| // 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 |
| 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<<k. |
| const uintSize = 32 << (^uint(0) >> 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 |
| |
| // Pick up a digit, put down a digit. |
| for ; r < a.nd; r++ { |
| c := uint(a.d[r]) |
| dig := n >> k |
| n -= dig << k |
| a.d[w] = byte(dig + '0') |
| w++ |
| n = n*10 + c - '0' |
| } |
| |
| // Put down extra digits. |
| for n > 0 { |
| dig := n >> k |
| n -= dig << k |
| 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 |
| } |
