In most assembly language texts, which bother to cover floating point arithmetic, this section would normally describe how to design your own floating point routines for addition, subtraction, multiplication, and division. This text will not do that for several reasons. First, to design a good floating point library requires a solid background in numerical analysis; a prerequisite this text does not assume of its readers. Second, the UCR Standard Library already provides a reasonable set of floating point routines in source code form; why waste space in this text when the sources are readily available elsewhere? Third, floating point units are quickly becoming standard equipment on all modern CPUs or motherboards; it makes no more sense to describe how to manually perform a floating point computation than it does to describe how to manually perform an integer computation. Therefore, this section will describe how to use the UCR Standard Library routines if you do not have an FPU available; a later section will describe the use of the floating point unit.

The UCR Standard Library provides a large number of routines to support floating point computation and I/O. This library uses the same memory format for 32, 64, and 80 bit floating point numbers as the 80x87 FPUs. The UCR Standard Library's floating point routines do not exactly follow the IEEE requirements with respect to error conditions and other degenerate cases, and it may produce slightly different results than an 80x87 FPU, but the results will be very close[5]. Since the UCR Standard Library uses the same memory format for 32, 64, and 80 bit numbers as the 80x87 FPUs, you can freely mix computations involving floating point between the FPU and the Standard Library routines.

The UCR Standard Library provides numerous routines to manipulate floating point numbes. The following sections describe each of these routines, by category.

14.3.1 Load and Store Routines

Since 80x86 CPUs without an FPU do not provide any 80-bit registers, the UCR Standard Library must use memory-based variables to hold floating point values during computation. The UCR Standard Library routines use two pseudo registers, an accumlator register and an operand register, when performing floating point operations. For example, the floating point addition routine adds the value in the floating point operand register to the floating point accumulator register, leaving the result in the accumulator. The load and store routines allow you to load floating point values into the floating point accumulator and operand registers as well as store the value of the floating point accumulator back to memory. The routines in this category include accop, xaccop, lsfpa, ssfpa, ldfpa, sdfpa, lefpa, sefpa, lefpal, lsfpo, ldfpo, lefpo, and lefpol.

The accop routine copies the value in the floating point accumulator to the floating point operand register. This routine is useful when you want to use the result of one computation as the second operand of a second computation.

The xaccop routine exchanges the values in the floating point accumuator and operand registers. Note that many floating point computations destory the value in the floating point operand register, so you cannot blindly assume that the routines preserve the operand register. Therefore, calling this routine only makes sense after performing some computation which you know does not affect the floating point operand register.

Lsfpa, ldfpa, and lefpa load the floating point accumulator with a single, double, or extended precision floating point value, respectively. The UCR Standard Library uses its own internal format for computations. These routines convert the specified values to the internal format during the load. On entry to each of these routines, es:di must contain the address of the variable you want to load into the floating point accumulator. The following code demonstrates how to call these routines:

rVar            real4   1.0
drVar           real8   2.0
xrVar           real10  3.0
                 .
                 .
                 .
                lesi    rVar
                lsfpa
                 .
                 .
                 .
                lesi    drVar
                ldfpa
                 .
                 .
                 .
                lesi    xrVar
                lefpa

The lsfpo, ldfpo, and lefpo routines are similar to the lsfpa, ldfpa, and lefpa routines except, of course, they load the floating point operand register rather than the floating point accumulator with the value at address es:di.

Lefpal and lefpol load the floating point accumulator or operand register with a literal 80 bit floating point constant appearing in the code stream. To use these two routines, simply follow the call with a real10 directive and the appropriate constant, e.g.,

                lefpal
                real10  1.0
                lefpol
                real10  2.0e5

The ssfpa, sdfpa, and sefpa routines store the value in the floating point accumulator into the memory based floating point variable whose address appears in es:di. There are no corresponding ssfpo, sdfpo, or sefpo routines because a result you would want to store should never appear in the floating point operand register. If you happen to get a value in the floating point operand that you want to store into memory, simply use the xaccop routine to swap the accumulator and operand registers, then use the store accumulator routines to save the result. The following code demonstrates the use of these routines:

rVar            real4   1.0
drVar           real8   2.0
xrVar           real10  3.0
                 .
                 .
                 .
                lesi    rVar
                ssfpa
                 .
                 .
                 .
                lesi    drVar
                sdfpa
                 .
                 .
                 .
                lesi    xrVar
                sefpa

14.3.2 Integer/Floating Point Conversion

The UCR Standard Library includes several routines to convert between binary integers and floating point values. These routines are itof, utof, ltof, ultof, ftoi, ftou, ftol, and ftoul. The first four routines convert signed and unsigned integers to floating point format, the last four routines truncate floating point values and convert them to an integer value.

Itof converts the signed 16-bit value in ax to a floating point value and leaves the result in the floating point accumulator. This routine does not affect the floating point operand register. Utof converts the unsigned integer in ax in a similar fashion. Ltof and ultof convert the 32 bit signed (ltof) or unsigned (ultof) integer in dx:ax to a floating point value, leaving the value in the floating point accumulator. These routines always succeed.

Ftoi converts the value in the floating point accumulator to a signed integer value, leaving the result in ax. Conversion is by truncation; this routine keeps the integer portion and throws away the fractional part. If an overflow occurs because the resulting integer portion does not fit into 16 bits, ftoi returns the carry flag set. If the conversion occurs without error, ftoi return the carry flag clear. Ftou works in a similar fashion, except it converts the floating point value to an unsigned integer in ax; it returns the carry set if the floating point value was negative.

Ftol and ftoul converts the value in the floating point accumulator to a 32 bit integer leaving the result in dx:ax. Ftol works on signed values, ftoul works with unsigned values. As with ftoi and ftou, these routines return the carry flag set if a conversion error occurs.

14.3.3 Floating Point Arithmetic

Floating point arithmetic is handled by the fpadd, fp sub, fpcmp, fpmul, and fpdiv routines. Fpadd adds the value in the floating point accumulator to the floating point accumulator. Fpsub subtracts the value in the floating point operand from the floating point accumulator. Fpmul multiplies the value in the floating accumulator by the floating point operand. Fpdiv divides the value in the floating point accumulator by the value in the floating point operand register. Fpcmp compares the value in the floating point accumulator against the floating point operand.

The UCR Standard Library arithmetic routines do very little error checking. For example, if arithmetic overflow occurs during addition, subtraction, multiplication, or division, the Standard Library simply sets the result to the largest legal value and returns. This is one of the major deviations from the IEEE floating point standard. Likewise, when underflow occurs the routines simply set the result to zero and return. If you divide any value by zero, the Standard Library routines simply set the result to the largest possible value and return. You may need to modify the standard library routines if you need to check for overflow, underflow, or division by zero in your programs.

The floating point comparison routine (fpcmp) compares the floating point accumulator against the floating point operand and returns -1, 0, or 1 in the ax register if the accumulator is less than, equal, or greater than the floating point operand. It also compares ax with zero immediately before returning so it sets the flags so you can use the jg, jge, jl, jle, je, and jne instructions immediately after calling fpcmp. Unlike fpadd, fpsub, fpmul, and fpdiv, fpcmp does not destroy the value in the floating point accumulator or the floating point operand register. Keep in mind the problems associated with comparing floating point numbers!

14.3.4 Float/Text Conversion and Printff

The UCR Standard Library provides three routines, ftoa, etoa, and atof, that let you convert floating point numbers to ASCII strings and vice versa; it also provides a special version of printf, printff, that includes the ability to print floating point values as well as other data types.

Ftoa converts a floating point number to an ASCII string which is a decimal representation of that floating point number. On entry, the floating point accumulator contains the number you want to convert to a string. The es:di register pair points at a buffer in memory where ftoa will store the string. The al register contains the field width (number of print positions). The ah register contains the number of positions to display to the right of the decimal point. If ftoa cannot display the number using the print format specified by al and ah, it will create a string of "#" characters, ah characters long. Es:di must point at a byte array containing at least al+1 characters and al should contain at least five. The field width and decimal length values in the al and ah registers are similar to the values appearing after floating point numbers in the Pascal write statement, e.g.,

		write(floatVal:al:ah);

Etoa outputs the floating point number in exponential form. As with ftoa, es:di points at the buffer where etoa will store the result. The al register must contain at least eight and is the field width for the number. If al contains less than eight, etoa will output a string of "#" characters. The string that es:di points at must contain at least al+1 characters. This conversion routine is similar to Pascal's write procedure when writing real values with a single field width specification:

write(realvar:al);

The Standard Library printff routine provides all the facilities of the standard printf routine plus the ability to handle floating point output. The printff routine includes several new format specifications to print floating point numbers in decimal form or using scientific notation. The specifications are

• %x.yF Prints a 32 bit floating point number in decimal form.

• %x.yGF Prints a 64 bit floating point number in decimal form.

• %x.yLF Prints an 80 bit floating point number in decimal form.

• %zE Prints a 32 bit floating point number using scientific notation.

• %zGE Prints a 64 bit floating point number using scientific notation.

• %zLE Prints an 80 bit floating point value using scientific notation.

In the format strings above, x and z are integer constants that denote the field width of the number to print. The y item is also an integer constant that specifies the number of positions to print after the decimal point. The x.y values are comparable to the values passed to ftoa in al and ah. The z value is comparable to the value etoa expects in the al register.

Other than the addition of these six new formats, the printff routine is identical to the printf routine. If you use the printff routine in your assembly language programs, you should not use the printf routine as well. Printff duplicates all the facilities of printf and using both would only waste memory.

[5] Note, by the way, that different floating point chips, especially across different CPU lines, but even within the Intel family, produce slightly different results. So the fact that the UCR Standard Library does not produce the exact same results as a particular FPU is not that important.

Chapter Fourteen: Floating Point Arithmetics (Part 2)
28 SEP 1996