75 lines
1.9 KiB
Lua
75 lines
1.9 KiB
Lua
--- Multi-precision arithmetic on 264-bit integers.
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--
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-- :::note Internal Module
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-- This module is meant for internal use within the library. Its API is unstable
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-- and subject to change without major version bumps.
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-- :::
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--
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-- <br />
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--
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-- @module[kind=internal] internal.mp
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--
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--- Carries a number in base 2²⁴.
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--
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-- @tparam {number...} a A number 0 <= a < 2 ^ (24 × (#a + 1)) as limbs in
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-- [-2⁵²..2⁵²].
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-- @treturn {number...} a as #a + 1 limbs in [0..2²⁴).
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--
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local function carry(a)
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local c = {unpack(a)}
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c[#c + 1] = 0
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for i = 1, #c - 1 do
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local val = c[i]
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local rem = val % 2 ^ 24
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local quot = (val - rem) / 2 ^ 24
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c[i + 1] = c[i + 1] + quot
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c[i] = rem
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end
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return c
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end
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--- Adds two numbers.
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--
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-- @tparam {number...} a An array limbs in [0..2²⁴).
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-- @tparam {number...} b An array of #a limbs in [0..2²⁴).
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-- @treturn {number...} a + b as #a + 1 limbs in [0..2²⁴).
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--
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local function add(a, b)
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local c = {}
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for i = 1, #a do
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c[i] = a[i] + b[i]
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end
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-- c's limbs fit in [-2²⁵..2²⁵], since addition adds at most one bit.
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return carry(c)
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end
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--- Multiplies two numbers.
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--
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-- @tparam {number...} a An array of 11 limbs in [0..2²⁴).
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-- @tparam {number...} b An array of 11 limbs in [0..2²⁴).
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-- @treturn {number...} a × b as 22 limbs in [0..2²⁴).
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--
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local function mul(a, b)
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local c = {}
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for i = 1, 21 do c[i] = 0 end
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for i = 1, 11 do
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for j = 1, 11 do
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local k = i + j - 1
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c[k] = c[k] + a[i] * b[j]
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end
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end
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-- {a, b} < 2²⁶⁴ means that c < 2⁵²⁸ = 2 ^ (24 × (21 + 1)).
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-- c's limbs are smaller than 2⁴⁸ × 11 < 2⁵², since multiplication doubles
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-- bit length, and 11 multiplied limbs are added together.
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return carry(c)
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end
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return {
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carry = carry,
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add = add,
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mul = mul,
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}
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