source: code/trunk/vendor/modernc.org/mathutil/mathutil.go@ 822

// 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
}