Model interval for exponentiation with a negative exponent AI-00476/04 1
89-06-16 ra WA
| !standard 04.05.07 (09) 89-06-16 AI-00476/04
!standard 04.05.06 (06)
!class ramification 87-05-15
| !status WG9-approved 89-06-16
!status ARG-approved 88-05-10 (reviewed)
!status ARG-approved (15-0-1) 87-09-15 (pending editorial review)
!status panel-approved (6-0-0) 87-09-15
!status work-item 87-05-15
!status received 86-10-13
!references 83-00828
!topic Model interval for exponentiation with a negative exponent
!summary 87-05-15
For exponentiation of a floating point value by a negative exponent, the
model interval of the dividend in the final division consists solely of the
model number 1.0.
!question 87-05-15
4.5.6(6) says:
Exponentiation with a positive exponent is equivalent to repeated
multiplication of the left operand by itself, as indicated by the
exponent and from left to right. For an operand of a floating
point type, the exponent can be negative, in which case the value
is the reciprocal of the value with the positive exponent.
Exponentiation by a zero exponent delivers the value one.
Exponentiation of a value of a floating point type is approximate
(see 4.5.7).
4.5.7(9) says:
For the result of exponentiation, the model interval defining the
bounds on the result is obtained by applying the above rules to
the sequence of multiplications defined by the exponent, and to
the final division in the case of a negative exponent.
For exponentiation of a floating point value by a negative exponent, what is
the model interval of the dividend in the final division mentioned in
4.5.7(9)?
!response 87-05-15
For exponentiation of a floating point value X by a negative exponent N, the
model interval of X ** N is determined by applying the rules of 4.5.7 to the
expression:
1.0 / (((X * X) * X) * ... * X)
where the number of multiplications is (ABS N) - 1.
Model interval for exponentiation with a negative exponent AI-00476/04 2
89-06-16 ra WA
The dividend (1.0) in the final division is a universal_real literal. Since
1.0 is a model number for all floating point types, by the rules of 4.5.7,
the model interval of the dividend consists solely of the model number 1.0.