posit 2022.0.1.0

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.

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
  eps :: p  -- Machine Epsilon near 1.0
  phi :: p
  gamma :: p -> p
  sinc :: p -> p
  expm1 :: p -> p
  hypot2 :: p -> p -> p
  hypot3 :: p -> p -> p -> p
  hypot4 :: p -> p -> p -> p -> p

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.