| 1 | // Copyright (c) 2014 The mathutil Authors. All rights reserved.
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| 2 | // Use of this source code is governed by a BSD-style
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| 3 | // license that can be found in the LICENSE file.
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| 4 |
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| 5 | package mathutil // import "modernc.org/mathutil"
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| 6 |
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| 7 | import (
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| 8 | "fmt"
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| 9 | "math"
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| 10 | "math/big"
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| 11 | )
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| 12 |
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| 13 | // FC32 is a full cycle PRNG covering the 32 bit signed integer range.
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| 14 | // In contrast to full cycle generators shown at e.g. http://en.wikipedia.org/wiki/Full_cycle,
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| 15 | // this code doesn't produce values at constant delta (mod cycle length).
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| 16 | // The 32 bit limit is per this implementation, the algorithm used has no intrinsic limit on the cycle size.
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| 17 | // Properties include:
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| 18 | // - Adjustable limits on creation (hi, lo).
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| 19 | // - Positionable/randomly accessible (Pos, Seek).
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| 20 | // - Repeatable (deterministic).
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| 21 | // - Can run forward or backward (Next, Prev).
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| 22 | // - For a billion numbers cycle the Next/Prev PRN can be produced in cca 100-150ns.
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| 23 | // That's like 5-10 times slower compared to PRNs generated using the (non FC) rand package.
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| 24 | type FC32 struct {
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| 25 | cycle int64 // On average: 3 * delta / 2, (HQ: 2 * delta)
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| 26 | delta int64 // hi - lo
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| 27 | factors [][]int64 // This trades some space for hopefully a bit of speed (multiple adding vs multiplying).
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| 28 | lo int
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| 29 | mods []int // pos % set
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| 30 | pos int64 // Within cycle.
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| 31 | primes []int64 // Ordered. ∏ primes == cycle.
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| 32 | set []int64 // Reordered primes (magnitude order bases) according to seed.
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| 33 | }
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| 34 |
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| 35 | // NewFC32 returns a newly created FC32 adjusted for the closed interval [lo, hi] or an Error if any.
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| 36 | // If hq == true then trade some generation time for improved (pseudo)randomness.
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| 37 | func NewFC32(lo, hi int, hq bool) (r *FC32, err error) {
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| 38 | if lo > hi {
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| 39 | return nil, fmt.Errorf("invalid range %d > %d", lo, hi)
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| 40 | }
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| 41 |
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| 42 | if uint64(hi)-uint64(lo) > math.MaxUint32 {
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| 43 | return nil, fmt.Errorf("range out of int32 limits %d, %d", lo, hi)
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| 44 | }
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| 45 |
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| 46 | delta := int64(hi) - int64(lo)
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| 47 | // Find the primorial covering whole delta
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| 48 | n, set, p := int64(1), []int64{}, uint32(2)
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| 49 | if hq {
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| 50 | p++
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| 51 | }
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| 52 | for {
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| 53 | set = append(set, int64(p))
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| 54 | n *= int64(p)
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| 55 | if n > delta {
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| 56 | break
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| 57 | }
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| 58 | p, _ = NextPrime(p)
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| 59 | }
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| 60 |
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| 61 | // Adjust the set so n ∊ [delta, 2 * delta] (HQ: [delta, 3 * delta])
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| 62 | // while keeping the cardinality of the set (correlates with the statistic "randomness quality")
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| 63 | // at max, i.e. discard atmost one member.
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| 64 | i := -1 // no candidate prime
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| 65 | if n > 2*(delta+1) {
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| 66 | for j, p := range set {
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| 67 | q := n / p
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| 68 | if q < delta+1 {
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| 69 | break
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| 70 | }
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| 71 |
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| 72 | i = j // mark the highest candidate prime set index
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| 73 | }
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| 74 | }
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| 75 | if i >= 0 { // shrink the inner cycle
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| 76 | n = n / set[i]
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| 77 | set = delete(set, i)
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| 78 | }
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| 79 | r = &FC32{
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| 80 | cycle: n,
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| 81 | delta: delta,
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| 82 | factors: make([][]int64, len(set)),
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| 83 | lo: lo,
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| 84 | mods: make([]int, len(set)),
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| 85 | primes: set,
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| 86 | }
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| 87 | r.Seed(1) // the default seed should be always non zero
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| 88 | return
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| 89 | }
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| 90 |
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| 91 | // Cycle reports the length of the inner FCPRNG cycle.
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| 92 | // Cycle is atmost the double (HQ: triple) of the generator period (hi - lo + 1).
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| 93 | func (r *FC32) Cycle() int64 {
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| 94 | return r.cycle
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| 95 | }
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| 96 |
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| 97 | // Next returns the first PRN after Pos.
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| 98 | func (r *FC32) Next() int {
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| 99 | return r.step(1)
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| 100 | }
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| 101 |
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| 102 | // Pos reports the current position within the inner cycle.
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| 103 | func (r *FC32) Pos() int64 {
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| 104 | return r.pos
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| 105 | }
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| 106 |
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| 107 | // Prev return the first PRN before Pos.
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| 108 | func (r *FC32) Prev() int {
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| 109 | return r.step(-1)
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| 110 | }
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| 111 |
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| 112 | // Seed uses the provided seed value to initialize the generator to a deterministic state.
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| 113 | // A zero seed produces a "canonical" generator with worse randomness than for most non zero seeds.
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| 114 | // Still, the FC property holds for any seed value.
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| 115 | func (r *FC32) Seed(seed int64) {
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| 116 | u := uint64(seed)
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| 117 | r.set = mix(r.primes, &u)
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| 118 | n := int64(1)
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| 119 | for i, p := range r.set {
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| 120 | k := make([]int64, p)
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| 121 | v := int64(0)
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| 122 | for j := range k {
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| 123 | k[j] = v
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| 124 | v += n
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| 125 | }
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| 126 | n *= p
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| 127 | r.factors[i] = mix(k, &u)
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| 128 | }
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| 129 | }
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| 130 |
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| 131 | // Seek sets Pos to |pos| % Cycle.
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| 132 | func (r *FC32) Seek(pos int64) { //vet:ignore
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| 133 | if pos < 0 {
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| 134 | pos = -pos
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| 135 | }
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| 136 | pos %= r.cycle
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| 137 | r.pos = pos
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| 138 | for i, p := range r.set {
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| 139 | r.mods[i] = int(pos % p)
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| 140 | }
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| 141 | }
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| 142 |
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| 143 | func (r *FC32) step(dir int) int {
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| 144 | for { // avg loops per step: 3/2 (HQ: 2)
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| 145 | y := int64(0)
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| 146 | pos := r.pos
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| 147 | pos += int64(dir)
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| 148 | switch {
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| 149 | case pos < 0:
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| 150 | pos = r.cycle - 1
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| 151 | case pos >= r.cycle:
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| 152 | pos = 0
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| 153 | }
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| 154 | r.pos = pos
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| 155 | for i, mod := range r.mods {
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| 156 | mod += dir
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| 157 | p := int(r.set[i])
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| 158 | switch {
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| 159 | case mod < 0:
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| 160 | mod = p - 1
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| 161 | case mod >= p:
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| 162 | mod = 0
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| 163 | }
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| 164 | r.mods[i] = mod
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| 165 | y += r.factors[i][mod]
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| 166 | }
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| 167 | if y <= r.delta {
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| 168 | return int(y) + r.lo
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| 169 | }
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| 170 | }
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| 171 | }
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| 172 |
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| 173 | func delete(set []int64, i int) (y []int64) {
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| 174 | for j, v := range set {
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| 175 | if j != i {
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| 176 | y = append(y, v)
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| 177 | }
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| 178 | }
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| 179 | return
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| 180 | }
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| 181 |
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| 182 | func mix(set []int64, seed *uint64) (y []int64) {
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| 183 | for len(set) != 0 {
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| 184 | *seed = rol(*seed)
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| 185 | i := int(*seed % uint64(len(set)))
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| 186 | y = append(y, set[i])
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| 187 | set = delete(set, i)
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| 188 | }
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| 189 | return
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| 190 | }
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| 191 |
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| 192 | func rol(u uint64) (y uint64) {
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| 193 | y = u << 1
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| 194 | if int64(u) < 0 {
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| 195 | y |= 1
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| 196 | }
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| 197 | return
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| 198 | }
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| 199 |
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| 200 | // FCBig is a full cycle PRNG covering ranges outside of the int32 limits.
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| 201 | // For more info see the FC32 docs.
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| 202 | // Next/Prev PRN on a 1e15 cycle can be produced in about 2 µsec.
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| 203 | type FCBig struct {
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| 204 | cycle *big.Int // On average: 3 * delta / 2, (HQ: 2 * delta)
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| 205 | delta *big.Int // hi - lo
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| 206 | factors [][]*big.Int // This trades some space for hopefully a bit of speed (multiple adding vs multiplying).
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| 207 | lo *big.Int
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| 208 | mods []int // pos % set
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| 209 | pos *big.Int // Within cycle.
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| 210 | primes []int64 // Ordered. ∏ primes == cycle.
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| 211 | set []int64 // Reordered primes (magnitude order bases) according to seed.
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| 212 | }
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| 213 |
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| 214 | // NewFCBig returns a newly created FCBig adjusted for the closed interval [lo, hi] or an Error if any.
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| 215 | // If hq == true then trade some generation time for improved (pseudo)randomness.
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| 216 | func NewFCBig(lo, hi *big.Int, hq bool) (r *FCBig, err error) {
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| 217 | if lo.Cmp(hi) > 0 {
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| 218 | return nil, fmt.Errorf("invalid range %d > %d", lo, hi)
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| 219 | }
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| 220 |
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| 221 | delta := big.NewInt(0)
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| 222 | delta.Add(delta, hi).Sub(delta, lo)
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| 223 |
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| 224 | // Find the primorial covering whole delta
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| 225 | n, set, pp, p := big.NewInt(1), []int64{}, big.NewInt(0), uint32(2)
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| 226 | if hq {
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| 227 | p++
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| 228 | }
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| 229 | for {
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| 230 | set = append(set, int64(p))
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| 231 | pp.SetInt64(int64(p))
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| 232 | n.Mul(n, pp)
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| 233 | if n.Cmp(delta) > 0 {
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| 234 | break
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| 235 | }
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| 236 | p, _ = NextPrime(p)
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| 237 | }
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| 238 |
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| 239 | // Adjust the set so n ∊ [delta, 2 * delta] (HQ: [delta, 3 * delta])
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| 240 | // while keeping the cardinality of the set (correlates with the statistic "randomness quality")
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| 241 | // at max, i.e. discard atmost one member.
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| 242 | dd1 := big.NewInt(1)
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| 243 | dd1.Add(dd1, delta)
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| 244 | dd2 := big.NewInt(0)
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| 245 | dd2.Lsh(dd1, 1)
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| 246 | i := -1 // no candidate prime
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| 247 | if n.Cmp(dd2) > 0 {
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| 248 | q := big.NewInt(0)
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| 249 | for j, p := range set {
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| 250 | pp.SetInt64(p)
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| 251 | q.Set(n)
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| 252 | q.Div(q, pp)
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| 253 | if q.Cmp(dd1) < 0 {
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| 254 | break
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| 255 | }
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| 256 |
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| 257 | i = j // mark the highest candidate prime set index
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| 258 | }
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| 259 | }
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| 260 | if i >= 0 { // shrink the inner cycle
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| 261 | pp.SetInt64(set[i])
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| 262 | n.Div(n, pp)
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| 263 | set = delete(set, i)
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| 264 | }
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| 265 | r = &FCBig{
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| 266 | cycle: n,
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| 267 | delta: delta,
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| 268 | factors: make([][]*big.Int, len(set)),
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| 269 | lo: lo,
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| 270 | mods: make([]int, len(set)),
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| 271 | pos: big.NewInt(0),
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| 272 | primes: set,
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| 273 | }
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| 274 | r.Seed(1) // the default seed should be always non zero
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| 275 | return
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| 276 | }
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| 277 |
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| 278 | // Cycle reports the length of the inner FCPRNG cycle.
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| 279 | // Cycle is atmost the double (HQ: triple) of the generator period (hi - lo + 1).
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| 280 | func (r *FCBig) Cycle() *big.Int {
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| 281 | return r.cycle
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| 282 | }
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| 283 |
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| 284 | // Next returns the first PRN after Pos.
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| 285 | func (r *FCBig) Next() *big.Int {
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| 286 | return r.step(1)
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| 287 | }
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| 288 |
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| 289 | // Pos reports the current position within the inner cycle.
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| 290 | func (r *FCBig) Pos() *big.Int {
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| 291 | return r.pos
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| 292 | }
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| 293 |
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| 294 | // Prev return the first PRN before Pos.
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| 295 | func (r *FCBig) Prev() *big.Int {
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| 296 | return r.step(-1)
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| 297 | }
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| 298 |
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| 299 | // Seed uses the provided seed value to initialize the generator to a deterministic state.
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| 300 | // A zero seed produces a "canonical" generator with worse randomness than for most non zero seeds.
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| 301 | // Still, the FC property holds for any seed value.
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| 302 | func (r *FCBig) Seed(seed int64) {
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| 303 | u := uint64(seed)
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| 304 | r.set = mix(r.primes, &u)
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| 305 | n := big.NewInt(1)
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| 306 | v := big.NewInt(0)
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| 307 | pp := big.NewInt(0)
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| 308 | for i, p := range r.set {
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| 309 | k := make([]*big.Int, p)
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| 310 | v.SetInt64(0)
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| 311 | for j := range k {
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| 312 | k[j] = big.NewInt(0)
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| 313 | k[j].Set(v)
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| 314 | v.Add(v, n)
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| 315 | }
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| 316 | pp.SetInt64(p)
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| 317 | n.Mul(n, pp)
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| 318 | r.factors[i] = mixBig(k, &u)
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| 319 | }
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| 320 | }
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| 321 |
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| 322 | // Seek sets Pos to |pos| % Cycle.
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| 323 | func (r *FCBig) Seek(pos *big.Int) {
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| 324 | r.pos.Set(pos)
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| 325 | r.pos.Abs(r.pos)
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| 326 | r.pos.Mod(r.pos, r.cycle)
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| 327 | mod := big.NewInt(0)
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| 328 | pp := big.NewInt(0)
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| 329 | for i, p := range r.set {
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| 330 | pp.SetInt64(p)
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| 331 | r.mods[i] = int(mod.Mod(r.pos, pp).Int64())
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| 332 | }
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| 333 | }
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| 334 |
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| 335 | func (r *FCBig) step(dir int) (y *big.Int) {
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| 336 | y = big.NewInt(0)
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| 337 | d := big.NewInt(int64(dir))
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| 338 | for { // avg loops per step: 3/2 (HQ: 2)
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| 339 | r.pos.Add(r.pos, d)
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| 340 | switch {
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| 341 | case r.pos.Sign() < 0:
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| 342 | r.pos.Add(r.pos, r.cycle)
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| 343 | case r.pos.Cmp(r.cycle) >= 0:
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| 344 | r.pos.SetInt64(0)
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| 345 | }
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| 346 | for i, mod := range r.mods {
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| 347 | mod += dir
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| 348 | p := int(r.set[i])
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| 349 | switch {
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| 350 | case mod < 0:
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| 351 | mod = p - 1
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| 352 | case mod >= p:
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| 353 | mod = 0
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| 354 | }
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| 355 | r.mods[i] = mod
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| 356 | y.Add(y, r.factors[i][mod])
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| 357 | }
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| 358 | if y.Cmp(r.delta) <= 0 {
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| 359 | y.Add(y, r.lo)
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| 360 | return
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| 361 | }
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| 362 | y.SetInt64(0)
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| 363 | }
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| 364 | }
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| 365 |
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| 366 | func deleteBig(set []*big.Int, i int) (y []*big.Int) {
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| 367 | for j, v := range set {
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| 368 | if j != i {
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| 369 | y = append(y, v)
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| 370 | }
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| 371 | }
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| 372 | return
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| 373 | }
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| 374 |
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| 375 | func mixBig(set []*big.Int, seed *uint64) (y []*big.Int) {
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| 376 | for len(set) != 0 {
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| 377 | *seed = rol(*seed)
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| 378 | i := int(*seed % uint64(len(set)))
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| 379 | y = append(y, set[i])
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| 380 | set = deleteBig(set, i)
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| 381 | }
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| 382 | return
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| 383 | }
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