0x10000' with maximal accuracy. Most of the time, this is */
/* used to multiply a given value by a 16.16 fixed float factor. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. Use a 16.16 factor here whenever */
/* possible (see note below). */
/* */
/* <Return> */
/* The result of `(a*b)/0x10000'. */
/* */
/* <Note> */
/* The optimization for FT_MulFix() is different. We could simply be */
/* happy by applying the same principles as with FT_MulDiv(), because */
/* */
/* c = 0x10000 < 176096 */
/* */
/* However, in most cases, we have a `b' with a value around 0x10000 */
/* which is greater than 46340. */
/* */
/* According to some testing, most cases have `a' < 2048, so a good */
/* idea is to use bounds like 2048 and 1048576 (=floor((2^31-1)/2048) */
/* for `a' and `b', respectively. */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_MulFix( FT_Long a,
FT_Long b )
{
FT_Long s;
FT_ULong ua, ub;
if ( a == 0 || b == 0x10000L )
return a;
s = a; a = ABS(a);
s ^= b; b = ABS(b);
ua = (FT_ULong)a;
ub = (FT_ULong)b;
if ( ua <= 2048 && ub <= 1048576L )
{
ua = ( ua * ub + 0x8000 ) >> 16;
}
else
{
FT_ULong al = ua & 0xFFFF;
ua = ( ua >> 16 ) * ub +
al * ( ub >> 16 ) +
( al * ( ub & 0xFFFF ) >> 16 );
}
return ( s < 0 ? -(FT_Long)ua : ua );
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_DivFix */
/* */
/* <Description> */
/* A very simple function used to perform the computation */
/* `(a*0x10000)/b' with maximal accuracy. Most of the time, this is */
/* used to divide a given value by a 16.16 fixed float factor. */
/* */
/* <Input> */
/* a :: The first multiplier. */
/* b :: The second multiplier. Use a 16.16 factor here whenever */
/* possible (see note below). */
/* */
/* <Return> */
/* The result of `(a*0x10000)/b'. */
/* */
/* <Note> */
/* The optimization for FT_DivFix() is simple: If (a << 16) fits into */
/* 32 bits, then the division is computed directly. Otherwise, we */
/* use a specialized version of the old FT_MulDiv64(). */
/* */
FT_EXPORT_FUNC( FT_Long ) FT_DivFix( FT_Long a,
FT_Long b )
{
FT_Int32 s;
FT_UInt32 q;
s = a; a = ABS(a);
s ^= b; b = ABS(b);
if ( b == 0 )
{
/* check for division by 0 */
q = 0x7FFFFFFFL;
}
else if ( ( a >> 16 ) == 0 )
{
/* compute result directly */
q = (FT_UInt32)( a << 16 ) / (FT_UInt32)b;
}
else
{
/* we need more bits; we have to do it by hand */
FT_Int64 temp, temp2;
temp.hi = (FT_Int32) (a >> 16);
temp.lo = (FT_UInt32)(a << 16);
temp2.hi = (FT_Int32)( b >> 31 );
temp2.lo = (FT_UInt32)( b / 2 );
FT_Add64( &temp, &temp2, &temp );
q = FT_Div64by32( &temp, b );
}
return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_Add64 */
/* */
/* <Description> */
/* Add two Int64 values. */
/* */
/* <Input> */
/* x :: A pointer to the first value to be added. */
/* y :: A pointer to the second value to be added. */
/* */
/* <Output> */
/* z :: A pointer to the result of `x + y'. */
/* */
/* <Note> */
/* Will be wrapped by the ADD_64() macro. */
/* */
FT_EXPORT_FUNC( void ) FT_Add64( FT_Int64* x,
FT_Int64* y,
FT_Int64* z )
{
register FT_UInt32 lo, hi;
lo = x->lo + y->lo;
hi = x->hi + y->hi + ( lo < x->lo );
z->lo = lo;
z->hi = hi;
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_MulTo64 */
/* */
/* <Description> */
/* Multiplies two Int32 integers. Returns an Int64 integer. */
/* */
/* <Input> */
/* x :: The first multiplier. */
/* y :: The second multiplier. */
/* */
/* <Output> */
/* z :: A pointer to the result of `x * y'. */
/* */
/* <Note> */
/* Will be wrapped by the MUL_64() macro. */
/* */
FT_EXPORT_FUNC( void ) FT_MulTo64( FT_Int32 x,
FT_Int32 y,
FT_Int64* z )
{
FT_Int32 s;
s = x; x = ABS( x );
s ^= y; y = ABS( y );
{
FT_UInt32 lo1, hi1, lo2, hi2, lo, hi, i1, i2;
lo1 = x & 0x0000FFFF; hi1 = x >> 16;
lo2 = y & 0x0000FFFF; hi2 = y >> 16;
lo = lo1 * lo2;
i1 = lo1 * hi2;
i2 = lo2 * hi1;
hi = hi1 * hi2;
/* Check carry overflow of i1 + i2 */
i1 += i2;
if ( i1 < i2 )
hi += 1L << 16;
hi += i1 >> 16;
i1 = i1 << 16;
/* Check carry overflow of i1 + lo */
lo += i1;
hi += ( lo < i1 );
z->lo = lo;
z->hi = hi;
}
if ( s < 0 )
{
z->lo = (FT_UInt32)-(FT_Int32)z->lo;
z->hi = ~z->hi + !( z->lo );
}
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_Div64by32 */
/* */
/* <Description> */
/* Divides an Int64 value by an Int32 value. Returns an Int32 */
/* integer. */
/* */
/* <Input> */
/* x :: A pointer to the dividend. */
/* y :: The divisor. */
/* */
/* <Return> */
/* The result of `x / y'. */
/* */
/* <Note> */
/* Will be wrapped by the DIV_64() macro. */
/* */
FT_EXPORT_FUNC( FT_Int32 ) FT_Div64by32( FT_Int64* x,
FT_Int32 y )
{
FT_Int32 s;
FT_UInt32 q, r, i, lo;
s = x->hi;
if ( s < 0 )
{
x->lo = (FT_UInt32)-(FT_Int32)x->lo;
x->hi = ~x->hi + !( x->lo );
}
s ^= y; y = ABS( y );
/* Shortcut */
if ( x->hi == 0 )
{
if ( y > 0 )
q = x->lo / y;
else
q = 0x7FFFFFFFL;
return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
}
r = x->hi;
lo = x->lo;
if ( r >= (FT_UInt32)y ) /* we know y is to be treated as unsigned here */
return ( s < 0 ? 0x80000001UL : 0x7FFFFFFFUL );
/* Return Max/Min Int32 if division overflow. */
/* This includes division by zero! */
q = 0;
for ( i = 0; i < 32; i++ )
{
r <<= 1;
q <<= 1;
r |= lo >> 31;
if ( r >= (FT_UInt32)y )
{
r -= y;
q |= 1;
}
lo <<= 1;
}
return ( s < 0 ? -(FT_Int32)q : (FT_Int32)q );
}
#ifdef FT_CONFIG_OPTION_OLD_CALCS
/* two helper functions for FT_Sqrt64() */
static
void FT_Sub64( FT_Int64* x,
FT_Int64* y,
FT_Int64* z )
{
register FT_UInt32 lo, hi;
lo = x->lo - y->lo;
hi = x->hi - y->hi - ( (FT_Int32)lo < 0 );
z->lo = lo;
z->hi = hi;
}
static
int ft_order64( FT_Int64* z )
{
FT_UInt32 i;
int j;
i = z->lo;
j = 0;
if ( z->hi )
{
i = z->hi;
j = 32;
}
while ( i > 0 )
{
i >>= 1;
j++;
}
return j - 1;
}
/*************************************************************************/
/* */
/* <Function> */
/* FT_Sqrt64 */
/* */
/* <Description> */
/* Computes the square root of a 64-bits value. That sounds stupid, */
/* but it is needed to obtain maximal accuracy in the TrueType */
/* bytecode interpreter. */
/* */
/* <Input> */
/* z :: A pointer to a 64-bit integer. */
/* */
/* <Return> */
/* The 32-bit square-root. */
/* */
FT_EXPORT_FUNC( FT_Int32 ) FT_Sqrt64( FT_Int64* l )
{
FT_Int64 l2;
FT_Int32 r, s;
if ( (FT_Int32)l->hi < 0 ||
( l->hi == 0 && l->lo == 0 ) )
return 0;
s = ft_order64( l );
if ( s == 0 )
return 1;
r = ft_square_roots[s];
do
{
s = r;
r = ( r + FT_Div64by32( l, r ) ) >> 1;
FT_MulTo64( r, r, &l2 );
FT_Sub64 ( l, &l2, &l2 );
} while ( r > s || (FT_Int32)l2.hi < 0 );
return r;
}
FT_EXPORT_DEF( FT_Int32 ) FT_SqrtFixed( FT_Int32 x )
{
FT_Int64 z;
z.hi = (FT_UInt32)((FT_Int32)(x) >> 16);
z.lo = (FT_UInt32)( x << 16 );
return FT_Sqrt64( &z );
}
#endif /* FT_CONFIG_OPTION_OLD_CALCS */
#endif /* FT_LONG64 */
/* END */
