8-30 Vol. 1

PROGRAMMING WITH THE X87 FPU

the exception handler has the option either of re-executing the store instruction after proper adjustment of the 
operand or of rounding the significand on the stack to the destination's precision as the standard requires. The 
exception handler should ultimately store a value into the destination location in memory if the program is to 
continue.

•

Destination is the register stack — The significand of the result is rounded according to current settings of 
the precision and rounding control bits in the x87 FPU control word and the exponent of the result is adjusted 
by dividing it by 2

24576

. (For instructions not affected by the precision field, the significand is rounded to 

double-extended precision.) The resulting value is stored in the destination operand. Condition code bit C1 in 
the x87 FPU status word (called in this situation the “round-up bit”) is set if the significand was rounded upward 
and cleared if the result was rounded toward 0. After the result is stored, the OE flag is set and a software 
exception handler is invoked. The scaling bias value 24,576 is equal to 3 ∗ 2

13

. Biasing the exponent by 24,576 

normally translates the number as nearly as possible to the middle of the double extended-precision floating-
point exponent range so that, if desired, it can be used in subsequent scaled operations with less risk of causing 
further exceptions.
When using the FSCALE instruction, massive overflow can occur, where the result is too large to be repre-
sented, even with a bias-adjusted exponent. Here, if overflow occurs again, after the result has been biased, a 
properly signed ∞ is stored in the destination operand.

8.5.5 

Numeric Underflow Exception (#U)

The x87 FPU detects a potential floating-point numeric underflow condition whenever the result of an arithmetic 
instruction is non-zero and tiny; that is, the magnitude of the rounded result with unbounded exponent is non-zero 
and less than the smallest possible normalized, finite value that will fit into the floating-point format of the destina-
tion operand. (See Section 4.9.1.5, “Numeric Underflow Exception (#U),” for additional information about the 
numeric underflow exception.)
Like numeric overflow, numeric underflow can occur on arithmetic operations where the result is stored in an x87 
FPU data register. It can also occur on store floating-point operations (with the FST and FSTP instructions), where 
a within-range value in a data register is stored in memory in the smaller single-precision or double-precision 
floating-point formats. A numeric underflow exception cannot occur when storing values in an integer or BCD 
integer format, because a value with magnitude less than 1 is always rounded to an integral value of 0 or 1, 
depending on the rounding mode in effect.
The flag (UE) for the numeric-underflow exception is bit 4 of the x87 FPU status word, and the mask bit (UM) is bit 
4 of the x87 FPU control word.
When a numeric-underflow condition occurs and the exception is masked, the x87 FPU performs the operation 
described in Section 4.9.1.5, “Numeric Underflow Exception (#U).”
When the exception is not masked, the action of the x87 FPU depends on whether the instruction is supposed to 
store the result in a memory location or on the x87 FPU resister stack.

•

Destination is a memory location — (Can occur only with a store instruction.) The UE flag is set and a 
software exception handler is invoked (see Section 8.7, “Handling x87 FPU Exceptions in Software”). The top-
of-stack pointer (TOP) and source and destination operands remain unchanged, and no result is stored in 
memory. 
Because the data in the stack is in double extended-precision format, the exception handler has the option 
either of re-exchanges the store instruction after proper adjustment of the operand or of rounding the 
significand on the stack to the destination's precision as the standard requires. The exception handler should 
ultimately store a value into the destination location in memory if the program is to continue.

•

Destination is the register stack — The significand of the result is rounded according to current settings of 
the precision and rounding control bits in the x87 FPU control word and the exponent of the result is adjusted 
by multiplying it by 2

24576

. (For instructions not affected by the precision field, the significand is rounded to 

double extended precision.) The resulting value is stored in the destination operand. Condition code bit C1 in 
the x87 FPU status register (acting here as a “round-up bit”) is set if the significand was rounded upward and 
cleared if the result was rounded toward 0. After the result is stored, the UE flag is set and a software exception 
handler is invoked. The scaling bias value 24,576 is the same as is used for the overflow exception and has the 
same effect, which is to translate the result as nearly as possible to the middle of the double extended-precision 
floating-point exponent range.