384 lines
9.1 KiB
Go
384 lines
9.1 KiB
Go
// Copyright (c) 2014 The mathutil Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package mathutil // import "modernc.org/mathutil"
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import (
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"fmt"
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"math"
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"math/big"
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)
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// FC32 is a full cycle PRNG covering the 32 bit signed integer range.
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// In contrast to full cycle generators shown at e.g. http://en.wikipedia.org/wiki/Full_cycle,
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// this code doesn't produce values at constant delta (mod cycle length).
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// The 32 bit limit is per this implementation, the algorithm used has no intrinsic limit on the cycle size.
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// Properties include:
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// - Adjustable limits on creation (hi, lo).
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// - Positionable/randomly accessible (Pos, Seek).
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// - Repeatable (deterministic).
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// - Can run forward or backward (Next, Prev).
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// - For a billion numbers cycle the Next/Prev PRN can be produced in cca 100-150ns.
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// That's like 5-10 times slower compared to PRNs generated using the (non FC) rand package.
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type FC32 struct {
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cycle int64 // On average: 3 * delta / 2, (HQ: 2 * delta)
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delta int64 // hi - lo
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factors [][]int64 // This trades some space for hopefully a bit of speed (multiple adding vs multiplying).
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lo int
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mods []int // pos % set
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pos int64 // Within cycle.
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primes []int64 // Ordered. ∏ primes == cycle.
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set []int64 // Reordered primes (magnitude order bases) according to seed.
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}
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// NewFC32 returns a newly created FC32 adjusted for the closed interval [lo, hi] or an Error if any.
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// If hq == true then trade some generation time for improved (pseudo)randomness.
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func NewFC32(lo, hi int, hq bool) (r *FC32, err error) {
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if lo > hi {
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return nil, fmt.Errorf("invalid range %d > %d", lo, hi)
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}
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if uint64(hi)-uint64(lo) > math.MaxUint32 {
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return nil, fmt.Errorf("range out of int32 limits %d, %d", lo, hi)
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}
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delta := int64(hi) - int64(lo)
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// Find the primorial covering whole delta
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n, set, p := int64(1), []int64{}, uint32(2)
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if hq {
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p++
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}
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for {
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set = append(set, int64(p))
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n *= int64(p)
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if n > delta {
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break
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}
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p, _ = NextPrime(p)
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}
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// Adjust the set so n ∊ [delta, 2 * delta] (HQ: [delta, 3 * delta])
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// while keeping the cardinality of the set (correlates with the statistic "randomness quality")
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// at max, i.e. discard atmost one member.
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i := -1 // no candidate prime
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if n > 2*(delta+1) {
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for j, p := range set {
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q := n / p
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if q < delta+1 {
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break
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}
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i = j // mark the highest candidate prime set index
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}
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}
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if i >= 0 { // shrink the inner cycle
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n = n / set[i]
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set = delete(set, i)
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}
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r = &FC32{
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cycle: n,
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delta: delta,
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factors: make([][]int64, len(set)),
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lo: lo,
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mods: make([]int, len(set)),
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primes: set,
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}
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r.Seed(1) // the default seed should be always non zero
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return
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}
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// Cycle reports the length of the inner FCPRNG cycle.
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// Cycle is atmost the double (HQ: triple) of the generator period (hi - lo + 1).
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func (r *FC32) Cycle() int64 {
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return r.cycle
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}
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// Next returns the first PRN after Pos.
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func (r *FC32) Next() int {
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return r.step(1)
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}
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// Pos reports the current position within the inner cycle.
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func (r *FC32) Pos() int64 {
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return r.pos
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}
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// Prev return the first PRN before Pos.
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func (r *FC32) Prev() int {
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return r.step(-1)
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}
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// Seed uses the provided seed value to initialize the generator to a deterministic state.
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// A zero seed produces a "canonical" generator with worse randomness than for most non zero seeds.
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// Still, the FC property holds for any seed value.
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func (r *FC32) Seed(seed int64) {
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u := uint64(seed)
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r.set = mix(r.primes, &u)
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n := int64(1)
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for i, p := range r.set {
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k := make([]int64, p)
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v := int64(0)
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for j := range k {
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k[j] = v
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v += n
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}
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n *= p
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r.factors[i] = mix(k, &u)
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}
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}
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// Seek sets Pos to |pos| % Cycle.
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func (r *FC32) Seek(pos int64) { //vet:ignore
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if pos < 0 {
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pos = -pos
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}
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pos %= r.cycle
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r.pos = pos
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for i, p := range r.set {
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r.mods[i] = int(pos % p)
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}
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}
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func (r *FC32) step(dir int) int {
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for { // avg loops per step: 3/2 (HQ: 2)
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y := int64(0)
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pos := r.pos
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pos += int64(dir)
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switch {
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case pos < 0:
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pos = r.cycle - 1
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case pos >= r.cycle:
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pos = 0
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}
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r.pos = pos
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for i, mod := range r.mods {
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mod += dir
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p := int(r.set[i])
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switch {
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case mod < 0:
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mod = p - 1
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case mod >= p:
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mod = 0
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}
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r.mods[i] = mod
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y += r.factors[i][mod]
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}
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if y <= r.delta {
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return int(y) + r.lo
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}
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}
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}
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func delete(set []int64, i int) (y []int64) {
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for j, v := range set {
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if j != i {
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y = append(y, v)
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}
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}
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return
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}
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func mix(set []int64, seed *uint64) (y []int64) {
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for len(set) != 0 {
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*seed = rol(*seed)
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i := int(*seed % uint64(len(set)))
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y = append(y, set[i])
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set = delete(set, i)
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}
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return
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}
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func rol(u uint64) (y uint64) {
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y = u << 1
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if int64(u) < 0 {
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y |= 1
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}
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return
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}
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// FCBig is a full cycle PRNG covering ranges outside of the int32 limits.
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// For more info see the FC32 docs.
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// Next/Prev PRN on a 1e15 cycle can be produced in about 2 µsec.
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type FCBig struct {
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cycle *big.Int // On average: 3 * delta / 2, (HQ: 2 * delta)
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delta *big.Int // hi - lo
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factors [][]*big.Int // This trades some space for hopefully a bit of speed (multiple adding vs multiplying).
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lo *big.Int
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mods []int // pos % set
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pos *big.Int // Within cycle.
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primes []int64 // Ordered. ∏ primes == cycle.
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set []int64 // Reordered primes (magnitude order bases) according to seed.
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}
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// NewFCBig returns a newly created FCBig adjusted for the closed interval [lo, hi] or an Error if any.
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// If hq == true then trade some generation time for improved (pseudo)randomness.
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func NewFCBig(lo, hi *big.Int, hq bool) (r *FCBig, err error) {
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if lo.Cmp(hi) > 0 {
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return nil, fmt.Errorf("invalid range %d > %d", lo, hi)
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}
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delta := big.NewInt(0)
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delta.Add(delta, hi).Sub(delta, lo)
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// Find the primorial covering whole delta
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n, set, pp, p := big.NewInt(1), []int64{}, big.NewInt(0), uint32(2)
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if hq {
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p++
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}
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for {
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set = append(set, int64(p))
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pp.SetInt64(int64(p))
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n.Mul(n, pp)
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if n.Cmp(delta) > 0 {
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break
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}
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p, _ = NextPrime(p)
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}
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// Adjust the set so n ∊ [delta, 2 * delta] (HQ: [delta, 3 * delta])
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// while keeping the cardinality of the set (correlates with the statistic "randomness quality")
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// at max, i.e. discard atmost one member.
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dd1 := big.NewInt(1)
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dd1.Add(dd1, delta)
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dd2 := big.NewInt(0)
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dd2.Lsh(dd1, 1)
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i := -1 // no candidate prime
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if n.Cmp(dd2) > 0 {
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q := big.NewInt(0)
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for j, p := range set {
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pp.SetInt64(p)
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q.Set(n)
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q.Div(q, pp)
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if q.Cmp(dd1) < 0 {
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break
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}
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i = j // mark the highest candidate prime set index
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}
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}
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if i >= 0 { // shrink the inner cycle
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pp.SetInt64(set[i])
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n.Div(n, pp)
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set = delete(set, i)
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}
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r = &FCBig{
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cycle: n,
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delta: delta,
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factors: make([][]*big.Int, len(set)),
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lo: lo,
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mods: make([]int, len(set)),
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pos: big.NewInt(0),
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primes: set,
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}
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r.Seed(1) // the default seed should be always non zero
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return
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}
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// Cycle reports the length of the inner FCPRNG cycle.
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// Cycle is atmost the double (HQ: triple) of the generator period (hi - lo + 1).
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func (r *FCBig) Cycle() *big.Int {
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return r.cycle
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}
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// Next returns the first PRN after Pos.
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func (r *FCBig) Next() *big.Int {
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return r.step(1)
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}
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// Pos reports the current position within the inner cycle.
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func (r *FCBig) Pos() *big.Int {
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return r.pos
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}
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// Prev return the first PRN before Pos.
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func (r *FCBig) Prev() *big.Int {
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return r.step(-1)
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}
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// Seed uses the provided seed value to initialize the generator to a deterministic state.
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// A zero seed produces a "canonical" generator with worse randomness than for most non zero seeds.
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// Still, the FC property holds for any seed value.
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func (r *FCBig) Seed(seed int64) {
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u := uint64(seed)
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r.set = mix(r.primes, &u)
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n := big.NewInt(1)
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v := big.NewInt(0)
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pp := big.NewInt(0)
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for i, p := range r.set {
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k := make([]*big.Int, p)
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v.SetInt64(0)
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for j := range k {
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k[j] = big.NewInt(0)
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k[j].Set(v)
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v.Add(v, n)
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}
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pp.SetInt64(p)
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n.Mul(n, pp)
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r.factors[i] = mixBig(k, &u)
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}
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}
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// Seek sets Pos to |pos| % Cycle.
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func (r *FCBig) Seek(pos *big.Int) {
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r.pos.Set(pos)
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r.pos.Abs(r.pos)
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r.pos.Mod(r.pos, r.cycle)
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mod := big.NewInt(0)
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pp := big.NewInt(0)
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for i, p := range r.set {
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pp.SetInt64(p)
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r.mods[i] = int(mod.Mod(r.pos, pp).Int64())
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}
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}
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func (r *FCBig) step(dir int) (y *big.Int) {
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y = big.NewInt(0)
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d := big.NewInt(int64(dir))
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for { // avg loops per step: 3/2 (HQ: 2)
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r.pos.Add(r.pos, d)
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switch {
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case r.pos.Sign() < 0:
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r.pos.Add(r.pos, r.cycle)
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case r.pos.Cmp(r.cycle) >= 0:
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r.pos.SetInt64(0)
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}
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for i, mod := range r.mods {
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mod += dir
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p := int(r.set[i])
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switch {
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case mod < 0:
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mod = p - 1
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case mod >= p:
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mod = 0
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}
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r.mods[i] = mod
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y.Add(y, r.factors[i][mod])
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}
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if y.Cmp(r.delta) <= 0 {
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y.Add(y, r.lo)
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return
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}
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y.SetInt64(0)
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}
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}
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func deleteBig(set []*big.Int, i int) (y []*big.Int) {
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for j, v := range set {
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if j != i {
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y = append(y, v)
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}
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}
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return
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}
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func mixBig(set []*big.Int, seed *uint64) (y []*big.Int) {
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for len(set) != 0 {
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*seed = rol(*seed)
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i := int(*seed % uint64(len(set)))
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y = append(y, set[i])
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set = deleteBig(set, i)
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}
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return
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}
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