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Getting efficient flonum arithmetic is much more complicated and harder than getting efficient fixnum arithmetic.

One of the main disadvantages of generic arithmetic is that not all
kinds of number fit in a machine register. Flonums have to be
*boxed* because a 64-bit IEEE floating-point number (the
representation that MIT/GNU Scheme uses) does not fit in a regular machine
word on 32-bit machines. Boxing is also done on 64-bit machines
because some extra bits are needed to distinguish floating-point
numbers from other objects like pairs and strings. Values are boxed
by storing them in a small record in the heap. Every floating-point
value that you see at the REPL is boxed. Floating-point
values are unboxed only for short periods of time when they are in the
machine’s floating-point unit and actual floating-point operations are
being performed.

Numerical calculations that happen to be using floating-point numbers cause many temporary floating-point numbers to be allocated. It is not uncommon for numerical programs to spend over half of their time creating and garbage collecting the boxed flonums.

Consider the following procedure for computing the distance of a point
(`x`,`y`) from the origin.

(define (distance x y) (sqrt (+ (* x x) (* y y))))

The call ‘`(distance 0.3 0.4)`’ returns a new, boxed flonum, 0.5.
The calculation also generates three intermediate boxed flonums. This
next version works only for flonum inputs, generates only one boxed
flonum (the result) and runs eight times faster:

(define (flo:distance x y) (flo:sqrt (flo:+ (flo:* x x) (flo:* y y))))

Note that `flo:`

operations are usually effective only within a
single arithmetic expression. If the expression contains conditionals
or calls to procedures then the values tend to get boxed anyway.

Flonum vectors are vectors that contain only floating-point values, in much the same way as a string is a ‘vector’ containing only character values.

Flonum vectors have the advantages of compact storage (about half that of a conventional vector of flonums on 32-bit machines and about one-third on 64-bit machines) and judicious use of flonum vectors can decrease flonum consing.

The disadvantages are that flonum vectors are incompatible with ordinary vectors, and if not used carefully, can increase flonum consing. Flonum vectors are a pain to use because they require you to make a decision about the representation and stick with it, and it might not be easy to ascertain whether the advantages in one part of the program outweigh the disadvantages in another.

The flonum vector operations are:

- procedure:
**flo:vector-cons***n*¶ Create a flonum vector of length

`n`. The contents of the vector are arbitrary and might not be valid floating-point numbers. The contents should not be used until initialized.

- procedure:
**flo:vector-ref***flonum-vector index*¶ - procedure:
**flo:vector-set!***flonum-vector index value*¶ - procedure:
**flo:vector-length***flonum-vector*¶ These operations are analogous to the ordinary vector operations.

The following operation causes no flonum consing because the flonum is loaded directly from the flonum vector into a floating-point machine register, added, and stored again. There is no need for a temporary boxed flonum.

(flo:vector-set! v 0 (flo:+ (flo:vector-ref v 0) 1.2))

In this next example, every time `g`

is called, a new boxed flonum
has to be created so that a valid Scheme object can be returned. If
`g`

is called more often than the elements of `v` are changed
then an ordinary vector might be more efficient.

(define (g i) (flo:vector-ref v i))

Pitfall 1: Make sure that your literals are floating-point constants:

(define (f1 a) (flo:+ a 1)) (define (f2 a) (flo:+ a 1.))

`f1`

will most likely cause a hardware error, and certainly give
the wrong answer. `f2`

is correct.

Pitfall 2:
It is tempting to insert calls to `inexact`

to coerce values into
flonums. This does not always work because complex numbers may be
exact or inexact too. Also, the current implementation of
`inexact`

is slow.

Pitfall 3:
A great deal of care has to be taken with the standard math procedures.
For example, when called with a flonum, both `sqrt`

and `asin`

can return a complex number (e.g with argument `-1.5`

).

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