Data/ByteString/Builder/RealFloat/TableGenerator.hs

    {-# LANGUAGE ExplicitForAll #-}
    {-# LANGUAGE MultiParamTypeClasses #-}
    {-# LANGUAGE ScopedTypeVariables #-}
    {-# LANGUAGE BangPatterns, MagicHash, UnboxedTuples #-}
    -- |
    -- Module      : Data.ByteString.Builder.RealFloat.TableGenerator
    -- Copyright   : (c) Lawrence Wu 2021
    -- License     : BSD-style
    -- Maintainer  : lawrencejwu@gmail.com
    --
    -- Constants and overview for compile-time table generation for Ryu internals
    --
    -- This module uses Haskell's arbitrary-precision `Integer` types to compute
    -- the necessary multipliers for efficient conversion to a decimal power base.
    --
    -- It also exposes constants relevant to the 32- and 64-bit tables (e.g maximum
    -- number of bits required to store the table values).

    module Data.ByteString.Builder.RealFloat.TableGenerator
      ( float_pow5_inv_bitcount
      , float_pow5_bitcount
      , double_pow5_bitcount
      , double_pow5_inv_bitcount
      , float_max_split
      , float_max_inv_split
      , double_max_split
      , double_max_inv_split
      ) where

    import GHC.Float (int2Double)

    -- The basic floating point conversion algorithm is as such:
    --
    -- Given floating point
    --
    --   f = (-1)^s * m_f * 2^e_f
    --
    -- which is IEEE encoded by `[s] [.. e ..] [.. m ..]`. `s` is the sign bit, `e`
    -- is the biased exponent, and `m` is the mantissa, let
    --
    --       | e /= 0            | e == 0
    --  -----+-------------------+-----------
    --   m_f | 2^len(m) + m      | m
    --   e_f | e - bias - len(m) | 1 - bias - len(m)
    --
    -- we compute the halfway points to the next smaller (`f-`) and larger (`f+`)
    -- floating point numbers as
    --
    --  lower halfway point u * 2^e2, u = 4 * m_f - (if m == 0 then 1 else 2)
    --                      v * 2^e2, v = 4 * m_f
    --  upper halfway point w * 2^e2, u = 4 * m_f + 2
    --  where e2 = ef - 2 (so u, v, w are integers)
    --
    --
    -- Then we compute (a, b, c) * 10^e10 = (u, v, w) * 2^e2 which is split into
    -- the case of
    --
    --   e2 >= 0   ==>    e10 = 0 , (a, b, c) = (u, v, w) * 2^e2
    --   e2 <  0   ==>    e10 = e2, (a, b, c) = (u, v, w) * 5^-e2
    --
    -- And finally we find the shortest representation from integers d0 and e0 such
    -- that
    --
    --  a * 10^e10 < d0 * 10^(e0+e10) < c * 10^e10
    --
    -- such that e0 is maximal (we allow equality to smaller or larger halfway
    -- point depending on rounding mode). This is found through iteratively
    -- dividing by 10 while a/10^j < c/10^j and doing some bookkeeping around
    -- zeros.
    --
    --
    --
    --
    -- The ryu algorithm removes the requirement for arbitrary precision arithmetic
    -- and improves the runtime significantly by skipping most of the iterative
    -- division by carefully selecting a point where certain invariants hold and
    -- precomputing a few tables.
    --
    -- Specifically, define `q` such that the correspondings values of a/10^q <
    -- c/10^q - 1. We can prove (not shown) that
    --
    --    if e2 >= 0, q = e2 * log_10(2)
    --    if e2 <  0, q = -e2 * log_10(5)
    --
    -- Then we can compute (a, b, c) / 10^q. Starting from (u, v, w) we have
    --
    --      (a, b, c) / 10^q                  (a, b, c) / 10^q
    --    = (u, v, w) * 2^e2 / 10^q    OR   = (u, v, w) * 5^-e2 / 10^q
    --
    -- And since q < e2,
    --
    --    = (u, v, w) * 2^e2-q / 5^q   OR   = (u, v, w) * 5^-e2-q / 2^q
    --
    -- While (u, v, w) are n-bit numbers, 5^q and whatnot are significantly larger,
    -- but we only need the top-most n bits of the result so we can choose `k` that
    -- reduce the number of bits required to ~2n. We then multiply by either
    --
    --    2^k / 5^q                    OR   5^-e2-q / 2^k
    --
    -- The required `k` is roughly linear in the exponent (we need more of the
    -- multiplication to be precise) but the number of bits to store the
    -- multiplicands above stays fixed.
    --
    -- Since the number of bits needed is relatively small for IEEE 32- and 64-bit
    -- floating types, we can compute appropriate values for `k` for the
    -- floating-point-type-specific bounds instead of each e2.
    --
    -- Finally, we need to do some final manual iterations potentially to do a
    -- final fixup of the skipped state

    -- | Bound for bits of @2^k / 5^q@ for floats
    float_pow5_inv_bitcount :: Int
    float_pow5_inv_bitcount = 59

    -- | Bound for bits of @5^-e2-q / 2^k@ for floats
    float_pow5_bitcount :: Int
    float_pow5_bitcount = 61

    -- | Bound for bits of @5^-e2-q / 2^k@ for doubles
    double_pow5_bitcount :: Int
    double_pow5_bitcount = 125

    -- | Bound for bits of @2^k / 5^q@ for doubles
    double_pow5_inv_bitcount :: Int
    double_pow5_inv_bitcount = 125

    -- NB: these tables are encoded directly into the source code in F2S and D2S
    --
    -- -- | Number of bits in a positive integer
    -- blen :: Integer -> Int
    -- blen 0 = 0
    -- blen 1 = 1
    -- blen n = 1 + blen (n `quot` 2)

    -- -- | Used for table generation of 2^k / 5^q + 1
    -- finv :: Int -> Int -> Integer
    -- finv bitcount i =
    --   let p = 5^i
    --    in (1 `shiftL` (blen p - 1 + bitcount)) `div` p + 1

    -- -- | Used for table generation of 5^-e2-q / 2^k
    -- fnorm :: Int -> Int -> Integer
    -- fnorm bitcount i =
    --   let p = 5^i
    --       s = blen p - bitcount
    --    in if s < 0 then p `shiftL` (-s) else p `shiftR` s

    -- -- | Generates a compile-time lookup table for floats as Word64
    -- gen_table_f :: Int -> (Int -> Integer) -> Q Exp
    -- gen_table_f n f = return $ ListE (fmap (LitE . IntegerL . f) [0..n])
    --
    -- -- | Generates a compile-time lookup table for doubles as Word128
    -- gen_table_d :: Int -> (Int -> Integer) -> Q Exp
    -- gen_table_d n f = return $ ListE (fmap ff [0..n])
    --   where
    --     ff :: Int -> Exp
    --     ff c = let r = f c
    --                hi = r `shiftR` 64
    --                lo = r .&. ((1 `shiftL` 64) - 1)
    --             in AppE (AppE (ConE 'Word128) (LitE . IntegerL $ hi)) (LitE . IntegerL $ lo)

    -- Given a specific floating-point type, determine the range of q for the < 0
    -- and >= 0 cases
    get_range :: forall ff. (RealFloat ff) => ff -> (Int, Int)
    get_range f =
      let (emin, emax) = floatRange f
          mantissaDigits = floatDigits f
          emin' = emin - mantissaDigits - 2
          emax' = emax - mantissaDigits - 2
       in ( (-emin') - floor (int2Double (-emin') * logBase 10 5)
          , floor (int2Double emax' * logBase 10 2))

    float_max_split :: Int     -- = 46
    float_max_inv_split :: Int -- = 30
    (float_max_split, float_max_inv_split) = get_range (undefined :: Float)

    -- we take a slightly different codepath s.t we need one extra entry
    double_max_split :: Int     -- = 325
    double_max_inv_split :: Int -- = 291
    (double_max_split, double_max_inv_split) =
        let (m, mi) = get_range (undefined :: Double)
         in (m + 1, mi)