// Copyright 2009 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. // Multiprecision decimal numbers. // For floating-point formatting only; not general purpose. // Only operations are assign and (binary) left/right shift. // Can do binary floating point in multiprecision decimal precisely // because 2 divides 10; cannot do decimal floating point // in multiprecision binary precisely. package strconv type decimal struct { d [800]byte // digits, big-endian representation nd int // number of digits used dp int // decimal point neg bool // negative flag trunc bool // discarded nonzero digits beyond d[:nd] } func (a *decimal) String() string { n := 10 + a.nd if a.dp > 0 { n += a.dp } if a.dp < 0 { n += -a.dp } buf := make([]byte, n) w := 0 switch { case a.nd == 0: return "0" case a.dp <= 0: // zeros fill space between decimal point and digits buf[w] = '0' w++ buf[w] = '.' w++ w += digitZero(buf[w : w+-a.dp]) w += copy(buf[w:], a.d[0:a.nd]) case a.dp < a.nd: // decimal point in middle of digits w += copy(buf[w:], a.d[0:a.dp]) buf[w] = '.' w++ w += copy(buf[w:], a.d[a.dp:a.nd]) default: // zeros fill space between digits and decimal point w += copy(buf[w:], a.d[0:a.nd]) w += digitZero(buf[w : w+a.dp-a.nd]) } return string(buf[0:w]) } func digitZero(dst []byte) int { for i := range dst { dst[i] = '0' } return len(dst) } // trim trailing zeros from number. // (They are meaningless; the decimal point is tracked // independent of the number of digits.) func trim(a *decimal) { for a.nd > 0 && a.d[a.nd-1] == '0' { a.nd-- } if a.nd == 0 { a.dp = 0 } } // Assign v to a. func (a *decimal) Assign(v uint64) { var buf [24]byte // Write reversed decimal in buf. n := 0 for v > 0 { v1 := v / 10 v -= 10 * v1 buf[n] = byte(v + '0') n++ v = v1 } // Reverse again to produce forward decimal in a.d. a.nd = 0 for n--; n >= 0; n-- { a.d[a.nd] = buf[n] a.nd++ } a.dp = a.nd trim(a) } // Maximum shift that we can do in one pass without overflow. // A uint has 32 or 64 bits, and we have to be able to accommodate 9<> 63) const maxShift = uintSize - 4 // Binary shift right (/ 2) by k bits. k <= maxShift to avoid overflow. func rightShift(a *decimal, k uint) { r := 0 // read pointer w := 0 // write pointer // Pick up enough leading digits to cover first shift. var n uint for ; n>>k == 0; r++ { if r >= a.nd { if n == 0 { // a == 0; shouldn't get here, but handle anyway. a.nd = 0 return } for n>>k == 0 { n = n * 10 r++ } break } c := uint(a.d[r]) n = n*10 + c - '0' } a.dp -= r - 1 var mask uint = (1 << k) - 1 // Pick up a digit, put down a digit. for ; r < a.nd; r++ { c := uint(a.d[r]) dig := n >> k n &= mask a.d[w] = byte(dig + '0') w++ n = n*10 + c - '0' } // Put down extra digits. for n > 0 { dig := n >> k n &= mask if w < len(a.d) { a.d[w] = byte(dig + '0') w++ } else if dig > 0 { a.trunc = true } n = n * 10 } a.nd = w trim(a) } // Cheat sheet for left shift: table indexed by shift count giving // number of new digits that will be introduced by that shift. // // For example, leftcheats[4] = {2, "625"}. That means that // if we are shifting by 4 (multiplying by 16), it will add 2 digits // when the string prefix is "625" through "999", and one fewer digit // if the string prefix is "000" through "624". // // Credit for this trick goes to Ken. type leftCheat struct { delta int // number of new digits cutoff string // minus one digit if original < a. } var leftcheats = []leftCheat{ // Leading digits of 1/2^i = 5^i. // 5^23 is not an exact 64-bit floating point number, // so have to use bc for the math. // Go up to 60 to be large enough for 32bit and 64bit platforms. /* seq 60 | sed 's/^/5^/' | bc | awk 'BEGIN{ print "\t{ 0, \"\" }," } { log2 = log(2)/log(10) printf("\t{ %d, \"%s\" },\t// * %d\n", int(log2*NR+1), $0, 2**NR) }' */ {0, ""}, {1, "5"}, // * 2 {1, "25"}, // * 4 {1, "125"}, // * 8 {2, "625"}, // * 16 {2, "3125"}, // * 32 {2, "15625"}, // * 64 {3, "78125"}, // * 128 {3, "390625"}, // * 256 {3, "1953125"}, // * 512 {4, "9765625"}, // * 1024 {4, "48828125"}, // * 2048 {4, "244140625"}, // * 4096 {4, "1220703125"}, // * 8192 {5, "6103515625"}, // * 16384 {5, "30517578125"}, // * 32768 {5, "152587890625"}, // * 65536 {6, "762939453125"}, // * 131072 {6, "3814697265625"}, // * 262144 {6, "19073486328125"}, // * 524288 {7, "95367431640625"}, // * 1048576 {7, "476837158203125"}, // * 2097152 {7, "2384185791015625"}, // * 4194304 {7, "11920928955078125"}, // * 8388608 {8, "59604644775390625"}, // * 16777216 {8, "298023223876953125"}, // * 33554432 {8, "1490116119384765625"}, // * 67108864 {9, "7450580596923828125"}, // * 134217728 {9, "37252902984619140625"}, // * 268435456 {9, "186264514923095703125"}, // * 536870912 {10, "931322574615478515625"}, // * 1073741824 {10, "4656612873077392578125"}, // * 2147483648 {10, "23283064365386962890625"}, // * 4294967296 {10, "116415321826934814453125"}, // * 8589934592 {11, "582076609134674072265625"}, // * 17179869184 {11, "2910383045673370361328125"}, // * 34359738368 {11, "14551915228366851806640625"}, // * 68719476736 {12, "72759576141834259033203125"}, // * 137438953472 {12, "363797880709171295166015625"}, // * 274877906944 {12, "1818989403545856475830078125"}, // * 549755813888 {13, "9094947017729282379150390625"}, // * 1099511627776 {13, "45474735088646411895751953125"}, // * 2199023255552 {13, "227373675443232059478759765625"}, // * 4398046511104 {13, "1136868377216160297393798828125"}, // * 8796093022208 {14, "5684341886080801486968994140625"}, // * 17592186044416 {14, "28421709430404007434844970703125"}, // * 35184372088832 {14, "142108547152020037174224853515625"}, // * 70368744177664 {15, "710542735760100185871124267578125"}, // * 140737488355328 {15, "3552713678800500929355621337890625"}, // * 281474976710656 {15, "17763568394002504646778106689453125"}, // * 562949953421312 {16, "88817841970012523233890533447265625"}, // * 1125899906842624 {16, "444089209850062616169452667236328125"}, // * 2251799813685248 {16, "2220446049250313080847263336181640625"}, // * 4503599627370496 {16, "11102230246251565404236316680908203125"}, // * 9007199254740992 {17, "55511151231257827021181583404541015625"}, // * 18014398509481984 {17, "277555756156289135105907917022705078125"}, // * 36028797018963968 {17, "1387778780781445675529539585113525390625"}, // * 72057594037927936 {18, "6938893903907228377647697925567626953125"}, // * 144115188075855872 {18, "34694469519536141888238489627838134765625"}, // * 288230376151711744 {18, "173472347597680709441192448139190673828125"}, // * 576460752303423488 {19, "867361737988403547205962240695953369140625"}, // * 1152921504606846976 } // Is the leading prefix of b lexicographically less than s? func prefixIsLessThan(b []byte, s string) bool { for i := 0; i < len(s); i++ { if i >= len(b) { return true } if b[i] != s[i] { return b[i] < s[i] } } return false } // Binary shift left (* 2) by k bits. k <= maxShift to avoid overflow. func leftShift(a *decimal, k uint) { delta := leftcheats[k].delta if prefixIsLessThan(a.d[0:a.nd], leftcheats[k].cutoff) { delta-- } r := a.nd // read index w := a.nd + delta // write index // Pick up a digit, put down a digit. var n uint for r--; r >= 0; r-- { n += (uint(a.d[r]) - '0') << k quo := n / 10 rem := n - 10*quo w-- if w < len(a.d) { a.d[w] = byte(rem + '0') } else if rem != 0 { a.trunc = true } n = quo } // Put down extra digits. for n > 0 { quo := n / 10 rem := n - 10*quo w-- if w < len(a.d) { a.d[w] = byte(rem + '0') } else if rem != 0 { a.trunc = true } n = quo } a.nd += delta if a.nd >= len(a.d) { a.nd = len(a.d) } a.dp += delta trim(a) } // Binary shift left (k > 0) or right (k < 0). func (a *decimal) Shift(k int) { switch { case a.nd == 0: // nothing to do: a == 0 case k > 0: for k > maxShift { leftShift(a, maxShift) k -= maxShift } leftShift(a, uint(k)) case k < 0: for k < -maxShift { rightShift(a, maxShift) k += maxShift } rightShift(a, uint(-k)) } } // If we chop a at nd digits, should we round up? func shouldRoundUp(a *decimal, nd int) bool { if nd < 0 || nd >= a.nd { return false } if a.d[nd] == '5' && nd+1 == a.nd { // exactly halfway - round to even // if we truncated, a little higher than what's recorded - always round up if a.trunc { return true } return nd > 0 && (a.d[nd-1]-'0')%2 != 0 } // not halfway - digit tells all return a.d[nd] >= '5' } // Round a to nd digits (or fewer). // If nd is zero, it means we're rounding // just to the left of the digits, as in // 0.09 -> 0.1. func (a *decimal) Round(nd int) { if nd < 0 || nd >= a.nd { return } if shouldRoundUp(a, nd) { a.RoundUp(nd) } else { a.RoundDown(nd) } } // Round a down to nd digits (or fewer). func (a *decimal) RoundDown(nd int) { if nd < 0 || nd >= a.nd { return } a.nd = nd trim(a) } // Round a up to nd digits (or fewer). func (a *decimal) RoundUp(nd int) { if nd < 0 || nd >= a.nd { return } // round up for i := nd - 1; i >= 0; i-- { c := a.d[i] if c < '9' { // can stop after this digit a.d[i]++ a.nd = i + 1 return } } // Number is all 9s. // Change to single 1 with adjusted decimal point. a.d[0] = '1' a.nd = 1 a.dp++ } // Extract integer part, rounded appropriately. // No guarantees about overflow. func (a *decimal) RoundedInteger() uint64 { if a.dp > 20 { return 0xFFFFFFFFFFFFFFFF } var i int n := uint64(0) for i = 0; i < a.dp && i < a.nd; i++ { n = n*10 + uint64(a.d[i]-'0') } for ; i < a.dp; i++ { n *= 10 } if shouldRoundUp(a, a.dp) { n++ } return n }