# Appendix A – The X3.274 subset

The full specification in the body of this document defines a decimal floating-point arithmetic which gives exact results and preserves exponents where possible. If insufficient precision is available for this, then numbers are handled according to the rules of IEEE 854. The use of IEEE 854 rules implies that special values (infinities and NaNs) are allowed, as subnormal values and the value –0.

For some applications and programming languages (especially those intended for use by people who are not mathematically sophisticated), it may be appropriate to provide an arithmetic where infinite, NaN, or subnormal results are always treated as errors, –0 results are hidden, and other (largely cosmetic) changes are provided to aid acceptance of results.

The arithmetic defined in ANSI X3.274 is such an arithmetic; this appendix describes the differences between this and the full specification. Implementations which support this subset explicitly might provide the subset behavior under the control of a parameter in the context[1]  or might provide a different interface (additional or parameterized methods, for example).

#### Simplified number set

In the subset arithmetic, a reduced set of number values is supported and (where appropriate) numbers with positive exponents have their exponent reduced to zero. Specifically:
• In the to-number conversion, if the coefficient for a finite number has the value zero, then the sign and the exponent are both set to 0.
• If the coefficient in a result has the value zero, then the sign is set to 0 and (unless the operation is rescale) the exponent is set to 0.[2]
• In the to-number conversion, strings which represent special values are not permitted. (That is, only finite numbers are accepted.)
• Subnormal numbers are not permitted. If the result from a conversion or operation would be subnormal then an Underflow error results (see below).
• After any operation and the rounding of its result (unless the operation is rescale), a result with a positive exponent is converted to an integer provided that the resulting coefficient would have no more than precision digits. In other words, in this case a positive exponent is reduced to 0 by multiplying the coefficient by 10exponent (which has the effect of suffixing exponent zeros).[3]

#### Operation differences

In the subset arithmetic, operands are rounded before use if necessary (as in Numerical Turing[4]  and Rexx), the Lost digits condition is added to the context, the results of some operations are trimmed, the rounding rule after a subtraction is less conservative, and raising 0 to the power 0 is not treated as an error. Specifically:
• If the number of decimal digits in the coefficient of an operand to an operation is greater than the current precision in the context then the operand is rounded to precision significant digits using the rounding algorithm described by the context before being used in the computation. In other words, an automatic ‘convert to shorter’ is applied before the operation.
• The Lost digits condition is added to the abstract context; it should be set to 0 in default contexts.
This condition is raised when non-zero digits are discarded before an operation. This can occur when an operand which has more leading significant digits in its coefficient than the precision setting is rounded to precision digits before use
Note that the lost digits test does not treat trailing decimal zeros in the coefficient as significant. For example, if precision had the value 5, then the operands
```  [0,12345,-5]
[0,12345,-2]
[0,12345,0]
[1,12345,0]
[0,123450000,-4]
[0,1234500000,0]
```
would not cause an exception (whereas [0,123451,-1] or [0,1234500001,0] would).
• After a divide or power operation is complete and the result has been rounded, any insignificant trailing zeros are removed. That is, if the exponent is not zero and the coefficient is a multiple of a positive power of ten then the coefficient is divided by that power of ten and the exponent increased accordingly. If the exponent was negative it will not be increased above zero.
• After an addition operation, the result is rounded to precision digits if necessary, taking into account any extra (carry) digit on the left after an addition, but otherwise counting from the position corresponding to the most significant digit of the operands being added or subtracted (rather than the most significant digit of the result).
• If both operands to a power operation are zero then the result is 1 (instead of being an error).
• If the right-hand operand to a power operation is negative, the left-hand operand is used as-is and the final result is inverted. The integer part of the right-hand operand must fit in precision digits.
• The integer part of the right-hand operand to a rescale operation must fit in precision digits.

#### Exceptional condition and rounding mode rules

In the subset arithmetic, exceptional conditions other than the informational conditions (Lost digits, Inexact, Rounded, and Subnormal) must be treated as errors, and results after these errors are undefined. Special values and subnormal numbers, therefore, are not part of the arithmetic.

In the subset, only the Lost digits trap enabler is required. Inexact, Rounded, and Subnormal trap enablers are optional, and the others are (in effect) always set. Similarly, the status bits in the context are optional.

Only the round-half-up rounding mode is required.

Footnotes:
 [1] The decNumber package, for example, provides the subset behavior if the extended bit is set to 0. [2] This rule, together with the to-number definition, ensures that numbers with values such as -0 or 0.0000 will not result from general operations in the subset arithmetic. This allows a concrete representation for the subset to comprise simply two integers in twos complement form. [3] The rule preserves integers as specified by ANSI X3.274, and in particular ensures that the results of the divide and divide-integer operations are identical when the result is an exact integer. [4] See: T. E. Hull, A. Abrham, M. S. Cohen, A. F. X. Curley, C. B. Hall, D. A. Penny, and J. T. M. Sawchuk, Numerical Turing, SIGNUM Newsletter, vol. 20 #3, pp26-34, ACM, May 1985.

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