15. The Floating-Point Coprocessor
Part of
the Hawk Manual
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15.1. Floating-Point Registers
15.1.1. Coprocessor Status Register — 0
15.1.2. Floating-Point-Low Register — 1
15.1.3. Floating-Point Accumulators — 2 and 3
15.2. Floating-Point Operations
15.2.1. Integer to Floating Conversion — 4 and 5
15.2.2. Floating Square Root — 6 and 7
15.2.3. Floating Add — 8 and 9
15.2.3. Floating Subtract — 10 and 11
15.2.3. Floating Multiply — 12 and 13
15.2.3. Floating Divide — 14 and 15
15.3. Examples
The floating-point coprocessor has 3 interface registers in addition to the COSTAT register that it shares with other coprocessors. These are referenced using the COGET and COSET instructions. These three 32-bit registers allow direct access to two 64-bit floating-point accumulators. The following constants should be defined by the assembly header file float.h to support access to these registers:
COSTAT = 0 ; this definition should be redundant FPLOW = 1 ; the low half of long floating operands FPA0 = 2 ; floating-point accumulator 0 FPA1 = 3 ; floating-point accumulator 1
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| unused | sl | u | 0 0 1 | unused | en | u | |||||||||||||||||||||||||
The floating-point coprocessor is coprocessor number 1. Thus it is enabled when COSTAT:1 (en) is set, and it is selected when COSTAT:10:8 (select) is set to 001.
COSTAT:12 (sl), part of the coprocessor operation field (coop), controls whether the coprocessor uses short (32-bit) or long (64-bit) floating point representations. The other bits in coop are unused, although they may be used in enhanced floating-point coprocessors.
The following constants should be defined by the assembly header file float.h to support use of the floating-point coprocessor; these can be added together (or ored together) to create a value to be placed in the status register:
FPENAB = #0002 ; enable the floating-point unit FPSEL = #0100 ; select the floating-point unit FPLONG = #1000 ; operate in long format FPSHORT = #0000 ; operate in short format (default)
Strictly speaking, there is no reason to explicitly specify short format, since it is the default, but an explicit statement of short mode makes programs easier to read. for program documentation since
To enable and select the floating-point coprocessor for long floating operations while turning off any other coprocessors in the system, use the following instruction sequence:
LIL R1,FPENAB+FPSEL+FPLONG
COSET R1,COSTAT
Most applications will not use multiple coprocessors concurrently. For those that keep several coprocessors enabled, to select the already enabled floating-point unit and and set it in long mode, use the following instruction sequence:
COGET R1,COSTAT
TRUNC R1,8
ADDI R1,R1,FPSEL+FPLONG
COSET R1,COSTAT
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| mantissa:31:0 | |||||||||||||||||||||||||||||||
The floating-point low (FPLOW) register holds the least significant 32-bits of a long floating-point operand. These are the low 32 bits of the mantissa. The contents of FPLOW are not defined when operating in 32-bit mode.
In 64-bit mode, FPLOW must be set before using COSET to set the high half of a floating-point accumulator, and it is automatically set as a side effect of using COGET to read the low half of a floating-point accumulator. The COGET instruction applied to FPLOW has an undefined effect on the condition codes.
The floating-point coprocessor contains two 64-bit floating-point accumulators, FPA0 and FPA1. The apparent format of these registers appears different in short floating-point mode, where they appear to be only 32 bits. In fact, the hardware always retains an 11 bit exponent and a 52-bit mantissa, at minimum, with the extra bits suppressed in short mode. IEEE standard floating-point format is supported. This means that the 53-bit mantissa is usually 54 bits, with a hidden one-bit (bit 52) except in the case of un-normalized numbers, where this hidden bit is zero, signified by a zero exponent.
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| s | exp:7:0 | mantissa:51:29 | |||||||||||||||||||||||||||||
In short mode, the floating-point operands are 32 bits, and the floating low register is not used. IEEE floating-point format is used. In short format, the 11-bit exponent is shortened to just 8 bits, and only the most significant 24 bits of the mantissa (including the hidden bit defined by the IEEE format) are used.
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| s | exp:10:0 | mantissa:51:32 | |||||||||||||||||||||||||||||||||||
In long mode, the floating-point operand registers are 64 bits, and the floating low bits are used to access the least significant 32 bits of the operand. IEEE floating-point format is used. In long format, the full 11-bit exponent is used, and the most significant 21 bits of the mantissa (including the hidden bit defined by the IEEE format) are given here, while the least significant 32 bits are found in FPLOW. To load FPA1 from with a 64-bit floating point number stored in R3 and R4, least significant half first, use:
COSET R3,FPLOW
COSET R4,FPA1 ; uses FPLOW
The corresponding code to get a 64-bit number from FPA1 into R3 and R4 is:
COGET R4,FPA1 ; sets FPLOW
COGET R3,FPLOW
Conversion between short and long format may be done by changing the value of the coprocessor operation field in COSTAT between the time a value is put into a floating-point operand register and the time it is retrieved.
| COGET | N = r[dst]:31 | — result is negative | ||
| Z = (r[dst] = 0) | — result is zero | |||
| V = 0 | ||||
| C = (exp = 111111111112) — not a number | ||||
In both long and short format, COGET sets the condition codes after reading FPA0 or FPA1 to report on the entire floating-point number, including that part (if any) loaded into FPLOW. N and Z indicate a negative or zero value, while C is used to report values that are NANs (not a number) or infinity. The overflow conditiion code, V, is always reset. As a result, the signed comparison instructions such as BGT and BLE will correctly report the relationship of the operand to zero.
Floating-point operations are initiated by putting values in coprocessor register numbers 4 through 15. That is, the coprocessor register number is used partially to select one of the floating-point accumulators and partly as a coprocessor operation code. Since 4 register numbers have already been used, as defined above, This allows for a total of 12 operations, or 6 operations that apply to each floating-point accumulator.
All of these operations are initiated by COSET instructions. COGET instructions using register numbers above 3 are undefined. Because the accumulators themselves have register numbers 2 and 3, the definitions provided in float.h are off by two from the actual values -- this allows these values to be added to the accumulator numbers to form the operation:
FPINT = 2 ; register numbers 4, 5 FPSQRT = 4 ; register numbers 6, 7 FPADD = 6 ; register numbers 8, 9 FPSUB = 8 ; register numbers 10, 11 FPMUL = 10 ; register numbers 12, 13 FPDIV = 12 ; register numbers 14, 15
Putting a value in floating-point unit register numbers 4 or 5 converts the integer operand to a normalized floating-point value in either FPA0 or FPA1. For example, if registers R3 and R4 contain a 64-bit signed integer, least significant word first, and if the floating-point unit is in long mode, the integer may be converted to floating-point in FPA1 as follows:
COSET R3,FPLOW
COSET R4,FPINT+FPA1
If register R3 contains a 32-bit unsigned integer, and if the floating-point unit is in long mode, the integer may be converted to floating-point in FPA1 as follows:
COSET R3,FPLOW
COSET R0,FPINT+FPA1
If register R3 contains a 32-bit signed integer, and if the floating-point unit is in short mode, the integer may be converted to floating-point in FPA1 as follows:
COSET R3,FPINT+FPA1
Floating to integer conversion, truncated or rounded, must be done by
software.
Putting a value in floating-point register numbers 6 or 7 puts the square root of the floating-point operand into either FPA0 or FPA1. For example, if R3 contains a short floating-point number, and if the floating-point unit is in short mode, the square root may be taken in FPA1 as follows:
COSET R3,FPSQRT+FPA1
Putting a floating-point value in floating-point registers 8 and 9 adds the operand to either FPA0 or FPA1. For example, if R3 and R4 contain a long floating-point number (least significant word first), and if the floating-point unit is in long mode, the number may be added to FPA1 as follows:
COSET R3,FPLOW
COSET R4,FPADD+FPA1
Putting a floating-point value in floating-point registers 10 and 11 subtracts the operand from either FPA0 or FPA1. For example, if R3 contains a short floating-point number, and if the floating-point unit is in short mode, the number may be subtracted from FPA1 as follows:
COSET R3,FPSUB+FPA1
Note that, unlike integer arithmetic, the subtract operation does not set the condition codes. To compare the two short floating point numbers in R3 and R4, use this sequence:
COSET R3,FPA1
COSET R4,FPSUB+FPA1
COGET R0,FPA1 ; sets the condition codes
Putting a floating-point value in floating-point registers 12 and 13 multiplies either FPA0 or FPA1 by the operand. For example, if R3 and R4 contain a long floating-point number (least significant word first), and if the floating-point unit is in long mode, FPA1 may be multiplied by the number as follows:
COSET R3,FPLOW
COSET R4,FPMUL+FPA1
Putting a floating-point value in floating-point registers 14 and 15 divides either FPA0 or FPA1 by the operand. For example, if R3 contains a short floating-point number, and if the floating-point unit is in short mode, from FPA1 may be divided by R3 as follows:
COSET R3,FPDIV+FPA1
Consider computing y = ax2 + bx + c, where y, a, b, c and x are short floating-point numbers in R3 to R7, in that order. Also, assume that coprocessors are turned off when not in use. The following code works:
LIL R3,FPENAB+FPSEL+FPSHORT
COSET R3,COSTAT ; turn on floating coprocessor
COSET R4,FPA1 ; FPA1 = a
COSET R7,FPMUL+FPA1 ; FPA1 = ax
COSET R5,FPADD+FPA1 ; FPA1 = ax + b
COSET R7,FPMUL+FPA1 ; FPA1 = (ax + b)x
COSET R6,FPADD+FPA1 ; FPA1 = (ax + b)x + c
COGET R3,FPA1
COSET R0,COSTAT ; turn off floating coprocessor
Saving and restoring the state of the coprocessor in an interrupt service routine requires care. If the coprocessor is not enabled, only the status needs saving. If enabled, it must be selected before accessing the registers to save them. Saving the registers in long mode is always safe. Note that FPLOW must always be saved because it need not represent the low half of either accumulator at the time of interrupt.
Subroutines that use the floating-point coprocessor can be written so that they restore the coprocessor status word to its former state on return, and so that they do not turn off any other coprocessors that may be in use while they turn on and select the floating-point coprocessor. The following code illustrates this, using a variable indexed off of R2 to hold the saved floating-point status, and using R3 and R4 as temporaries:
COGET R3,COSTAT
STORE R3,R2,SVCOSTAT ; save old COSTAT
TRUNC R3,8 ; clear coop and select fields
LIL R4,FPENAB + FPSEL ...
OR R3,R4 ; set enable, select and mode bits
COSET R3,COSTAT ; update COSTAT
... code using floating point ...
LOAD R3,R2,SVCOSTAT
COSET R3,COSTAT ; restore COSTAT
Many interrupt and trap service routines make no use of the floating-point coprocessor and return to the same code that was interrupted or that caused the trap. For such trap service routines, there is no need to take any action with regard to the floating-point coprocessor. When, however, a trap or interrupt service routine needs to do floating-point computation or performs a context switch, the entire state of the floating-point coprocessor must be saved. Here is code to save the entire coprocessor state into registers R8 through R13:
COGET R8,COSTAT ; COSTAT into R8
BITTST R8,FPENBIT
BBR NOFPSV ; if floating-point coprocessor enabled
EXTB R9,R8,R0 ; move just the enable bits
ADDI R9,R9,FPSEL+FPLONG
COSET R9,COSTAT ; select floting-point long mode
COGET R9,FPLOW ; FPLOW into R9
COGET R11,FPA0
COGET R10,FPLOW ; FPA0 into R10 (low) and R11 (high)
COGET R13,FPA1
COGET R12,FPLOW ; FPA1 into R12 (low) and R13 (high)
NOFPSV: ; endif
The registers cannot be restored in the same order they were saved because of side effects of COSET and because the value needed in COSTAT during saving is not the same as the saved value of COSTAT.
Note that, in addition to defining values for fields of COSTAT, the file float.h should define the following:
F 3.1415 ; like W, but with a floating value
F 3E2 ; equivalent to 300.0
F +1.0e-2 ; equivalent to 0.01
LIF -123.45E+7 ; like LIW, but with a floating value
Decimal floating point values have an optional sign on both the mantissa and the optional exponent, which comes after the letter E or e. The fractional part of the mantissa is also optional.