retroforth/doc/book/Programming-Techniques-Working-With-Floating-Point

# Working With Floating Point

Some RETRO systems include support for floating point numbers.
When present, this is built over the system `libm` using the
C `double` type.

Floating point values are typically 64 bit IEEE 754 double
precision (1 bit for the sign, 11 bits for the exponent, and
the remaining 52 bits for the value), i.e. 15 decimal digits
of precision.

## Prefix

Floating point numbers start with a `.`

Examples:

   Token    Value
    .1       1.0
    .0.5     0.5
    .-.4    -0.4
    .1.3     1.3

## Namespace

Floating point words are in the `f:` namespace. There is also
a related `e:` namespace for *encoded values*, which allows
storing of floats in standard memory.

## Operation

Floating point values exist on a separate stack, and are bigger
than the standard memory cells, so can not be directly stored
and fetched from memory.

The floating point system also provides an alternate stack that
can be used to temporarily store values.

The following words exist for arranging values on the floating
point stack. These are direct analogs to the non-prefiexd words
for dealing with the data stack.

- `f:nip`
- `f:over`
- `f:depth`
- `f:drop`
- `f:drop-pair`
- `f:dup`
- `f:dup-pair`
- `f:dump-stack`
- `f:tuck`
- `f:swap`
- `f:rot`

For the secondary floating point stack, the following words are
provided:

- `f:push`
- `f:pop`
- `f:adepth`
- `f:dump-astack`

## Constants

    | Name     | Returns           |
    | -------- | ----------------- |
    | `f:E`    | Euler's number    |
    | `f:-INF` | Negative infinity |
    | `f:INF`  | Positive infinity |
    | `f:NAN`  | Not a Number      |
    | `f:PI`   | PI                |

## Comparisons

The basic set of comparators are the same as those for
operating on integers. These are:

- `f:-eq?`
- `f:between?`
- `f:eq?`
- `f:gt?`
- `f:lt?`
- `f:negative?`
- `f:positive?`
- `f:case`

There are also a few additions for comparing to special values
like infinity and NaN.

- `f:-inf?`
- `f:inf?`
- `f:nan?`

## Basic Math

- `f:*`
- `f:+`
- `f:-`
- `f:/`
- `f:abs`
- `f:floor`
- `f:inc`
- `f:limit`
- `f:max`
- `f:min`
- `f:negate`
- `f:power`
- `f:ceiling`
- `f:dec`
- `f:log`
- `f:sqrt`
- `f:square`
- `f:round`
- `f:sign`
- `f:signed-sqrt`
- `f:signed-square`

## Geometry

RETRO provides a small number of words for doing geometric
related calculations.

| Word     | Returns      |
| -------- | ------------ |
| `f:acos` | arc cosine   |
| `f:asin` | arc sine     |
| `f:atan` | arc tangent  |
| `f:cos`  | cosine       |
| `f:sin`  | sine         |
| `f:tan`  | tangent      |

## Storage and Retrieval

By leveraging the encoded value functions, RETRO is able to
allow storage of floating point values in memory. This does
have a tradeoff in accuracy as the memory cells are considerably
smaller than a full floating point size.

You can use `f:fetch` to fetch a floating point value and
`f:store` to store one.

If you need more precision, try Kiyoshi Yoneda's FloatVar
example (`example/FloatVar.forth`), which includes words to
store and retrieve values using multiple cells.

- `f:to-number`
- `f:to-string`

## I/O

The floating point vocabulary has a single I/O word, `f:put`,
for the display of floating point numbers.

## Encoded Values

RETRO provides a means of encoding and decoding floating point
values into standard integer cells. This is based on the paper
"Encoding floating point values to shorter integers" by Kiyoshi
Yoneda and Charles Childers.

- `f:E1`
- `f:to-e`
- `e:-INF`
- `e:-inf?`
- `e:INF`
- `e:MAX`
- `e:MIN`
- `e:NAN`
- `e:clip`
- `e:inf?`
- `e:max?`
- `e:min?`
- `e:n?`
- `e:nan?`
- `e:put`
- `e:to-f`
- `e:zero?`