June 96 - 64-Bit Integer Math on 680x0 Machines
DALE SEMCHISHEN
When
an application has to perform integer arithmetic with numbers larger than 32
bits on both the PowerPC and 680x0 platforms, you could use the floating-point
types of the SANE and PowerPC Numerics libraries. But if all you really need is
a larger integer, a better choice is to use the existing 64-bit math routines
available on the PowerPC platform and write an equivalent library for the 680x0
Macintosh. This article presents just such a library.
Developers of PowerPC applications that need 64-bit math can simply call the
various "wide" Toolbox routines. These routines perform addition, subtraction,
multiplication, division, square root, and a few other operations. On the
680x0-based Macintosh, some of these same routines are available in QuickDraw
GX. But if you can't assume your customers have QuickDraw GX installed, you
need a library that supports 64-bit math.
The Wide library presented in this article works on both platforms and has
exactly the same interface and types as the wide routines in the Toolbox on
PowerPC machines. The library also provides some new routines such as 32-bit to
64-bit add and subtract and a 64-bit-to-string conversion function. The library
is included on this issue's CD, along with its source code.
All the routines use the 64-bit data type defined in the header file Types.h,
which is the standard type used for signed 64-bit integers on both the PowerPC
and 680x0 Macintosh:
struct wide {
Sint32 hi; /* upper 32 bits (signed) */
Uint32 lo; /* lower 32 bits (unsigned) */
};
typedef struct wide wide, *WidePtr;
Before plunging into the Wide library, let's see what 64-bit math routines I'll
be talking about. First, I'll introduce those that are available on PowerPC
machines, then those you'll find on a 680x0 Macintosh with QuickDraw GX, and
finally the routines in the Wide library.
In the header file FixMath.h, the routines listed in Table 1 are defined for
64-bit math on the PowerPC platform.
On 680x0 machines that have QuickDraw GX installed, all the wide routines for
the PowerPC platform listed in Table 1 are available, with the exception of
WideBitShift. The QuickDraw GX header file GXTypes.h defines the wide routine
types and function prototypes in exactly the same way that the header file
FixMath.h does for PowerPC machines.
In addition, QuickDraw GX on 680x0 machines has a routine that the PowerPC
platform doesn't have: WideScale. This function returns the bit number of the
highest-order nonzero bit in a 64-bit number. The Wide library implements this
function on the PowerPC platform.
The Wide 64-bit integer math library on this issue's CD provides all the wide
routines that are available on PowerPC machines and on 680x0 machines with
QuickDraw GX, plus a few extras. The extra routines, which are available on
both the PowerPC and 680x0 platforms, are listed in Table 2.
WideAssign32,
WideAdd32, WideSubtract32. These routines are self-explanatory.
WideToDecStr.
This routine converts a signed 64-bit integer to the SANE string type decimal,
which is also defined by the PowerPC Numerics library. This string structure is
a good intermediate format for final conversion to a string format of your
choosing.
Since WideToDecStr calls the SANE library to generate the string, SANE must be
linked with your 680x0 application. The SANE library is included with all the
major development systems.
To convert the string returned by WideToDecStr to a Pascal string, call the
SANE routine dec2str.
If you want to generate a localized number, take a look at the article
"International Number Formatting" in develop Issue 16. You could call the
LocalizeNumberString function from that article after converting the output of
WideToDecStr to a Pascal string, or you could modify LocalizeNumberString to
accept the output of WideToDecStr.*
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WideInit.
The library is self-initializing; the first time you call any wide routine,
WideInit is also called. If the execution speed of your first runtime call to a
wide routine is important, you have the option of calling WideInit during your
application's startup to avoid that overhead.
The purpose of WideInit is to determine what processor is being used, or
emulated; it calls Gestalt to make this determination. If your Macintosh has a
68020-68040 CPU (68020, 68030, or 68040), the library will use the 64-bit
multiply and divide instructions available on that processor; otherwise, the
library will have to call software subroutines for those operations. On 68000
machines, such as the Macintosh Plus and SE, the processor's multiply
instruction is limited to 32 bits and the library has no choice but to use the
slower algorithmic approach for multiplication and division.
The library can be compiled on the 680x0 and PowerPC platforms using either the
Metrowerks CodeWarrior or Symantec C development system. The library tests
which development system is compiling it and, if it's not CodeWarrior or
Symantec, the preprocessor displays an error message saying the library needs
to be ported to your environment. This is necessary because there's some inline
assembly language in the source file, as discussed later in this section, and
different C compilers handle assembly language differently.
While the interface routines to our 64-bit library are the same on the PowerPC
and 680x0 machines, when you compile the library a different subset of routines
is linked in, depending on your environment:
- If you build the library for a 680x0 machine without QuickDraw GX headers,
all the Wide library routines are defined.
- If you build the library for a 680x0 machine and include the QuickDraw
GX header file GXTypes.h or GXMath.h before the Wide library's Wide.h header
file, the extra routines and the WideBitShift routine are defined. The other
wide routines are already available via the QuickDraw GX traps.
- When you compile for the PowerPC platform, only the five extra routines
(WideAssign32, WideAdd32, WideSubtract32, WideToDecStr, and WideInit) are
defined in the library. All the other wide routines already exist in the
PowerPC Toolbox. Additionally, if GXTypes.h or GXMath.h isn't included,
WideScale is defined.
Table 3 summarizes where the wide routines can be
found on the different platforms.
Note that the Wide library decides at compile time which routines to use. When
QuickDraw GX header files are not included, the Wide library routines are
called. If your application needs to make a runtime decision about whether to
use QuickDraw GX, you'll need to make some changes to the library. One solution
is to rename the Wide library routines and remove the conditional compilation
tests for QuickDraw GX from the source. Then at run time you can decide which
version to call -- the QuickDraw GX routines if they're available, or the
internal Wide library routinesif not.
The Wide library was compiled with version 2.1 of Apple's universal headers.
The latest headers are available on this issue's CD. You should make sure you
have a recent version of these headers, because the library uses the constant
GENERATING68K. If the header file ConditionalMacros.h doesn't contain this
constant, your version of the universal headers is too old.
Some of the routines in the library are written in assembly language to take
advantage of the 64-bit multiply and divide instructions on 68020-68040
machines, because on these machines the C language will use only 32-bit
multiply and divide instructions. On PowerPC machines, the Wide library doesn't
need assembly language because the 64-bit multiply and divide routines are
provided by the Toolbox.
The library's source file Wide.c contains both C and assembly language. It has
been successfully compiled by Symantec C 7.0.4 and CodeWarrior 7. If you want
to compile the library on any other development system, you may have to do a
little work porting it. Most of the changes will be confined to the conditional
compilation statements at the beginning of Wide.c where the differences in SANE
types and inline assembly language are handled.
Now let's look at a couple of the more interesting routines in the Wide library
to see how they work. See the source code on the CD for full implementations of
all the routines.
WideMultiply (Listing 1) performs a 32-by-32-bit multiply and produces a 64-bit
result. The first and second parameters are the two signed 32-bit integers to
be multiplied together. The return value is a pointer to the 64-bit result
that's also returned via the third parameter.
Listing 1. The multiply routine
wide *WideMultiply (
long multiplicand, /* in: first value to multiply */
long multiplier, /* in: second value to multiply */
wide *target_ptr) /* out: 64 bits to be assigned */
{
/* Initialize Wide library if not already done. */
if (!gWide_Initialized) WideInit();
/* If the 64-bit multiply instruction is available... */
if (gWide_64instr) {
/* Execute the assembly-language instruction MULS.L */
Wide_MulS64(multiplicand, multiplier, target_ptr);
}
else {
/* Call the Toolbox to perform the multiply. */
LongMul(multiplicand, multiplier, (Int64Bit *) target_ptr);
}
return (target_ptr);
}
WideMultiply
first tests whether the library has been initialized yet; if not, it calls
WideInit. Next the routine tests whether the 64-bit multiply instruction is
available on the current CPU by examining the global variable gWide_64instr
(which was set by the initialization routine WideInit). If the instruction is
available, WideMultiply calls the assembly-language function Wide_MulS64 to
take advantage of it (as described later); otherwise, WideMultiply calls the
Toolbox routine LongMul to perform the multiplication, as would be the case on
68000 machines.
The WideSquareRoot function (Listing 2) takes a 64-bit unsigned number as input
and returns a 32-bit unsigned result. All possible results can be expressed in
32 bits, so overflow isn't possible.
Listing 2. The square root routine
unsigned long WideSquareRoot (
const wide *source_ptr) /* in: value to take the square root of */
{
wide work_integer;
Extended_80 extended_80_number;
/* Initialize Wide library if not already done. */
if (!gWide_Initialized) WideInit();
/* Convert "wide" number to "extended" format. */
Wide_ToExtended(&extended_80_number, source_ptr);
/* If compiling with CodeWarrior, the parameter to sqrt is a
pointer instead of a value, as defined in PowerPC Numerics. */
#ifdef __MWERKS__
Sqrt(&extended_80_number);
#else
extended_80_number = sqrt(extended_80_number);
#endif
/* Convert "extended" format to "wide" number. */
Wide_FromExtended(&work_integer, &extended_80_number);
/* OK to ignore work_integer.hi as it's always 0. */
return (work_integer.lo);
}
For
this routine I decided to let the SANE library do the work of generating the
square root. The routine converts the 64-bit input number to an 80-bit
floating-point number and then calls the SANE library function
sqrt to
calculate the square root. Finally, WideSquareRoot converts the resulting
80-bit floating-point number back to a 64-bit integer and returns the low-order
half of the result.
When a 64-bit integer is converted to an 80-bit floating-point number, no loss
in precision occurs. An 80-bit floating-point number is made up of three parts
-- the sign (1 bit), the exponent (15 bits), and the fractional part (64 bits).
As you can see, a 64-bit integer exactly fits in the fractional part.
Two differences between the CodeWarrior and Symantec development systems that
show up in the Wide library's WideSquareRoot function are the 80-bit
floating-point types and the parameters of the SANE library's square root
function. Under CodeWarrior, the Wide library internal type Extended_80 is
defined as the type extended80, and Sqrt returns the result to the same
location as the input number. Under Symantec C, Extended_80 is defined as the
type extended, and sqrt returns the result as a function return value.
The Wide library uses internal assembly-language routines to execute 64-bit
multiply and divide instructions on machines that support those instructions.
In case you're interested, here are the details.
Symantec and CodeWarrior handle the asm keyword differently, so I used some
preprocessor commands (#defines) to handle the differences between the two
development systems. Near the beginning of the Wide.c source file there are
four #defines that differ depending on which development system you're using,
as shown in Table 4.
Wide_MulS64 (Listing 3) is an internal assembly-language routine that
WideMultiply calls to execute the 64-bit multiply instruction on the
68020-68040 CPUs. It starts with ASM_FUNC_HEAD, as mentioned in Table 4. The
three definitions at the start of the function (MULTIPLICAND, MULTIPLIER, and
OUT_PTR) are the byte offsets to the parameters. Although in Symantec C it's
possible to refer to function parameters by name via A6, this isn't possible in
CodeWarrior. I had to give up accessing the parameters by name and use #defines
instead.
Listing 3. 64-bit multiply instruction
ASM_FUNC_HEAD static void Wide_MulS64 (
long multiplicand, /* in: first value to multiply */
long multiplier, /* in: second value to multiply */
wide *out_ptr) /* out: 64 bits to be assigned */
{
#define MULTIPLICAND 8
#define MULTIPLIER 12
#define OUT_PTR 16
ASM_BEGIN
MOVE.L MULTIPLICAND(A6),D0 //
DC.W 0x4C2E,0x0C01,0x000C // MULS.L multiplier(A6),D1-D0
MOVE.L OUT_PTR(A6),A0 //
MOVE.L D0,WIDE_LO(A0) //
MOVE.L D1,WIDE_HI(A0) //
ASM_END
ASM_FUNC_TAIL
}
To
execute the 64-bit multiply instruction I had to define it with a DC.W
directive that generates the desired object code. This was necessary because
the Symantec C inline assembler supports only the 32-bit multiply instruction
and won't recognize the 64-bit assembly opcode.
If the 64-bit divide instruction isn't available, the library calls the
internal assembly-language routine Wide_DivideU (Listing 4) to perform the
division using an algorithm. The algorithm is basically a binary version of the
paper and pencil method of doing long division that all of us learned in
school. It's a loop that executes once for each bit in the size of the divisor,
which is 32 in our case. The Wide_DivideU subroutine actually handles only
unsigned division, but the library function that calls it will take care of
converting the input parameters to positive values and, if required, converting
the result to a negative value.
Listing 4. 64-bit unsigned division algorithm
ASM_FUNC_HEAD static void Wide_DivideU (
wide *dividend_ptr, /* in/out: 64 bits to be divided */
long divisor, /* in: value to divide by */
long *remainder_ptr) /* out: the remainder of the division */
{
#define DIVIDEND_PTR 8
#define DIVISOR 12
#define REMAINDER_PTR 16
ASM_BEGIN
MOVEM.L D2-D7,-(SP) // save work registers
CLR.L D0 //
CLR.L D1 // D0-D1 is the quotient accumulator
MOVE.L DIVIDEND_PTR(A6),A0 //
MOVE.L WIDE_HI(A0),D2 //
MOVE.L WIDE_LO(A0),D3 // D2-D3 = remainder accumulator
CLR.L D4 //
MOVE.L D2,D5 // D5 = copy of dividend.hi
MOVE.L DIVISOR(A6),D6 // D6 = copy of divisor
MOVEQ.L #31,D7 // FOR number of bits in divisor
@divloop:
LSL.L #1,D0 // shift quotient.hi accum left once
LSL.L #1,D1 // shift quotient.lo accum left once
LSL.L #1,D4 //
LSL.L #1,D3 //
ROXL.L #1,D2 // shift remainder accum left once
SUB.L D6,D2 // remainder -= divisor
BCS @div50 // If CS, remainder is negative
BSET #0,D1 // quotient.lo |= 1
BRA.S @div77 //
@div50:
ADD.L D6,D2 // remainder += divisor
@div77:
BTST D7,D5 //
BEQ @div90 // If EQ, bit not set in dividend.hi
BSET #0,D4 //
@div90:
CMP.L D6,D4 //
BCS @div99 // If CS, divisor < D4
SUB.L D6,D4 // D4 -= divisor
BSET #0,D0 // quotient.hi |= 1
@div99:
DBF D7,@divloop // loop until D7 == -1
MOVE.L DIVIDEND_PTR(A6),A0 // output the remainder
MOVE.L D0,WIDE_HI(A0) //
MOVE.L D1,WIDE_LO(A0) //
MOVE.L REMAINDER_PTR(A6),A0 // output the remainder
MOVE.L D2,(A0) //
MOVEM.L (SP)+,D2-D7 // restore work registers
ASM_END
ASM_FUNC_TAIL
}
The
top of the assembly-language loop starts at the @divloop label. For each loop,
the algorithm shifts the quotient and the remainder left one bit position
before trying to subtract the divisor from the remainder. If the subtraction
can be done, the least-significant bit in quotient.lo is set; otherwise, the
subtraction is undone by the add instruction near the @div50 label. Then, if
the divisor is greater than the loop bits that are accumulating in register D4,
the least-significant bit in quotient.hi is set.
Notice that the first assembly-language statement in Wide_DivideU is a MOVEM.L
instruction that saves on the stack all the registers that the division loop
uses; the last instruction is a MOVEM.L instruction that restores these
registers. Fortunately, this subroutine can place all its working variables in
registers and avoid the stack for its loop, thus improving performance.
There you have it. Now 64-bit integer math can be handled with the same API on
both the 680x0 and PowerPC platforms. Having the same function-level interface
on these two very different processors makes life a lot easier for application
programmers. Don't you wish all libraries had the same interface regardless of
the CPU or system software version?
DALE SEMCHISHEN (Dale_Semchishen@mindlink.net) lives in Vancouver, British
Columbia, with his wife Josephine. He works for Glenayre Technologies as a
paging software developer (they make the control systems that send messages to
your belt beeper). Recently, he had to accept the fact that the world is
changing when his retired father started talking about his Internet
provider.*
Thanks to our technical reviewers Dave Evans, Quinn "The Eskimo!", and Dave
Radcliffe. Special thanks to Dave Johnson for software testing.*
