#     ## Issue COMPLEX-RATIONAL-RESULT Writeup

```Issue:         COMPLEX-RATIONAL-RESULT

References:    CLtL p.203

Category:      CLARIFICATION

Edit history:  Version 1, 20-Mar-89, by Moon
Version 2, 08-Apr-89, by Steele
Version 3, 28-Apr-89, by Steele

Problem description:

Referring to irrational and transcendental functions, CLtL says:

When the arguments to a function in this section are all rational and
the true mathematical result is also (mathematically) rational, then
unless otherwise noted an implementation is free to return either an
accurate result of type rational or a single-precision floating-point
approximation.  If the arguments are all rational but the result cannot
be expressed as a rational number, then a single-precision
floating-point approximation is always returned.

Referring to EXPT, CLtL says:

If the base-number is of type RATIONAL and the power-number is an
INTEGER, the calculation will be exact and the result will be of
type RATIONAL; otherwise a floating-point approximation may result.

What about arguments of type (complex rational)?

Proposal (COMPLEX-RATIONAL-RESULT:EXTEND):

Extend the paragraph quoted above as follows to cover the components
of complex numbers.  If the arguments to a function are all of type
(OR RATIONAL (COMPLEX RATIONAL)) and the true mathematical result is
(mathematically) a complex number with rational real and imaginary
parts, then unless otherwise noted an implementation is free to return
either an accurate result of type (OR RATIONAL (COMPLEX RATIONAL)) or
a single-precision floating-point approximation of type SINGLE-FLOAT
(permissible only if the imaginary part of the true mathematical
result is zero) or (COMPLEX SINGLE-FLOAT). If the arguments are
all of type (OR RATIONAL (COMPLEX RATIONAL)) but the result cannot be
expressed as a rational or complex rational number, then the returned
value will be of type SINGLE-FLOAT (permissible only if the imaginary
part of the true mathematical result is zero) or (COMPLEX SINGLE-FLOAT).

For EXPT of a (COMPLEX RATIONAL) to an integer power, the
calculation must be exact and the result will be of type
(OR RATIONAL (COMPLEX RATIONAL)).

Examples:

[a]  (log #c(-16 16) #c(2 2)) => 3 or approximately #c(3.0 0.0)
or approximately 3.0 (unlikely)
[b]  (abs #c(3/5 4/5)) => 1 or approximately 1.0
[c]  (expt #c(2 2) 3) => #c(-16 16)
[d]  (expt #c(2 2) 4) => -64

Rationale:

This seems most consistent with the treatment of real numbers.

Current practice:

[a]			[b]	[c]		[d]
Symbolics Genera 7.4	#c(3.00... 1.40...e-7)	1	#c(-16 16)	-64
Sun Common Lisp 3.0.1	#c(3.0 2.61...e-16)	1.0	#c(-16 16)	-64
KCL of 9/16/86 on VAX	#c(3.0s0 -1.40...s-7)	1.0s0	#c(-16 16)	-64
Allegro CL (Mac II)	#c(3.0 2.05...e-16)	1.0	#c(-16 16)	-64

Cost to Implementors:

Only EXPT would have to change, since the type of the other results
is at the discretion of the implementation.

Cost to Users:

Probably none, but it is hard to predict.

Slightly less self-consistent language.

Performance impact:

None of any significance.

Benefits/Esthetics:

More self-consistent language.

Discussion:

None.
```      