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|
"""
b, e10 = reduce_shortest(f[, maxsignif])
Reduce to shortest decimal representation where `abs(f) == b * 10^e10` and `b` is an
integer. If a `maxsignif` argument is provided, then `b < maxsignif`.
"""
@inline function reduce_shortest(f::T, maxsignif=nothing) where {T}
U = uinttype(T)
uf = reinterpret(U, f)
m = uf & significand_mask(T)
e = ((uf & exponent_mask(T)) >> significand_bits(T)) % Int
## Step 1
# mf * 2^ef == f
mf = (one(U) << significand_bits(T)) | m
ef = e - exponent_bias(T) - significand_bits(T)
f_isinteger = mf & ((one(U) << -ef) - one(U)) == 0
if ef > 0 || ef < -Base.significand_bits(T) || !f_isinteger
# fixup subnormals
if e == 0
ef = 1 - exponent_bias(T) - significand_bits(T)
mf = m
end
## Step 2
# u * 2^e2 == (f + prevfloat(f))/2
# v * 2^e2 == f
# w * 2^e2 == (f + nextfloat(f))/2
e2 = ef - 2
mf_iseven = iseven(mf) # trailing bit of significand is zero
v = U(4) * mf
w = v + U(2)
u_shift_half = m == 0 && e > 1 # if first element of binade, other than first normal one
u = v - U(2) + u_shift_half
## Step 3
# a == floor(u * 2^e2 / 10^e10), exact if a_allzero
# b == floor(v * 2^e2 / 10^e10), exact if b_allzero
# c == floor(w * 2^e2 / 10^e10)
a_allzero = false
b_allzero = false
b_lastdigit = 0x00
if e2 >= 0
q = log10pow2(e2) - (T == Float64 ? (e2 > 3) : 0)
e10 = q
k = pow5_inv_bitcount(T) + pow5bits(q) - 1
i = -e2 + q + k
a, b, c = mulshiftinvsplit(T, u, v, w, q, i)
if T == Float32 || T == Float16
if q != 0 && div(c - 1, 10) <= div(a, 10)
l = pow5_inv_bitcount(T) + pow5bits(q - 1) - 1
mul = pow5invsplit_lookup(T, q-1)
b_lastdigit = (mulshift(v, mul, -e2 + q - 1 + l) % 10) % UInt8
end
end
if q <= qinvbound(T)
if ((v % UInt32) - 5 * div(v, 5)) == 0
b_allzero = pow5(v, q)
elseif mf_iseven
a_allzero = pow5(u, q)
else
c -= pow5(w, q)
end
end
else
q = log10pow5(-e2) - (T == Float64 ? (-e2 > 1) : 0)
e10 = q + e2
i = -e2 - q
k = pow5bits(i) - pow5_bitcount(T)
j = q - k
a, b, c = mulshiftsplit(T, u, v, w, i, j)
if T == Float32 || T == Float16
if q != 0 && div(c - 1, 10) <= div(a, 10)
j = q - 1 - (pow5bits(i + 1) - pow5_bitcount(T))
mul = pow5split_lookup(T, i+1)
b_lastdigit = (mulshift(v, mul, j) % 10) % UInt8
end
end
if q <= 1
b_allzero = true
if mf_iseven
a_allzero = !u_shift_half
else
c -= 1
end
elseif q < qbound(T)
b_allzero = pow2(v, q - (T != Float64))
end
end
## Step 4: reduction
if a_allzero || b_allzero
# a) slow loop
while true
c_div10 = div(c, 10)
a_div10 = div(a, 10)
if c_div10 <= a_div10
break
end
a_mod10 = (a % UInt32) - UInt32(10) * (a_div10 % UInt32)
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
a_allzero &= a_mod10 == 0
b_allzero &= b_lastdigit == 0
b_lastdigit = b_mod10 % UInt8
b = b_div10
c = c_div10
a = a_div10
e10 += 1
end
if a_allzero
while true
a_div10 = div(a, 10)
a_mod10 = (a % UInt32) - UInt32(10) * (a_div10 % UInt32)
if a_mod10 != 0 && (maxsignif === nothing || b < maxsignif)
break
end
c_div10 = div(c, 10)
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
b_allzero &= b_lastdigit == 0
b_lastdigit = b_mod10 % UInt8
b = b_div10
c = c_div10
a = a_div10
e10 += 1
end
end
if b_allzero && b_lastdigit == 5 && iseven(b)
b_lastdigit = UInt8(4)
end
roundup = (b == a && (!mf_iseven || !a_allzero)) || b_lastdigit >= 5
else
# b) specialized for common case (99% Float64, 96% Float32)
roundup = b_lastdigit >= 5
c_div100 = div(c, 100)
a_div100 = div(a, 100)
if c_div100 > a_div100
b_div100 = div(b, 100)
b_mod100 = (b % UInt32) - UInt32(100) * (b_div100 % UInt32)
roundup = b_mod100 >= 50
b = b_div100
c = c_div100
a = a_div100
e10 += 2
end
while true
c_div10 = div(c, 10)
a_div10 = div(a, 10)
if c_div10 <= a_div10
break
end
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
roundup = b_mod10 >= 5
b = b_div10
c = c_div10
a = a_div10
e10 += 1
end
roundup = (b == a || roundup)
end
if maxsignif !== nothing && b > maxsignif
# reduce to max significant digits
while true
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
if b <= maxsignif
break
end
b = b_div10
roundup = (b_allzero && iseven(b)) ? b_mod10 > 5 : b_mod10 >= 5
b_allzero &= b_mod10 == 0
e10 += 1
end
b = b + roundup
# remove trailing zeros
while true
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
if b_mod10 != 0
break
end
b = b_div10
e10 += 1
end
else
b = b + roundup
end
else
# c) specialized f an integer < 2^53
b = mf >> -ef
e10 = 0
if maxsignif !== nothing && b > maxsignif
b_allzero = true
# reduce to max significant digits
while true
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
if b <= maxsignif
break
end
b = b_div10
roundup = (b_allzero && iseven(b)) ? b_mod10 > 5 : b_mod10 >= 5
b_allzero &= b_mod10 == 0
e10 += 1
end
b = b + roundup
end
while true
b_div10 = div(b, 10)
b_mod10 = (b % UInt32) - UInt32(10) * (b_div10 % UInt32)
if b_mod10 != 0
break
end
b = b_div10
e10 += 1
end
end
return b, e10
end
@inline function writeshortest(buf::Vector{UInt8}, pos, x::T,
plus=false, space=false, hash=true,
precision=-1, expchar=UInt8('e'), padexp=false, decchar=UInt8('.'),
typed=false, compact=false) where {T}
@assert 0 < pos <= length(buf)
neg = signbit(x)
# special cases
if x == 0
if neg
buf[pos] = UInt8('-')
pos += 1
elseif plus
buf[pos] = UInt8('+')
pos += 1
elseif space
buf[pos] = UInt8(' ')
pos += 1
end
buf[pos] = UInt8('0')
pos += 1
if hash
buf[pos] = decchar
pos += 1
end
if precision == -1
buf[pos] = UInt8('0')
pos += 1
if typed && x isa Float32
buf[pos] = UInt8('f')
buf[pos + 1] = UInt8('0')
pos += 2
end
return pos
end
while precision > 1
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
if typed && x isa Float32
buf[pos] = UInt8('f')
buf[pos + 1] = UInt8('0')
pos += 2
end
return pos
elseif isnan(x)
buf[pos] = UInt8('N')
buf[pos + 1] = UInt8('a')
buf[pos + 2] = UInt8('N')
if typed
if x isa Float32
buf[pos + 3] = UInt8('3')
buf[pos + 4] = UInt8('2')
elseif x isa Float16
buf[pos + 3] = UInt8('1')
buf[pos + 4] = UInt8('6')
end
end
return pos + 3 + (typed && x isa Union{Float32, Float16} ? 2 : 0)
elseif !isfinite(x)
if neg
buf[pos] = UInt8('-')
end
buf[pos + neg] = UInt8('I')
buf[pos + neg + 1] = UInt8('n')
buf[pos + neg + 2] = UInt8('f')
if typed
if x isa Float32
buf[pos + neg + 3] = UInt8('3')
buf[pos + neg + 4] = UInt8('2')
elseif x isa Float16
buf[pos + neg + 3] = UInt8('1')
buf[pos + neg + 4] = UInt8('6')
end
end
return pos + neg + 3 + (typed && x isa Union{Float32, Float16} ? 2 : 0)
end
output, nexp = reduce_shortest(x, compact ? 999_999 : nothing)
if typed && x isa Float16
buf[pos] = UInt8('F')
buf[pos + 1] = UInt8('l')
buf[pos + 2] = UInt8('o')
buf[pos + 3] = UInt8('a')
buf[pos + 4] = UInt8('t')
buf[pos + 5] = UInt8('1')
buf[pos + 6] = UInt8('6')
buf[pos + 7] = UInt8('(')
pos += 8
end
if neg
buf[pos] = UInt8('-')
pos += 1
elseif plus
buf[pos] = UInt8('+')
pos += 1
elseif space
buf[pos] = UInt8(' ')
pos += 1
end
olength = decimallength(output)
exp_form = true
pt = nexp + olength
if -4 < pt <= (precision == -1 ? (T == Float16 ? 3 : 6) : precision) &&
!(pt >= olength && abs(mod(x + 0.05, 10^(pt - olength)) - 0.05) > 0.05)
exp_form = false
if pt <= 0
buf[pos] = UInt8('0')
pos += 1
buf[pos] = decchar
pos += 1
for _ = 1:abs(pt)
buf[pos] = UInt8('0')
pos += 1
end
# elseif pt >= olength
# nothing to do at this point
# else
# nothing to do at this point
end
else
pos += 1
end
i = 0
ptr = pointer(buf)
ptr2 = pointer(DIGIT_TABLE)
if (output >> 32) != 0
q = output ÷ 100000000
output2 = (output % UInt32) - UInt32(100000000) * (q % UInt32)
output = q
c = output2 % UInt32(10000)
output2 = div(output2, UInt32(10000))
d = output2 % UInt32(10000)
c0 = (c % 100) << 1
c1 = (c ÷ 100) << 1
d0 = (d % 100) << 1
d1 = (d ÷ 100) << 1
memcpy(ptr, pos + olength - 2, ptr2, c0 + 1, 2)
memcpy(ptr, pos + olength - 4, ptr2, c1 + 1, 2)
memcpy(ptr, pos + olength - 6, ptr2, d0 + 1, 2)
memcpy(ptr, pos + olength - 8, ptr2, d1 + 1, 2)
i += 8
end
output2 = output % UInt32
while output2 >= 10000
c = output2 % UInt32(10000)
output2 = div(output2, UInt32(10000))
c0 = (c % 100) << 1
c1 = (c ÷ 100) << 1
memcpy(ptr, pos + olength - i - 2, ptr2, c0 + 1, 2)
memcpy(ptr, pos + olength - i - 4, ptr2, c1 + 1, 2)
i += 4
end
if output2 >= 100
c = (output2 % UInt32(100)) << 1
output2 = div(output2, UInt32(100))
memcpy(ptr, pos + olength - i - 2, ptr2, c + 1, 2)
i += 2
end
if output2 >= 10
c = output2 << 1
buf[pos + 1] = DIGIT_TABLE[c + 2]
buf[pos - exp_form] = DIGIT_TABLE[c + 1]
else
buf[pos - exp_form] = UInt8('0') + (output2 % UInt8)
end
if !exp_form
if pt <= 0
pos += olength
precision -= olength
while hash && precision > 0
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
elseif pt >= olength
pos += olength
precision -= olength
for _ = 1:nexp
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
if hash
buf[pos] = decchar
pos += 1
if precision < 0
buf[pos] = UInt8('0')
pos += 1
end
while precision > 0
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
end
else
pointoff = olength - abs(nexp)
memmove(ptr, pos + pointoff + 1, ptr, pos + pointoff, olength - pointoff + 1)
buf[pos + pointoff] = decchar
pos += olength + 1
precision -= olength
while hash && precision > 0
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
end
if typed && x isa Float32
buf[pos] = UInt8('f')
buf[pos + 1] = UInt8('0')
pos += 2
end
else
if olength > 1 || hash
buf[pos] = decchar
pos += olength
precision -= olength
end
if hash && olength == 1
buf[pos] = UInt8('0')
pos += 1
end
while hash && precision > 0
buf[pos] = UInt8('0')
pos += 1
precision -= 1
end
buf[pos] = expchar
pos += 1
exp2 = nexp + olength - 1
if exp2 < 0
buf[pos] = UInt8('-')
pos += 1
exp2 = -exp2
elseif padexp
buf[pos] = UInt8('+')
pos += 1
end
if exp2 >= 100
c = exp2 % 10
memcpy(ptr, pos, ptr2, 2 * div(exp2, 10) + 1, 2)
buf[pos + 2] = UInt8('0') + (c % UInt8)
pos += 3
elseif exp2 >= 10
memcpy(ptr, pos, ptr2, 2 * exp2 + 1, 2)
pos += 2
else
if padexp
buf[pos] = UInt8('0')
pos += 1
end
buf[pos] = UInt8('0') + (exp2 % UInt8)
pos += 1
end
end
if typed && x isa Float16
buf[pos] = UInt8(')')
pos += 1
end
return pos
end
|
