math/big: implement precise Float to decimal conversion (core functionality)

Change-Id: Ic0153397922ded28a5cb362e86ecdfec42e92163
Reviewed-on: https://go-review.googlesource.com/3752Reviewed-by: Alan Donovan <adonovan@google.com>

src/math/big/decimal.go

+// Copyright 2015 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.
+
+// This file implements multi-precision decimal numbers.
+// The implementation is for float to decimal conversion only;
+// not general purpose use.
+// The only operations are precise conversion from binary to
+// decimal and rounding.
+//
+// The key observation and some code (shr) is borrowed from
+// strconv/decimal.go: conversion of binary fractional values can be done
+// precisely in multi-precision decimal because 2 divides 10 (required for
+// >> of mantissa); but conversion of decimal floating-point values cannot
+// be done precisely in binary representation.
+//
+// In contrast to strconv/decimal.go, only right shift is implemented in
+// decimal format - left shift can be done precisely in binary format.
+
+package big
+
+// A decimal represents a floating-point number in decimal representation.
+// The value of a decimal x is x.mant * 10 ** x.exp with 0.5 <= x.mant < 1,
+// with the most-significant mantissa digit at index 0.
+type decimal struct {
+	mant []byte // mantissa ASCII digits, big-endian
+	exp  int    // exponent, valid if len(mant) > 0
+}
+
+// Maximum shift amount that can be done in one pass without overflow.
+// A Word has _W bits and (1<<maxShift - 1)*10 + 9 must fit into Word.
+const maxShift = _W - 4
+
+// TODO(gri) Since we know the desired decimal precision when converting
+// a floating-point number, we may be able to limit the number of decimal
+// digits that need to be computed by init by providing an additional
+// precision argument and keeping track of when a number was truncated early
+// (equivalent of "sticky bit" in binary rounding).
+
+// Init initializes x to the decimal representation of m << shift (for
+// shift >= 0), or m >> -shift (for shift < 0).
+func (x *decimal) init(m nat, shift int) {
+	// special case 0
+	if len(m) == 0 {
+		x.mant = x.mant[:0]
+		return
+	}
+
+	// Optimization: If we need to shift right, first remove any trailing
+	// zero bits from m to reduce shift amount that needs to be done in
+	// decimal format (since that is likely slower).
+	if shift < 0 {
+		ntz := m.trailingZeroBits()
+		s := uint(-shift)
+		if s >= ntz {
+			s = ntz // shift at most ntz bits
+		}
+		m = nat(nil).shr(m, s)
+		shift += int(s)
+	}
+
+	// Do any shift left in binary representation.
+	if shift > 0 {
+		m = nat(nil).shl(m, uint(shift))
+		shift = 0
+	}
+
+	// Convert mantissa into decimal representation.
+	s := m.decimalString() // TODO(gri) avoid string conversion here
+	n := len(s)
+	x.exp = n
+	// Trim trailing zeros; instead the exponent is tracking
+	// the decimal point independent of the number of digits.
+	for n > 0 && s[n-1] == 0 {
+		n--
+	}
+	x.mant = make([]byte, n)
+	copy(x.mant, s)
+
+	// Do any (remaining) shift right in decimal representation.
+	if shift < 0 {
+		for shift < -maxShift {
+			x.shr(maxShift)
+			shift += maxShift
+		}
+		x.shr(uint(-shift))
+	}
+}
+
+// Possibly optimization: The current implementation of nat.string takes
+// a charset argument. When a right shift is needed, we could provide
+// "\x00\x01...\x09" instead of "012..9" (as in nat.decimalString) and
+// avoid the repeated +'0' and -'0' operations in decimal.shr (and do a
+// single +'0' pass at the end).
+
+// shr implements x >> s, for s <= maxShift.
+func (x *decimal) shr(s uint) {
+	// Division by 1<<s using shift-and-subtract algorithm.
+
+	// pick up enough leading digits to cover first shift
+	r := 0 // read index
+	var n Word
+	for n>>s == 0 && r < len(x.mant) {
+		ch := Word(x.mant[r])
+		r++
+		n = n*10 + ch - '0'
+	}
+	if n == 0 {
+		// x == 0; shouldn't get here, but handle anyway
+		x.mant = x.mant[:0]
+		return
+	}
+	for n>>s == 0 {
+		r++
+		n *= 10
+	}
+	x.exp += 1 - r
+
+	// read a digit, write a digit
+	w := 0 // write index
+	for r < len(x.mant) {
+		ch := Word(x.mant[r])
+		r++
+		d := n >> s
+		n -= d << s
+		x.mant[w] = byte(d + '0')
+		w++
+		n = n*10 + ch - '0'
+	}
+
+	// write extra digits that still fit
+	for n > 0 && w < len(x.mant) {
+		d := n >> s
+		n -= d << s
+		x.mant[w] = byte(d + '0')
+		w++
+		n = n * 10
+	}
+	x.mant = x.mant[:w] // the number may be shorter (e.g. 1024 >> 10)
+
+	// append additional digits that didn't fit
+	for n > 0 {
+		d := n >> s
+		n -= d << s
+		x.mant = append(x.mant, byte(d+'0'))
+		n = n * 10
+	}
+
+	// remove trailing zeros
+	w = len(x.mant)
+	for w > 0 && x.mant[w-1] == '0' {
+		w--
+	}
+	x.mant = x.mant[:w]
+}
+
+func (x *decimal) String() string {
+	if len(x.mant) == 0 {
+		return "0"
+	}
+
+	var buf []byte
+	switch {
+	case x.exp <= 0:
+		// 0.00ddd
+		buf = append(buf, "0."...)
+		buf = appendZeros(buf, -x.exp)
+		buf = append(buf, x.mant...)
+
+	case /* 0 < */ x.exp < len(x.mant):
+		// dd.ddd
+		buf = append(buf, x.mant[:x.exp]...)
+		buf = append(buf, '.')
+		buf = append(buf, x.mant[x.exp:]...)
+
+	default: // len(x.mant) <= x.exp
+		// ddd00
+		buf = append(buf, x.mant...)
+		buf = appendZeros(buf, x.exp-len(x.mant))
+	}
+
+	return string(buf)
+}
+
+// appendZeros appends n 0 digits to buf and returns buf.
+func appendZeros(buf []byte, n int) []byte {
+	for ; n > 0; n-- {
+		buf = append(buf, '0')
+	}
+	return buf
+}

src/math/big/decimal_test.go

+// Copyright 2015 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.
+
+package big
+
+import "testing"
+
+func TestDecimalString(t *testing.T) {
+	for _, test := range []struct {
+		x    decimal
+		want string
+	}{
+		{want: "0"},
+		{decimal{nil, 1000}, "0"}, // exponent of 0 is ignored
+		{decimal{[]byte("12345"), 0}, "0.12345"},
+		{decimal{[]byte("12345"), -3}, "0.00012345"},
+		{decimal{[]byte("12345"), +3}, "123.45"},
+		{decimal{[]byte("12345"), +10}, "1234500000"},
+	} {
+		if got := test.x.String(); got != test.want {
+			t.Errorf("%v == %s; want %s", test.x, got, test.want)
+		}
+	}
+}
+
+func TestDecimalInit(t *testing.T) {
+	for _, test := range []struct {
+		x     Word
+		shift int
+		want  string
+	}{
+		{0, 0, "0"},
+		{0, -100, "0"},
+		{0, 100, "0"},
+		{1, 0, "1"},
+		{1, 10, "1024"},
+		{1, 100, "1267650600228229401496703205376"},
+		{1, -100, "0.0000000000000000000000000000007888609052210118054117285652827862296732064351090230047702789306640625"},
+		{12345678, 8, "3160493568"},
+		{12345678, -8, "48225.3046875"},
+		{195312, 9, "99999744"},
+		{1953125, 9, "1000000000"},
+	} {
+		var d decimal
+		d.init(nat{test.x}.norm(), test.shift)
+		if got := d.String(); got != test.want {
+			t.Errorf("%d << %d == %s; want %s", test.x, test.shift, got, test.want)
+		}
+	}
+}