# posit 2022.2.0.0

The Posit Standard 2022,
and Posit Standard 3.2,
where Real numbers are approximated by Maybe Rational. The Posit
Numbers are a drop in replacement for `Float`

or `Double`

mapped to a
2's complement integer type; smoothly and with tapering precision, in a
similar way to the projective real line. The 'posit' library implements
the following standard classes:

- Show
- Eq
- Ord -- compare as an integer representation
- Num -- Addition, subtraction, multiplication, and other operations
- Enum -- Successor and Predecessor
- Fractional -- division, divide by zero is Not a Real (NaR) number
- Real
- Bounded
- FusedOps -- dot product and others
- Convertable -- Conversions between different posit formats
- AltShow
- Read
- Storable -- Formats for binary data, for computation and data interchange
- Random
- Uniform
- RealFrac
- RealFloat
- Floating -- Mathematical functions such as logarithm, exponential, trigonometric, and hyperbolic functions. Warning! May induce trance.

The Posits are indexed by the type (es :: ES) where exponent size and
word size are related. In `posit-3.2`

es is instantiated as Z, I,
II, III, IV, V. In `posit-2022`

es is instantiated as Z_2022, I_2022,
II_2022, III_2022, IV_2022, V_2022. The word size (in bits) of the
value is `= 8 * 2^es`

, that is `2^es`

bytes. The Types: 'Posit8',
'Posit16', 'Posit32', 'Posit64', 'Posit128', and 'Posit256' as well as,
'P8', 'P16', 'P32', 'P64', 'P128', and 'P256' are implemented and
include a couple of auxiliary classes, like AltShow, AltFloating, and
FusedOps. So, 3.2 scales by dynamic range, 2022 scales by precision.

```
class AltShow a where
-- Display the Posit in its Binary Representation
displayBinary :: a -> String
-- Display the Posit in its Integral Representation
displayIntegral :: a -> String
-- Display the Posit as a Rational
displayRational :: a -> String
-- Display the Posit as a Decimal until the Repented occurs
displayDecimal :: a -> String
```

```
class AltFloating p where
machEps :: p -- Machine Epsilon near 1.0
approxEq :: p -> p -> Bool
goldenRatio :: p
hypot2 :: p -> p -> p
hypot3 :: p -> p -> p -> p
hypot4 :: p -> p -> p -> p -> p
```

The 'FusedOps' class has been updated with rewrite rules, enabled by the
`--flag posit:do-rewrite`

flag. When the flag is enabled, the `ghc`

compiler will match the pattern and then automatically rewrite with the
fused operation.

```
class Num a => FusedOps a where
-- |Fused Multiply Add: (a * b) + c
fma :: a -> a -> a -> a
-- |Fused Add Multiply: (a + b) * c
fam :: a -> a -> a -> a
-- |Fused Multiply Multiply Subtract: (a * b) - (c * d)
fmms :: a -> a -> a -> a -> a
-- |Fused Sum of 3 values: a + b + c
fsum3 :: a -> a -> a -> a
-- |Fused Sum of 4 values: a + b + c + d
fsum4 :: a -> a -> a -> a -> a
-- |Fused Sum of a List of Posits
fsumL :: Foldable t => t a -> a
-- |Fused Dot Product of 3 element vector: (a1 * b1) + (a2 * b2) + (a3 * b3)
fdot3 :: a -> a -> a -> a -> a -> a -> a
-- |Fused Dot Product of 4 element veector: (a0 * b0) + (a1 * b1) + (a2 * b2) + (a3 * b3)
fdot4 :: a -> a -> a -> a -> a -> a -> a -> a -> a
-- |Fused Dot Product of Two Lists
fdotL :: Foldable t => t a -> t a -> a
-- |Fused Subtract Multiply: a - (b * c)
fsm :: a -> a -> a -> a
```

The Posit type is 'Convertible' between other Posit lengths.

```
class Convertible a b where
convert :: a -> b
```

The Posit Library is built on top of two of the most excellent libraries:
data-dword, and
scientific. The
'data-dword' library provides the underlining machine word
representation, it can provide 2^es word size, 2's complement fixed
length integers. The 'scientific' library provides 'read' and 'show'
instances.

The Elementary Functions for this posit library are implemented natively
using the next larger posit number. P16 and Posit16 numbers are tested
exhaustively against Double Precision numbers as the orical. The results
are below:

Number of Accurate Bits `exp`

:

Number of Accurate Bits `log`

:

Number of Accurate Bits `sqrt`

:

Number of Accurate Bits `sin`

:

Number of Accurate Bits `cos`

:

Number of Accurate Bits `asin`

:

Number of Accurate Bits `acos`

:

Number of Accurate Bits `atan`

:

Number of Accurate Bits `sinh`

:

Number of Accurate Bits `cosh`

:

Number of Accurate Bits `asinh`

:

Number of Accurate Bits `acosh`

:

Number of Accurate Bits `atanh`

: