gtsocial-umbx

mathutil.go (30753B)

---

 // Copyright (c) 2014 The mathutil Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style
 // license that can be found in the LICENSE file.
 
 // Package mathutil provides utilities supplementing the standard 'math' and
 // 'math/rand' packages.
 //
 // Release history and compatibility issues
 //
 // 2020-12-20 v1.2.1 fixes MulOverflowInt64.
 //
 // 2020-12-19 Added {Add,Sub,Mul}OverflowInt{8,16,32,64}
 //
 // 2018-10-21 Added BinaryLog
 //
 // 2018-04-25: New functions for determining Max/Min of nullable values. Ex:
 //  func MaxPtr(a, b *int) *int {
 //  func MinPtr(a, b *int) *int {
 //  func MaxBytePtr(a, b *byte) *byte {
 //  func MinBytePtr(a, b *byte) *byte {
 //  ...
 //
 // 2017-10-14: New variadic functions for Max/Min. Ex:
 //  func MaxVal(val int, vals ...int) int {
 //  func MinVal(val int, vals ...int) int {
 //  func MaxByteVal(val byte, vals ...byte) byte {
 //  func MinByteVal(val byte, vals ...byte) byte {
 //  ...
 //
 // 2016-10-10: New functions QuadPolyDiscriminant and QuadPolyFactors.
 //
 // 2013-12-13: The following functions have been REMOVED
 //
 // 	func Uint64ToBigInt(n uint64) *big.Int
 // 	func Uint64FromBigInt(n *big.Int) (uint64, bool)
 //
 // 2013-05-13: The following functions are now DEPRECATED
 //
 // 	func Uint64ToBigInt(n uint64) *big.Int
 // 	func Uint64FromBigInt(n *big.Int) (uint64, bool)
 //
 // These functions will be REMOVED with Go release 1.1+1.
 //
 // 2013-01-21: The following functions have been REMOVED
 //
 // 	func MaxInt() int
 // 	func MinInt() int
 // 	func MaxUint() uint
 // 	func UintPtrBits() int
 //
 // They are now replaced by untyped constants
 //
 // 	MaxInt
 // 	MinInt
 // 	MaxUint
 // 	UintPtrBits
 //
 // Additionally one more untyped constant was added
 //
 // 	IntBits
 //
 // This change breaks any existing code depending on the above removed
 // functions.  They should have not been published in the first place, that was
 // unfortunate. Instead, defining such architecture and/or implementation
 // specific integer limits and bit widths as untyped constants improves
 // performance and allows for static dead code elimination if it depends on
 // these values. Thanks to minux for pointing it out in the mail list
 // (https://groups.google.com/d/msg/golang-nuts/tlPpLW6aJw8/NT3mpToH-a4J).
 //
 // 2012-12-12: The following functions will be DEPRECATED with Go release
 // 1.0.3+1 and REMOVED with Go release 1.0.3+2, b/c of
 // http://code.google.com/p/go/source/detail?r=954a79ee3ea8
 //
 // 	func Uint64ToBigInt(n uint64) *big.Int
 // 	func Uint64FromBigInt(n *big.Int) (uint64, bool)
 package mathutil // import "modernc.org/mathutil"
 
 import (
 	"math"
 	"math/big"
 )
 
 // Architecture and/or implementation specific integer limits and bit widths.
 const (
 	MaxInt      = 1<<(IntBits-1) - 1
 	MinInt      = -MaxInt - 1
 	MaxUint     = 1<<IntBits - 1
 	IntBits     = 1 << (^uint(0)>>32&1 + ^uint(0)>>16&1 + ^uint(0)>>8&1 + 3)
 	UintPtrBits = 1 << (^uintptr(0)>>32&1 + ^uintptr(0)>>16&1 + ^uintptr(0)>>8&1 + 3)
 )
 
 var (
 	_1 = big.NewInt(1)
 	_2 = big.NewInt(2)
 )
 
 // GCDByte returns the greatest common divisor of a and b. Based on:
 // http://en.wikipedia.org/wiki/Euclidean_algorithm#Implementations
 func GCDByte(a, b byte) byte {
 	for b != 0 {
 		a, b = b, a%b
 	}
 	return a
 }
 
 // GCDUint16 returns the greatest common divisor of a and b.
 func GCDUint16(a, b uint16) uint16 {
 	for b != 0 {
 		a, b = b, a%b
 	}
 	return a
 }
 
 // GCDUint32 returns the greatest common divisor of a and b.
 func GCDUint32(a, b uint32) uint32 {
 	for b != 0 {
 		a, b = b, a%b
 	}
 	return a
 }
 
 // GCDUint64 returns the greatest common divisor of a and b.
 func GCDUint64(a, b uint64) uint64 {
 	for b != 0 {
 		a, b = b, a%b
 	}
 	return a
 }
 
 // ISqrt returns floor(sqrt(n)). Typical run time is few hundreds of ns.
 func ISqrt(n uint32) (x uint32) {
 	if n == 0 {
 		return
 	}
 
 	if n >= math.MaxUint16*math.MaxUint16 {
 		return math.MaxUint16
 	}
 
 	var px, nx uint32
 	for x = n; ; px, x = x, nx {
 		nx = (x + n/x) / 2
 		if nx == x || nx == px {
 			break
 		}
 	}
 	return
 }
 
 // SqrtUint64 returns floor(sqrt(n)). Typical run time is about 0.5 µs.
 func SqrtUint64(n uint64) (x uint64) {
 	if n == 0 {
 		return
 	}
 
 	if n >= math.MaxUint32*math.MaxUint32 {
 		return math.MaxUint32
 	}
 
 	var px, nx uint64
 	for x = n; ; px, x = x, nx {
 		nx = (x + n/x) / 2
 		if nx == x || nx == px {
 			break
 		}
 	}
 	return
 }
 
 // SqrtBig returns floor(sqrt(n)). It panics on n < 0.
 func SqrtBig(n *big.Int) (x *big.Int) {
 	switch n.Sign() {
 	case -1:
 		panic(-1)
 	case 0:
 		return big.NewInt(0)
 	}
 
 	var px, nx big.Int
 	x = big.NewInt(0)
 	x.SetBit(x, n.BitLen()/2+1, 1)
 	for {
 		nx.Rsh(nx.Add(x, nx.Div(n, x)), 1)
 		if nx.Cmp(x) == 0 || nx.Cmp(&px) == 0 {
 			break
 		}
 		px.Set(x)
 		x.Set(&nx)
 	}
 	return
 }
 
 // Log2Byte returns log base 2 of n. It's the same as index of the highest
 // bit set in n.  For n == 0 -1 is returned.
 func Log2Byte(n byte) int {
 	return log2[n]
 }
 
 // Log2Uint16 returns log base 2 of n. It's the same as index of the highest
 // bit set in n.  For n == 0 -1 is returned.
 func Log2Uint16(n uint16) int {
 	if b := n >> 8; b != 0 {
 		return log2[b] + 8
 	}
 
 	return log2[n]
 }
 
 // Log2Uint32 returns log base 2 of n. It's the same as index of the highest
 // bit set in n.  For n == 0 -1 is returned.
 func Log2Uint32(n uint32) int {
 	if b := n >> 24; b != 0 {
 		return log2[b] + 24
 	}
 
 	if b := n >> 16; b != 0 {
 		return log2[b] + 16
 	}
 
 	if b := n >> 8; b != 0 {
 		return log2[b] + 8
 	}
 
 	return log2[n]
 }
 
 // Log2Uint64 returns log base 2 of n. It's the same as index of the highest
 // bit set in n.  For n == 0 -1 is returned.
 func Log2Uint64(n uint64) int {
 	if b := n >> 56; b != 0 {
 		return log2[b] + 56
 	}
 
 	if b := n >> 48; b != 0 {
 		return log2[b] + 48
 	}
 
 	if b := n >> 40; b != 0 {
 		return log2[b] + 40
 	}
 
 	if b := n >> 32; b != 0 {
 		return log2[b] + 32
 	}
 
 	if b := n >> 24; b != 0 {
 		return log2[b] + 24
 	}
 
 	if b := n >> 16; b != 0 {
 		return log2[b] + 16
 	}
 
 	if b := n >> 8; b != 0 {
 		return log2[b] + 8
 	}
 
 	return log2[n]
 }
 
 // ModPowByte computes (b^e)%m. It panics for m == 0 || b == e == 0.
 //
 // See also: http://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method
 func ModPowByte(b, e, m byte) byte {
 	if b == 0 && e == 0 {
 		panic(0)
 	}
 
 	if m == 1 {
 		return 0
 	}
 
 	r := uint16(1)
 	for b, m := uint16(b), uint16(m); e > 0; b, e = b*b%m, e>>1 {
 		if e&1 == 1 {
 			r = r * b % m
 		}
 	}
 	return byte(r)
 }
 
 // ModPowUint16 computes (b^e)%m. It panics for m == 0 || b == e == 0.
 func ModPowUint16(b, e, m uint16) uint16 {
 	if b == 0 && e == 0 {
 		panic(0)
 	}
 
 	if m == 1 {
 		return 0
 	}
 
 	r := uint32(1)
 	for b, m := uint32(b), uint32(m); e > 0; b, e = b*b%m, e>>1 {
 		if e&1 == 1 {
 			r = r * b % m
 		}
 	}
 	return uint16(r)
 }
 
 // ModPowUint32 computes (b^e)%m. It panics for m == 0 || b == e == 0.
 func ModPowUint32(b, e, m uint32) uint32 {
 	if b == 0 && e == 0 {
 		panic(0)
 	}
 
 	if m == 1 {
 		return 0
 	}
 
 	r := uint64(1)
 	for b, m := uint64(b), uint64(m); e > 0; b, e = b*b%m, e>>1 {
 		if e&1 == 1 {
 			r = r * b % m
 		}
 	}
 	return uint32(r)
 }
 
 // ModPowUint64 computes (b^e)%m. It panics for m == 0 || b == e == 0.
 func ModPowUint64(b, e, m uint64) (r uint64) {
 	if b == 0 && e == 0 {
 		panic(0)
 	}
 
 	if m == 1 {
 		return 0
 	}
 
 	return modPowBigInt(big.NewInt(0).SetUint64(b), big.NewInt(0).SetUint64(e), big.NewInt(0).SetUint64(m)).Uint64()
 }
 
 func modPowBigInt(b, e, m *big.Int) (r *big.Int) {
 	r = big.NewInt(1)
 	for i, n := 0, e.BitLen(); i < n; i++ {
 		if e.Bit(i) != 0 {
 			r.Mod(r.Mul(r, b), m)
 		}
 		b.Mod(b.Mul(b, b), m)
 	}
 	return
 }
 
 // ModPowBigInt computes (b^e)%m. Returns nil for e < 0. It panics for m == 0 || b == e == 0.
 func ModPowBigInt(b, e, m *big.Int) (r *big.Int) {
 	if b.Sign() == 0 && e.Sign() == 0 {
 		panic(0)
 	}
 
 	if m.Cmp(_1) == 0 {
 		return big.NewInt(0)
 	}
 
 	if e.Sign() < 0 {
 		return
 	}
 
 	return modPowBigInt(big.NewInt(0).Set(b), big.NewInt(0).Set(e), m)
 }
 
 var uint64ToBigIntDelta big.Int
 
 func init() {
 	uint64ToBigIntDelta.SetBit(&uint64ToBigIntDelta, 63, 1)
 }
 
 var uintptrBits int
 
 func init() {
 	x := uint64(math.MaxUint64)
 	uintptrBits = BitLenUintptr(uintptr(x))
 }
 
 // UintptrBits returns the bit width of an uintptr at the executing machine.
 func UintptrBits() int {
 	return uintptrBits
 }
 
 // AddUint128_64 returns the uint128 sum of uint64 a and b.
 func AddUint128_64(a, b uint64) (hi uint64, lo uint64) {
 	lo = a + b
 	if lo < a {
 		hi = 1
 	}
 	return hi, lo
 }
 
 // MulUint128_64 returns the uint128 bit product of uint64 a and b.
 func MulUint128_64(a, b uint64) (hi, lo uint64) {
 	/*
 		2^(2 W) ahi bhi + 2^W alo bhi + 2^W ahi blo + alo blo
 
 		FEDCBA98 76543210 FEDCBA98 76543210
 		                  ---- alo*blo ----
 		         ---- alo*bhi ----
 		         ---- ahi*blo ----
 		---- ahi*bhi ----
 	*/
 	const w = 32
 	const m = 1<<w - 1
 	ahi, bhi, alo, blo := a>>w, b>>w, a&m, b&m
 	lo = alo * blo
 	mid1 := alo * bhi
 	mid2 := ahi * blo
 	c1, lo := AddUint128_64(lo, mid1<<w)
 	c2, lo := AddUint128_64(lo, mid2<<w)
 	_, hi = AddUint128_64(ahi*bhi, mid1>>w+mid2>>w+c1+c2)
 	return
 }
 
 // PowerizeBigInt returns (e, p) such that e is the smallest number for which p
 // == b^e is greater or equal n. For n < 0 or b < 2 (0, nil) is returned.
 //
 // NOTE: Run time for large values of n (above about 2^1e6 ~= 1e300000) can be
 // significant and/or unacceptabe.  For any smaller values of n the function
 // typically performs in sub second time.  For "small" values of n (cca bellow
 // 2^1e3 ~= 1e300) the same can be easily below 10 µs.
 //
 // A special (and trivial) case of b == 2 is handled separately and performs
 // much faster.
 func PowerizeBigInt(b, n *big.Int) (e uint32, p *big.Int) {
 	switch {
 	case b.Cmp(_2) < 0 || n.Sign() < 0:
 		return
 	case n.Sign() == 0 || n.Cmp(_1) == 0:
 		return 0, big.NewInt(1)
 	case b.Cmp(_2) == 0:
 		p = big.NewInt(0)
 		e = uint32(n.BitLen() - 1)
 		p.SetBit(p, int(e), 1)
 		if p.Cmp(n) < 0 {
 			p.Mul(p, _2)
 			e++
 		}
 		return
 	}
 
 	bw := b.BitLen()
 	nw := n.BitLen()
 	p = big.NewInt(1)
 	var bb, r big.Int
 	for {
 		switch p.Cmp(n) {
 		case -1:
 			x := uint32((nw - p.BitLen()) / bw)
 			if x == 0 {
 				x = 1
 			}
 			e += x
 			switch x {
 			case 1:
 				p.Mul(p, b)
 			default:
 				r.Set(_1)
 				bb.Set(b)
 				e := x
 				for {
 					if e&1 != 0 {
 						r.Mul(&r, &bb)
 					}
 					if e >>= 1; e == 0 {
 						break
 					}
 
 					bb.Mul(&bb, &bb)
 				}
 				p.Mul(p, &r)
 			}
 		case 0, 1:
 			return
 		}
 	}
 }
 
 // PowerizeUint32BigInt returns (e, p) such that e is the smallest number for
 // which p == b^e is greater or equal n. For n < 0 or b < 2 (0, nil) is
 // returned.
 //
 // More info: see PowerizeBigInt.
 func PowerizeUint32BigInt(b uint32, n *big.Int) (e uint32, p *big.Int) {
 	switch {
 	case b < 2 || n.Sign() < 0:
 		return
 	case n.Sign() == 0 || n.Cmp(_1) == 0:
 		return 0, big.NewInt(1)
 	case b == 2:
 		p = big.NewInt(0)
 		e = uint32(n.BitLen() - 1)
 		p.SetBit(p, int(e), 1)
 		if p.Cmp(n) < 0 {
 			p.Mul(p, _2)
 			e++
 		}
 		return
 	}
 
 	var bb big.Int
 	bb.SetInt64(int64(b))
 	return PowerizeBigInt(&bb, n)
 }
 
 /*
 ProbablyPrimeUint32 returns true if n is prime or n is a pseudoprime to base a.
 It implements the Miller-Rabin primality test for one specific value of 'a' and
 k == 1.
 
 Wrt pseudocode shown at
 http://en.wikipedia.org/wiki/Miller-Rabin_primality_test#Algorithm_and_running_time
 
  Input: n > 3, an odd integer to be tested for primality;
  Input: k, a parameter that determines the accuracy of the test
  Output: composite if n is composite, otherwise probably prime
  write n − 1 as 2^s·d with d odd by factoring powers of 2 from n − 1
  LOOP: repeat k times:
     pick a random integer a in the range [2, n − 2]
     x ← a^d mod n
     if x = 1 or x = n − 1 then do next LOOP
     for r = 1 .. s − 1
        x ← x^2 mod n
        if x = 1 then return composite
        if x = n − 1 then do next LOOP
     return composite
  return probably prime
 
 ... this function behaves like passing 1 for 'k' and additionally a
 fixed/non-random 'a'.  Otherwise it's the same algorithm.
 
 See also: http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html
 */
 func ProbablyPrimeUint32(n, a uint32) bool {
 	d, s := n-1, 0
 	for ; d&1 == 0; d, s = d>>1, s+1 {
 	}
 	x := uint64(ModPowUint32(a, d, n))
 	if x == 1 || uint32(x) == n-1 {
 		return true
 	}
 
 	for ; s > 1; s-- {
 		if x = x * x % uint64(n); x == 1 {
 			return false
 		}
 
 		if uint32(x) == n-1 {
 			return true
 		}
 	}
 	return false
 }
 
 // ProbablyPrimeUint64_32 returns true if n is prime or n is a pseudoprime to
 // base a. It implements the Miller-Rabin primality test for one specific value
 // of 'a' and k == 1.  See also ProbablyPrimeUint32.
 func ProbablyPrimeUint64_32(n uint64, a uint32) bool {
 	d, s := n-1, 0
 	for ; d&1 == 0; d, s = d>>1, s+1 {
 	}
 	x := ModPowUint64(uint64(a), d, n)
 	if x == 1 || x == n-1 {
 		return true
 	}
 
 	bx, bn := big.NewInt(0).SetUint64(x), big.NewInt(0).SetUint64(n)
 	for ; s > 1; s-- {
 		if x = bx.Mod(bx.Mul(bx, bx), bn).Uint64(); x == 1 {
 			return false
 		}
 
 		if x == n-1 {
 			return true
 		}
 	}
 	return false
 }
 
 // ProbablyPrimeBigInt_32 returns true if n is prime or n is a pseudoprime to
 // base a. It implements the Miller-Rabin primality test for one specific value
 // of 'a' and k == 1.  See also ProbablyPrimeUint32.
 func ProbablyPrimeBigInt_32(n *big.Int, a uint32) bool {
 	var d big.Int
 	d.Set(n)
 	d.Sub(&d, _1) // d <- n-1
 	s := 0
 	for ; d.Bit(s) == 0; s++ {
 	}
 	nMinus1 := big.NewInt(0).Set(&d)
 	d.Rsh(&d, uint(s))
 
 	x := ModPowBigInt(big.NewInt(int64(a)), &d, n)
 	if x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0 {
 		return true
 	}
 
 	for ; s > 1; s-- {
 		if x = x.Mod(x.Mul(x, x), n); x.Cmp(_1) == 0 {
 			return false
 		}
 
 		if x.Cmp(nMinus1) == 0 {
 			return true
 		}
 	}
 	return false
 }
 
 // ProbablyPrimeBigInt returns true if n is prime or n is a pseudoprime to base
 // a. It implements the Miller-Rabin primality test for one specific value of
 // 'a' and k == 1.  See also ProbablyPrimeUint32.
 func ProbablyPrimeBigInt(n, a *big.Int) bool {
 	var d big.Int
 	d.Set(n)
 	d.Sub(&d, _1) // d <- n-1
 	s := 0
 	for ; d.Bit(s) == 0; s++ {
 	}
 	nMinus1 := big.NewInt(0).Set(&d)
 	d.Rsh(&d, uint(s))
 
 	x := ModPowBigInt(a, &d, n)
 	if x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0 {
 		return true
 	}
 
 	for ; s > 1; s-- {
 		if x = x.Mod(x.Mul(x, x), n); x.Cmp(_1) == 0 {
 			return false
 		}
 
 		if x.Cmp(nMinus1) == 0 {
 			return true
 		}
 	}
 	return false
 }
 
 // Max returns the larger of a and b.
 func Max(a, b int) int {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // Min returns the smaller of a and b.
 func Min(a, b int) int {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxPtr returns a pointer to the larger of a and b, or nil.
 func MaxPtr(a, b *int) *int {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinPtr returns a pointer to the smaller of a and b, or nil.
 func MinPtr(a, b *int) *int {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxVal returns the largest argument passed.
 func MaxVal(val int, vals ...int) int {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinVal returns the smallest argument passed.
 func MinVal(val int, vals ...int) int {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // Clamp returns a value restricted between lo and hi.
 func Clamp(v, lo, hi int) int {
 	return Min(Max(v, lo), hi)
 }
 
 // UMax returns the larger of a and b.
 func UMax(a, b uint) uint {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // UMin returns the smaller of a and b.
 func UMin(a, b uint) uint {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // UMaxPtr returns a pointer to the larger of a and b, or nil.
 func UMaxPtr(a, b *uint) *uint {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // UMinPtr returns a pointer to the smaller of a and b, or nil.
 func UMinPtr(a, b *uint) *uint {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // UMaxVal returns the largest argument passed.
 func UMaxVal(val uint, vals ...uint) uint {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // UMinVal returns the smallest argument passed.
 func UMinVal(val uint, vals ...uint) uint {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // UClamp returns a value restricted between lo and hi.
 func UClamp(v, lo, hi uint) uint {
 	return UMin(UMax(v, lo), hi)
 }
 
 // MaxByte returns the larger of a and b.
 func MaxByte(a, b byte) byte {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinByte returns the smaller of a and b.
 func MinByte(a, b byte) byte {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxBytePtr returns a pointer to the larger of a and b, or nil.
 func MaxBytePtr(a, b *byte) *byte {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinBytePtr returns a pointer to the smaller of a and b, or nil.
 func MinBytePtr(a, b *byte) *byte {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxByteVal returns the largest argument passed.
 func MaxByteVal(val byte, vals ...byte) byte {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinByteVal returns the smallest argument passed.
 func MinByteVal(val byte, vals ...byte) byte {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampByte returns a value restricted between lo and hi.
 func ClampByte(v, lo, hi byte) byte {
 	return MinByte(MaxByte(v, lo), hi)
 }
 
 // MaxInt8 returns the larger of a and b.
 func MaxInt8(a, b int8) int8 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt8 returns the smaller of a and b.
 func MinInt8(a, b int8) int8 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt8Ptr returns a pointer to the larger of a and b, or nil.
 func MaxInt8Ptr(a, b *int8) *int8 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt8Ptr returns a pointer to the smaller of a and b, or nil.
 func MinInt8Ptr(a, b *int8) *int8 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt8Val returns the largest argument passed.
 func MaxInt8Val(val int8, vals ...int8) int8 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinInt8Val returns the smallest argument passed.
 func MinInt8Val(val int8, vals ...int8) int8 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampInt8 returns a value restricted between lo and hi.
 func ClampInt8(v, lo, hi int8) int8 {
 	return MinInt8(MaxInt8(v, lo), hi)
 }
 
 // MaxUint16 returns the larger of a and b.
 func MaxUint16(a, b uint16) uint16 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint16 returns the smaller of a and b.
 func MinUint16(a, b uint16) uint16 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint16Ptr returns a pointer to the larger of a and b, or nil.
 func MaxUint16Ptr(a, b *uint16) *uint16 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint16Ptr returns a pointer to the smaller of a and b, or nil.
 func MinUint16Ptr(a, b *uint16) *uint16 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint16Val returns the largest argument passed.
 func MaxUint16Val(val uint16, vals ...uint16) uint16 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinUint16Val returns the smallest argument passed.
 func MinUint16Val(val uint16, vals ...uint16) uint16 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampUint16 returns a value restricted between lo and hi.
 func ClampUint16(v, lo, hi uint16) uint16 {
 	return MinUint16(MaxUint16(v, lo), hi)
 }
 
 // MaxInt16 returns the larger of a and b.
 func MaxInt16(a, b int16) int16 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt16 returns the smaller of a and b.
 func MinInt16(a, b int16) int16 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt16Ptr returns a pointer to the larger of a and b, or nil.
 func MaxInt16Ptr(a, b *int16) *int16 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt16Ptr returns a pointer to the smaller of a and b, or nil.
 func MinInt16Ptr(a, b *int16) *int16 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt16Val returns the largest argument passed.
 func MaxInt16Val(val int16, vals ...int16) int16 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinInt16Val returns the smallest argument passed.
 func MinInt16Val(val int16, vals ...int16) int16 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampInt16 returns a value restricted between lo and hi.
 func ClampInt16(v, lo, hi int16) int16 {
 	return MinInt16(MaxInt16(v, lo), hi)
 }
 
 // MaxUint32 returns the larger of a and b.
 func MaxUint32(a, b uint32) uint32 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint32 returns the smaller of a and b.
 func MinUint32(a, b uint32) uint32 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint32Ptr returns a pointer to the larger of a and b, or nil.
 func MaxUint32Ptr(a, b *uint32) *uint32 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint32Ptr returns a pointer to the smaller of a and b, or nil.
 func MinUint32Ptr(a, b *uint32) *uint32 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint32Val returns the largest argument passed.
 func MaxUint32Val(val uint32, vals ...uint32) uint32 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinUint32Val returns the smallest argument passed.
 func MinUint32Val(val uint32, vals ...uint32) uint32 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampUint32 returns a value restricted between lo and hi.
 func ClampUint32(v, lo, hi uint32) uint32 {
 	return MinUint32(MaxUint32(v, lo), hi)
 }
 
 // MaxInt32 returns the larger of a and b.
 func MaxInt32(a, b int32) int32 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt32 returns the smaller of a and b.
 func MinInt32(a, b int32) int32 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt32Ptr returns a pointer to the larger of a and b, or nil.
 func MaxInt32Ptr(a, b *int32) *int32 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt32Ptr returns a pointer to the smaller of a and b, or nil.
 func MinInt32Ptr(a, b *int32) *int32 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt32Val returns the largest argument passed.
 func MaxInt32Val(val int32, vals ...int32) int32 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinInt32Val returns the smallest argument passed.
 func MinInt32Val(val int32, vals ...int32) int32 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampInt32 returns a value restricted between lo and hi.
 func ClampInt32(v, lo, hi int32) int32 {
 	return MinInt32(MaxInt32(v, lo), hi)
 }
 
 // MaxUint64 returns the larger of a and b.
 func MaxUint64(a, b uint64) uint64 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint64 returns the smaller of a and b.
 func MinUint64(a, b uint64) uint64 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint64Ptr returns a pointer to the larger of a and b, or nil.
 func MaxUint64Ptr(a, b *uint64) *uint64 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinUint64Ptr returns a pointer to the smaller of a and b, or nil.
 func MinUint64Ptr(a, b *uint64) *uint64 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxUint64Val returns the largest argument passed.
 func MaxUint64Val(val uint64, vals ...uint64) uint64 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinUint64Val returns the smallest argument passed.
 func MinUint64Val(val uint64, vals ...uint64) uint64 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampUint64 returns a value restricted between lo and hi.
 func ClampUint64(v, lo, hi uint64) uint64 {
 	return MinUint64(MaxUint64(v, lo), hi)
 }
 
 // MaxInt64 returns the larger of a and b.
 func MaxInt64(a, b int64) int64 {
 	if a > b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt64 returns the smaller of a and b.
 func MinInt64(a, b int64) int64 {
 	if a < b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt64Ptr returns a pointer to the larger of a and b, or nil.
 func MaxInt64Ptr(a, b *int64) *int64 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a > *b {
 		return a
 	}
 
 	return b
 }
 
 // MinInt64Ptr returns a pointer to the smaller of a and b, or nil.
 func MinInt64Ptr(a, b *int64) *int64 {
 	if a == nil {
 		return b
 	}
 	if b == nil {
 		return a
 	}
 	if *a < *b {
 		return a
 	}
 
 	return b
 }
 
 // MaxInt64Val returns the largest argument passed.
 func MaxInt64Val(val int64, vals ...int64) int64 {
 	res := val
 	for _, v := range vals {
 		if v > res {
 			res = v
 		}
 	}
 	return res
 }
 
 // MinInt64Val returns the smallest argument passed.
 func MinInt64Val(val int64, vals ...int64) int64 {
 	res := val
 	for _, v := range vals {
 		if v < res {
 			res = v
 		}
 	}
 	return res
 }
 
 // ClampInt64 returns a value restricted between lo and hi.
 func ClampInt64(v, lo, hi int64) int64 {
 	return MinInt64(MaxInt64(v, lo), hi)
 }
 
 // ToBase produces n in base b. For example
 //
 // 	ToBase(2047, 22) -> [1, 5, 4]
 //
 //	1 * 22^0           1
 //	5 * 22^1         110
 //	4 * 22^2        1936
 //	                ----
 //	                2047
 //
 // ToBase panics for bases < 2.
 func ToBase(n *big.Int, b int) []int {
 	var nn big.Int
 	nn.Set(n)
 	if b < 2 {
 		panic("invalid base")
 	}
 
 	k := 1
 	switch nn.Sign() {
 	case -1:
 		nn.Neg(&nn)
 		k = -1
 	case 0:
 		return []int{0}
 	}
 
 	bb := big.NewInt(int64(b))
 	var r []int
 	rem := big.NewInt(0)
 	for nn.Sign() != 0 {
 		nn.QuoRem(&nn, bb, rem)
 		r = append(r, k*int(rem.Int64()))
 	}
 	return r
 }
 
 // CheckAddInt64 returns the a+b and an indicator that the result is greater
 // than math.MaxInt64.
 func CheckAddInt64(a, b int64) (sum int64, gt bool) {
 	return a + b, a > 0 && b > math.MaxInt64-a || a < 0 && b < math.MinInt64-a
 }
 
 // CheckSubInt64 returns a-b and an indicator that the result is less than than
 // math.MinInt64.
 func CheckSubInt64(a, b int64) (sum int64, lt bool) {
 	return a - b, a > 0 && a-math.MaxInt64 > b || a < 0 && a-math.MinInt64 < b
 }
 
 // AddOverflowInt8 returns a + b and an indication whether the addition
 // overflowed the int8 range.
 func AddOverflowInt8(a, b int8) (r int8, ovf bool) {
 	r = a + b
 	if a > 0 && b > 0 {
 		return r, uint8(r) > math.MaxInt8
 	}
 
 	if a < 0 && b < 0 {
 		return r, uint8(r) <= math.MaxInt8
 	}
 
 	return r, false
 }
 
 // AddOverflowInt16 returns a + b and an indication whether the addition
 // overflowed the int16 range.
 func AddOverflowInt16(a, b int16) (r int16, ovf bool) {
 	r = a + b
 	if a > 0 && b > 0 {
 		return r, uint16(r) > math.MaxInt16
 	}
 
 	if a < 0 && b < 0 {
 		return r, uint16(r) <= math.MaxInt16
 	}
 
 	return r, false
 }
 
 // AddOverflowInt32 returns a + b and an indication whether the addition
 // overflowed the int32 range.
 func AddOverflowInt32(a, b int32) (r int32, ovf bool) {
 	r = a + b
 	if a > 0 && b > 0 {
 		return r, uint32(r) > math.MaxInt32
 	}
 
 	if a < 0 && b < 0 {
 		return r, uint32(r) <= math.MaxInt32
 	}
 
 	return r, false
 }
 
 // AddOverflowInt64 returns a + b and an indication whether the addition
 // overflowed the int64 range.
 func AddOverflowInt64(a, b int64) (r int64, ovf bool) {
 	r = a + b
 	if a > 0 && b > 0 {
 		return r, uint64(r) > math.MaxInt64
 	}
 
 	if a < 0 && b < 0 {
 		return r, uint64(r) <= math.MaxInt64
 	}
 
 	return r, false
 }
 
 // SubOverflowInt8 returns a - b and an indication whether the subtraction
 // overflowed the int8 range.
 func SubOverflowInt8(a, b int8) (r int8, ovf bool) {
 	r = a - b
 	if a >= 0 && b < 0 {
 		return r, uint8(r) >= math.MaxInt8+1
 	}
 
 	if a < 0 && b > 0 {
 		return r, uint8(r) <= math.MaxInt8
 	}
 
 	return r, false
 }
 
 // SubOverflowInt16 returns a - b and an indication whether the subtraction
 // overflowed the int16 range.
 func SubOverflowInt16(a, b int16) (r int16, ovf bool) {
 	r = a - b
 	if a >= 0 && b < 0 {
 		return r, uint16(r) >= math.MaxInt16+1
 	}
 
 	if a < 0 && b > 0 {
 		return r, uint16(r) <= math.MaxInt16
 	}
 
 	return r, false
 }
 
 // SubOverflowInt32 returns a - b and an indication whether the subtraction
 // overflowed the int32 range.
 func SubOverflowInt32(a, b int32) (r int32, ovf bool) {
 	r = a - b
 	if a >= 0 && b < 0 {
 		return r, uint32(r) >= math.MaxInt32+1
 	}
 
 	if a < 0 && b > 0 {
 		return r, uint32(r) <= math.MaxInt32
 	}
 
 	return r, false
 }
 
 // SubOverflowInt64 returns a - b and an indication whether the subtraction
 // overflowed the int64 range.
 func SubOverflowInt64(a, b int64) (r int64, ovf bool) {
 	r = a - b
 	if a >= 0 && b < 0 {
 		return r, uint64(r) >= math.MaxInt64+1
 	}
 
 	if a < 0 && b > 0 {
 		return r, uint64(r) <= math.MaxInt64
 	}
 
 	return r, false
 }
 
 // MulOverflowInt8 returns a * b and an indication whether the product
 // overflowed the int8 range.
 func MulOverflowInt8(a, b int8) (r int8, ovf bool) {
 	if a == 0 || b == 0 {
 		return 0, false
 	}
 
 	z := int16(a) * int16(b)
 	return int8(z), z < math.MinInt8 || z > math.MaxInt8
 }
 
 // MulOverflowInt16 returns a * b and an indication whether the product
 // overflowed the int16 range.
 func MulOverflowInt16(a, b int16) (r int16, ovf bool) {
 	if a == 0 || b == 0 {
 		return 0, false
 	}
 
 	z := int32(a) * int32(b)
 	return int16(z), z < math.MinInt16 || z > math.MaxInt16
 }
 
 // MulOverflowInt32 returns a * b and an indication whether the product
 // overflowed the int32 range.
 func MulOverflowInt32(a, b int32) (r int32, ovf bool) {
 	if a == 0 || b == 0 {
 		return 0, false
 	}
 
 	z := int64(a) * int64(b)
 	return int32(z), z < math.MinInt32 || z > math.MaxInt32
 }
 
 // MulOverflowInt64 returns a * b and an indication whether the product
 // overflowed the int64 range.
 func MulOverflowInt64(a, b int64) (r int64, ovf bool) {
 	// https://groups.google.com/g/golang-nuts/c/h5oSN5t3Au4/m/KaNQREhZh0QJ
 	const mostPositive = 1<<63 - 1
 	const mostNegative = -(mostPositive + 1)
 	r = a * b
 	if a == 0 || b == 0 || a == 1 || b == 1 {
 		return r, false
 	}
 
 	if a == mostNegative || b == mostNegative {
 		return r, true
 	}
 
 	return r, r/b != a
 }