exp.jl

# magic rounding constant: 1.5*2^52 Adding, then subtracting it from a float rounds it to an Int.
# This works because eps(MAGIC_ROUND_CONST(T)) == one(T), so adding it to a smaller number aligns the lsb to the 1s place.
# Values for which this trick doesn't work are going to have outputs of 0 or Inf.
MAGIC_ROUND_CONST(::Type{Float64}) = 6.755399441055744e15
MAGIC_ROUND_CONST(::Type{Float32}) = 1.048576f7

# max, min, and subnormal arguments
# max_exp = T(exponent_bias(T)*log(base, big(2)) + log(base, 2 - big(2.0)^-significand_bits(T)))
MAX_EXP(n::Val{2}, ::Type{Float32}) = 128.0f0
MAX_EXP(n::Val{2}, ::Type{Float64}) = 1024.0
MAX_EXP(n::Val{:ℯ}, ::Type{Float32}) = 88.72284f0
MAX_EXP(n::Val{:ℯ}, ::Type{Float64}) = 709.782712893384
MAX_EXP(n::Val{10}, ::Type{Float32}) = 38.53184f0
MAX_EXP(n::Val{10}, ::Type{Float64}) = 308.25471555991675

# min_exp = T(-(exponent_bias(T)+significand_bits(T)) * log(base, big(2)))
MIN_EXP(n::Val{2}, ::Type{Float32}) = -150.0f0
MIN_EXP(n::Val{2}, ::Type{Float64}) = -1075.0
MIN_EXP(n::Val{:ℯ}, ::Type{Float32}) = -103.97208f0
MIN_EXP(n::Val{:ℯ}, ::Type{Float64}) = -745.1332191019412
MIN_EXP(n::Val{10}, ::Type{Float32}) = -45.1545f0
MIN_EXP(n::Val{10}, ::Type{Float64}) = -323.60724533877976

# subnorm_exp = abs(log(base, floatmin(T)))
# these vals are positive since it's easier to take abs(x) than -abs(x)
SUBNORM_EXP(n::Val{2}, ::Type{Float32}) = 126.00001f0
SUBNORM_EXP(n::Val{2}, ::Type{Float64}) = 1022.0
SUBNORM_EXP(n::Val{:ℯ}, ::Type{Float32}) = 87.33655f0
SUBNORM_EXP(n::Val{:ℯ}, ::Type{Float64}) = 708.3964185322641
SUBNORM_EXP(n::Val{10}, ::Type{Float32}) = 37.92978f0
SUBNORM_EXP(n::Val{10}, ::Type{Float64}) = 307.6526555685887

# 256/log(base, 2) (For Float64 reductions)
LogBo256INV(::Val{2}, ::Type{Float64}) = 256.0
LogBo256INV(::Val{:ℯ}, ::Type{Float64}) = 369.3299304675746
LogBo256INV(::Val{10}, ::Type{Float64}) = 850.4135922911647

# -log(base, 2)/256 in upper and lower bits
# Upper is truncated to only have 34 bits of significand since N has at most
# ceil(log2(-MIN_EXP(base, Float64)*LogBo256INV(Val(2), Float64))) = 19 bits.
# This ensures no rounding when multiplying LogBo256U*N for FMAless hardware
LogBo256U(::Val{2}, ::Type{Float64}) = -0.00390625
LogBo256U(::Val{:ℯ}, ::Type{Float64}) = -0.002707606173999011
LogBo256U(::Val{10}, ::Type{Float64}) = -0.0011758984204561784
LogBo256L(::Val{2}, ::Type{Float64}) = 0.0
LogBo256L(::Val{:ℯ}, ::Type{Float64}) = -6.327543041662719e-14
LogBo256L(::Val{10}, ::Type{Float64}) = -1.0624811566412999e-13

# 1/log(base, 2) (For Float32 reductions)
LogBINV(::Val{2}, ::Type{Float32}) = 1.0f0
LogBINV(::Val{:ℯ}, ::Type{Float32}) = 1.442695f0
LogBINV(::Val{10}, ::Type{Float32}) = 3.321928f0

# -log(base, 2) in upper and lower bits
# Upper is truncated to only have 16 bits of significand since N has at most
# ceil(log2(-MIN_EXP(n, Float32)*LogBINV(Val(2), Float32))) = 8 bits.
# This ensures no rounding when multiplying LogBU*N for FMAless hardware
LogBU(::Val{2}, ::Type{Float32}) = -1.0f0
LogBU(::Val{:ℯ}, ::Type{Float32}) = -0.69314575f0
LogBU(::Val{10}, ::Type{Float32}) = -0.3010254f0
LogBL(::Val{2}, ::Type{Float32}) = 0.0f0
LogBL(::Val{:ℯ}, ::Type{Float32}) = -1.4286068f-6
LogBL(::Val{10}, ::Type{Float32}) = -4.605039f-6

# Range reduced kernels
@inline function expm1b_kernel(::Val{2}, x::Float64)
    return x * evalpoly(x, (0.6931471805599393, 0.24022650695910058,
                            0.05550411502333161, 0.009618129548366803))
end
@inline function expm1b_kernel(::Val{:ℯ}, x::Float64)
    return x * evalpoly(x, (0.9999999999999912, 0.4999999999999997,
                            0.1666666857598779, 0.04166666857598777))
end

@inline function expm1b_kernel(::Val{10}, x::Float64)
    return x * evalpoly(x, (2.3025850929940255, 2.6509490552391974,
                            2.034678825384765, 1.1712552025835192))
end

@inline function expb_kernel(::Val{2}, x::Float32)
    return evalpoly(x, (1.0f0, 0.6931472f0, 0.2402265f0,
                        0.05550411f0, 0.009618025f0,
                        0.0013333423f0, 0.00015469732f0, 1.5316464f-5))
end
@inline function expb_kernel(::Val{:ℯ}, x::Float32)
    return evalpoly(x, (1.0f0, 1.0f0, 0.5f0, 0.16666667f0,
                        0.041666217f0, 0.008333249f0,
                        0.001394858f0, 0.00019924171f0))
end
@inline function expb_kernel(::Val{10}, x::Float32)
    return evalpoly(x, (1.0f0, 2.3025851f0, 2.650949f0,
                        2.0346787f0, 1.1712426f0, 0.53937745f0,
                        0.20788547f0, 0.06837386f0))
end

# Table stores data with 60 sig figs by using the fact that the first 12 bits of all the
# values would be the same if stored as regular Float64.
# This only gains 8 bits since the least significant 4 bits of the exponent
# of the small part are not the same for all table entries
const JU_MASK = typemax(UInt64)>>12
const JL_MASK = typemax(UInt64)>>8
const JU_CONST = 0x3FF0000000000000
const JL_CONST = 0x3C00000000000000


#function make_table(size)
#    t_array = zeros(UInt64, size);
#    for j in 1:size
#        val = 2.0^(BigFloat(j-1)/size)
#        valU = Float64(val, RoundDown)
#        valL = Float64(val-valU)
#        valU = reinterpret(UInt64, valU) & JU_MASK
#        valL = ((reinterpret(UInt64, valL) & JL_MASK)>>44)<<52
#        t_array[j] = valU | valL
#    end
#    return Tuple(t_array)
#end
#const J_TABLE = make_table(256);
const J_TABLE = (0x0000000000000000, 0xaac00b1afa5abcbe, 0x9b60163da9fb3335, 0xab502168143b0280, 0xadc02c9a3e778060,
                 0x656037d42e11bbcc, 0xa7a04315e86e7f84, 0x84c04e5f72f654b1, 0x8d7059b0d3158574, 0xa510650a0e3c1f88,
                 0xa8d0706b29ddf6dd, 0x83207bd42b72a836, 0x6180874518759bc8, 0xa4b092bdf66607df, 0x91409e3ecac6f383,
                 0x85d0a9c79b1f3919, 0x98a0b5586cf9890f, 0x94f0c0f145e46c85, 0x9010cc922b7247f7, 0xa210d83b23395deb,
                 0x4030e3ec32d3d1a2, 0xa5b0efa55fdfa9c4, 0xae40fb66affed31a, 0x8d41073028d7233e, 0xa4911301d0125b50,
                 0xa1a11edbab5e2ab5, 0xaf712abdc06c31cb, 0xae8136a814f204aa, 0xa661429aaea92ddf, 0xa9114e95934f312d,
                 0x82415a98c8a58e51, 0x58f166a45471c3c2, 0xab9172b83c7d517a, 0x70917ed48695bbc0, 0xa7718af9388c8de9,
                 0x94a1972658375d2f, 0x8e51a35beb6fcb75, 0x97b1af99f8138a1c, 0xa351bbe084045cd3, 0x9001c82f95281c6b,
                 0x9e01d4873168b9aa, 0xa481e0e75eb44026, 0xa711ed5022fcd91c, 0xa201f9c18438ce4c, 0x8dc2063b88628cd6,
                 0x935212be3578a819, 0x82a21f49917ddc96, 0x8d322bdda27912d1, 0x99b2387a6e756238, 0x8ac2451ffb82140a,
                 0x8ac251ce4fb2a63f, 0x93e25e85711ece75, 0x82b26b4565e27cdd, 0x9e02780e341ddf29, 0xa2d284dfe1f56380,
                 0xab4291ba7591bb6f, 0x86129e9df51fdee1, 0xa352ab8a66d10f12, 0xafb2b87fd0dad98f, 0xa572c57e39771b2e,
                 0x9002d285a6e4030b, 0x9d12df961f641589, 0x71c2ecafa93e2f56, 0xaea2f9d24abd886a, 0x86f306fe0a31b715,
                 0x89531432edeeb2fd, 0x8a932170fc4cd831, 0xa1d32eb83ba8ea31, 0x93233c08b26416ff, 0xab23496266e3fa2c,
                 0xa92356c55f929ff0, 0xa8f36431a2de883a, 0xa4e371a7373aa9ca, 0xa3037f26231e7549, 0xa0b38cae6d05d865,
                 0xa3239a401b7140ee, 0xad43a7db34e59ff6, 0x9543b57fbfec6cf4, 0xa083c32dc313a8e4, 0x7fe3d0e544ede173,
                 0x8ad3dea64c123422, 0xa943ec70df1c5174, 0xa413fa4504ac801b, 0x8bd40822c367a024, 0xaf04160a21f72e29,
                 0xa3d423fb27094689, 0xab8431f5d950a896, 0x88843ffa3f84b9d4, 0x48944e086061892d, 0xae745c2042a7d231,
                 0x9c946a41ed1d0057, 0xa1e4786d668b3236, 0x73c486a2b5c13cd0, 0xab1494e1e192aed1, 0x99c4a32af0d7d3de,
                 0xabb4b17dea6db7d6, 0x7d44bfdad5362a27, 0x9054ce41b817c114, 0x98e4dcb299fddd0d, 0xa564eb2d81d8abfe,
                 0xa5a4f9b2769d2ca6, 0x7a2508417f4531ee, 0xa82516daa2cf6641, 0xac65257de83f4eee, 0xabe5342b569d4f81,
                 0x879542e2f4f6ad27, 0xa8a551a4ca5d920e, 0xa7856070dde910d1, 0x99b56f4736b527da, 0xa7a57e27dbe2c4ce,
                 0x82958d12d497c7fd, 0xa4059c0827ff07cb, 0x9635ab07dd485429, 0xa245ba11fba87a02, 0x3c45c9268a5946b7,
                 0xa195d84590998b92, 0x9ba5e76f15ad2148, 0xa985f6a320dceb70, 0xa60605e1b976dc08, 0x9e46152ae6cdf6f4,
                 0xa636247eb03a5584, 0x984633dd1d1929fd, 0xa8e6434634ccc31f, 0xa28652b9febc8fb6, 0xa226623882552224,
                 0xa85671c1c70833f5, 0x60368155d44ca973, 0x880690f4b19e9538, 0xa216a09e667f3bcc, 0x7a36b052fa75173e,
                 0xada6c012750bdabe, 0x9c76cfdcddd47645, 0xae46dfb23c651a2e, 0xa7a6ef9298593ae4, 0xa9f6ff7df9519483,
                 0x59d70f7466f42e87, 0xaba71f75e8ec5f73, 0xa6f72f8286ead089, 0xa7a73f9a48a58173, 0x90474fbd35d7cbfd,
                 0xa7e75feb564267c8, 0x9b777024b1ab6e09, 0x986780694fde5d3f, 0x934790b938ac1cf6, 0xaaf7a11473eb0186,
                 0xa207b17b0976cfda, 0x9f17c1ed0130c132, 0x91b7d26a62ff86f0, 0x7057e2f336cf4e62, 0xabe7f3878491c490,
                 0xa6c80427543e1a11, 0x946814d2add106d9, 0xa1582589994cce12, 0x9998364c1eb941f7, 0xa9c8471a4623c7ac,
                 0xaf2857f4179f5b20, 0xa01868d99b4492ec, 0x85d879cad931a436, 0x99988ac7d98a6699, 0x9d589bd0a478580f,
                 0x96e8ace5422aa0db, 0x9ec8be05bad61778, 0xade8cf3216b5448b, 0xa478e06a5e0866d8, 0x85c8f1ae99157736,
                 0x959902fed0282c8a, 0xa119145b0b91ffc5, 0xab2925c353aa2fe1, 0xae893737b0cdc5e4, 0xa88948b82b5f98e4,
                 0xad395a44cbc8520e, 0xaf296bdd9a7670b2, 0xa1797d829fde4e4f, 0x7ca98f33e47a22a2, 0xa749a0f170ca07b9,
                 0xa119b2bb4d53fe0c, 0x7c79c49182a3f090, 0xa579d674194bb8d4, 0x7829e86319e32323, 0xaad9fa5e8d07f29d,
                 0xa65a0c667b5de564, 0x9c6a1e7aed8eb8bb, 0x963a309bec4a2d33, 0xa2aa42c980460ad7, 0xa16a5503b23e255c,
                 0x650a674a8af46052, 0x9bca799e1330b358, 0xa58a8bfe53c12e58, 0x90fa9e6b5579fdbf, 0x889ab0e521356eba,
                 0xa81ac36bbfd3f379, 0x97ead5ff3a3c2774, 0x97aae89f995ad3ad, 0xa5aafb4ce622f2fe, 0xa21b0e07298db665,
                 0x94db20ce6c9a8952, 0xaedb33a2b84f15fa, 0xac1b468415b749b0, 0xa1cb59728de55939, 0x92ab6c6e29f1c52a,
                 0xad5b7f76f2fb5e46, 0xa24b928cf22749e3, 0xa08ba5b030a10649, 0xafcbb8e0b79a6f1e, 0x823bcc1e904bc1d2,
                 0xafcbdf69c3f3a206, 0xa08bf2c25bd71e08, 0xa89c06286141b33c, 0x811c199bdd85529c, 0xa48c2d1cd9fa652b,
                 0x9b4c40ab5fffd07a, 0x912c544778fafb22, 0x928c67f12e57d14b, 0xa86c7ba88988c932, 0x71ac8f6d9406e7b5,
                 0xaa0ca3405751c4da, 0x750cb720dcef9069, 0xac5ccb0f2e6d1674, 0xa88cdf0b555dc3f9, 0xa2fcf3155b5bab73,
                 0xa1ad072d4a07897b, 0x955d1b532b08c968, 0xa15d2f87080d89f1, 0x93dd43c8eacaa1d6, 0x82ed5818dcfba487,
                 0x5fed6c76e862e6d3, 0xa77d80e316c98397, 0x9a0d955d71ff6075, 0x9c2da9e603db3285, 0xa24dbe7cd63a8314,
                 0x92ddd321f301b460, 0xa1ade7d5641c0657, 0xa72dfc97337b9b5e, 0xadae11676b197d16, 0xa42e264614f5a128,
                 0xa30e3b333b16ee11, 0x839e502ee78b3ff6, 0xaa7e653924676d75, 0x92de7a51fbc74c83, 0xa77e8f7977cdb73f,
                 0xa0bea4afa2a490d9, 0x948eb9f4867cca6e, 0xa1becf482d8e67f0, 0x91cee4aaa2188510, 0x9dcefa1bee615a27,
                 0xa66f0f9c1cb64129, 0x93af252b376bba97, 0xacdf3ac948dd7273, 0x99df50765b6e4540, 0x9faf6632798844f8,
                 0xa12f7bfdad9cbe13, 0xaeef91d802243c88, 0x874fa7c1819e90d8, 0xacdfbdba3692d513, 0x62efd3c22b8f71f1, 0x74afe9d96b2a23d9)

@inline function table_unpack(ind)
    j = @inbounds J_TABLE[ind]
    jU = reinterpret(Float64, JU_CONST | (j&JU_MASK))
    jL = reinterpret(Float64, JL_CONST | (j>>8))
    return jU, jL
end

# Method for Float64
# 1. Argument reduction: Reduce x to an r so that |r| <= log(b, 2)/512. Given x, base b,
#    find r and integers k, j such that
#       x = (k + j/256)*log(b, 2) + r,  0 <= j < 256, |r| <= log(b,2)/512.
#
# 2. Approximate b^r-1 by 3rd-degree minimax polynomial p_b(r) on the interval [-log(b,2)/512, log(b,2)/512].
#    Since the bounds on r are very tight, this is sufficient to be accurate to floating point epsilon.
#
# 3. Scale back: b^x = 2^k * 2^(j/256) * (1 + p_b(r))
#    Since the range of possible j is small, 2^(j/256) is stored for all possible values in slightly extended precision.

# Method for Float32
# 1. Argument reduction: Reduce x to an r so that |r| <= log(b, 2)/2. Given x, base b,
#    find r and integer N such that
#       x = N*log(b, 2) + r,  |r| <= log(b,2)/2.
#
# 2. Approximate b^r by 7th-degree minimax polynomial p_b(r) on the interval [-log(b,2)/2, log(b,2)/2].
# 3. Scale back: b^x = 2^N * p_b(r)

# For both, a little extra care needs to be taken if b^r is subnormal.
# The solution is to do the scaling back in 2 steps as just messing with the exponent wouldn't work.
for (func, base) in (:exp2=>Val(2), :exp=>Val(:ℯ), :exp10=>Val(10))
    @eval begin
        function ($func)(x::Real)
            xf = float(x)
            x === xf && throw(MethodError($func, (x,)))
            return ($func)(xf)
        end
        function ($func)(x::T) where T<:Float64
            N_float = muladd(x, LogBo256INV($base, T), MAGIC_ROUND_CONST(T))
            N = reinterpret(uinttype(T), N_float) % Int32
            N_float -=  MAGIC_ROUND_CONST(T) #N_float now equals round(x*LogBo256INV($base, T))
            r = muladd(N_float, LogBo256U($base, T), x)
            r = muladd(N_float, LogBo256L($base, T), r)
            k = N >> 8
            jU, jL = table_unpack(N&255 +1)
            small_part =  muladd(jU, expm1b_kernel($base, r), jL) + jU

            if !(abs(x) <= SUBNORM_EXP($base, T))
                x >= MAX_EXP($base, T) && return Inf
                x <= MIN_EXP($base, T) && return 0.0
                if k <= -53
                    # The UInt64 forces promotion. (Only matters for 32 bit systems.)
                    twopk = (k + UInt64(53)) << 52
                    return reinterpret(T, twopk + reinterpret(UInt64, small_part))*(2.0^-53)
                end
            end
            twopk = Int64(k) << 52
            return reinterpret(T, twopk + reinterpret(Int64, small_part))
        end

        function ($func)(x::T) where T<:Float32
            N_float = round(x*LogBINV($base, T))
            N = unsafe_trunc(Int32, N_float)
            r = muladd(N_float, LogBU($base, T), x)
            r = muladd(N_float, LogBL($base, T), r)
            small_part = expb_kernel($base, r)
            if !(abs(x) <= SUBNORM_EXP($base, T))
                x > MAX_EXP($base, T) && return Inf32
                x < MIN_EXP($base, T) && return 0.0f0
                if N<=Int32(-24)
                    twopk = reinterpret(T, (N+Int32(151)) << Int32(23))
                    return (twopk*small_part)*(2f0^(-24))
                end
                N == exponent_max(T) && return small_part * T(2.0) * T(2.0)^(exponent_max(T) - 1)
            end
            twopk = reinterpret(T, (N+Int32(127)) << Int32(23))
            return twopk*small_part
        end
    end
end
@doc """
    exp(x)

Compute the natural base exponential of `x`, in other words ``e^x``.

# Examples
```jldoctest
julia> exp(1.0)
2.718281828459045
```
""" exp(x::Real)