Source File
	ratconv.go

Belonging Package
	math/big

// 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 rat-to-string conversion functions.

package big

import (
	"errors"
	"fmt"
	"io"
	"strconv"
	"strings"
)

func ratTok(ch rune) bool {
	return strings.ContainsRune("+-/0123456789.eE", ch)
}

var ratZero Rat
var _ fmt.Scanner = &ratZero // *Rat must implement fmt.Scanner

// Scan is a support routine for fmt.Scanner. It accepts the formats
// 'e', 'E', 'f', 'F', 'g', 'G', and 'v'. All formats are equivalent.
func (z *Rat) Scan(s fmt.ScanState, ch rune) error {
	tok, err := s.Token(true, ratTok)
	if err != nil {
		return err
	}
	if !strings.ContainsRune("efgEFGv", ch) {
		return errors.New("Rat.Scan: invalid verb")
	}
	if _, ok := z.SetString(string(tok)); !ok {
		return errors.New("Rat.Scan: invalid syntax")
	}
	return nil
}

// SetString sets z to the value of s and returns z and a boolean indicating
// success. s can be given as a (possibly signed) fraction "a/b", or as a
// floating-point number optionally followed by an exponent.
// If a fraction is provided, both the dividend and the divisor may be a
// decimal integer or independently use a prefix of “0b”, “0” or “0o”,
// or “0x” (or their upper-case variants) to denote a binary, octal, or
// hexadecimal integer, respectively. The divisor may not be signed.
// If a floating-point number is provided, it may be in decimal form or
// use any of the same prefixes as above but for “0” to denote a non-decimal
// mantissa. A leading “0” is considered a decimal leading 0; it does not
// indicate octal representation in this case.
// An optional base-10 “e” or base-2 “p” (or their upper-case variants)
// exponent may be provided as well, except for hexadecimal floats which
// only accept an (optional) “p” exponent (because an “e” or “E” cannot
// be distinguished from a mantissa digit). If the exponent's absolute value
// is too large, the operation may fail.
// The entire string, not just a prefix, must be valid for success. If the
// operation failed, the value of z is undefined but the returned value is nil.
func (z *Rat) SetString(s string) (*Rat, bool) {
	if len(s) == 0 {
		return nil, false
	}
	// len(s) > 0

	// parse fraction a/b, if any
	if sep := strings.Index(s, "/"); sep >= 0 {
		if _, ok := z.a.SetString(s[:sep], 0); !ok {
			return nil, false
		}
		r := strings.NewReader(s[sep+1:])
		var err error
		if z.b.abs, _, _, err = z.b.abs.scan(r, 0, false); err != nil {
			return nil, false
		}
		// entire string must have been consumed
		if _, err = r.ReadByte(); err != io.EOF {
			return nil, false
		}
		if len(z.b.abs) == 0 {
			return nil, false
		}
		return z.norm(), true
	}

	// parse floating-point number
	r := strings.NewReader(s)

	// sign
	neg, err := scanSign(r)
	if err != nil {
		return nil, false
	}

	// mantissa
	var base int
	var fcount int // fractional digit count; valid if <= 0
	z.a.abs, base, fcount, err = z.a.abs.scan(r, 0, true)
	if err != nil {
		return nil, false
	}

	// exponent
	var exp int64
	var ebase int
	exp, ebase, err = scanExponent(r, true, true)
	if err != nil {
		return nil, false
	}

	// there should be no unread characters left
	if _, err = r.ReadByte(); err != io.EOF {
		return nil, false
	}

	// special-case 0 (see also issue #16176)
	if len(z.a.abs) == 0 {
		return z.norm(), true
	}
	// len(z.a.abs) > 0

	// The mantissa may have a radix point (fcount <= 0) and there
	// may be a nonzero exponent exp. The radix point amounts to a
	// division by base**(-fcount), which equals a multiplication by
	// base**fcount. An exponent means multiplication by ebase**exp.
	// Multiplications are commutative, so we can apply them in any
	// order. We only have powers of 2 and 10, and we split powers
	// of 10 into the product of the same powers of 2 and 5. This
	// may reduce the size of shift/multiplication factors or
	// divisors required to create the final fraction, depending
	// on the actual floating-point value.

	// determine binary or decimal exponent contribution of radix point
	var exp2, exp5 int64
	if fcount < 0 {
		// The mantissa has a radix point ddd.dddd; and
		// -fcount is the number of digits to the right
		// of '.'. Adjust relevant exponent accordingly.
		d := int64(fcount)
		switch base {
		case 10:
			exp5 = d
			fallthrough // 10**e == 5**e * 2**e
		case 2:
			exp2 = d
		case 8:
			exp2 = d * 3 // octal digits are 3 bits each
		case 16:
			exp2 = d * 4 // hexadecimal digits are 4 bits each
		default:
			panic("unexpected mantissa base")
		}
		// fcount consumed - not needed anymore
	}

	// take actual exponent into account
	switch ebase {
	case 10:
		exp5 += exp
		fallthrough // see fallthrough above
	case 2:
		exp2 += exp
	default:
		panic("unexpected exponent base")
	}
	// exp consumed - not needed anymore

	// apply exp5 contributions
	// (start with exp5 so the numbers to multiply are smaller)
	if exp5 != 0 {
		n := exp5
		if n < 0 {
			n = -n
			if n < 0 {
				// This can occur if -n overflows. -(-1 << 63) would become
				// -1 << 63, which is still negative.
				return nil, false
			}
		}
		if n > 1e6 {
			return nil, false // avoid excessively large exponents
		}
		pow5 := z.b.abs.expNN(natFive, nat(nil).setWord(Word(n)), nil, false) // use underlying array of z.b.abs
		if exp5 > 0 {
			z.a.abs = z.a.abs.mul(z.a.abs, pow5)
			z.b.abs = z.b.abs.setWord(1)
		} else {
			z.b.abs = pow5
		}
	} else {
		z.b.abs = z.b.abs.setWord(1)
	}

	// apply exp2 contributions
	if exp2 < -1e7 || exp2 > 1e7 {
		return nil, false // avoid excessively large exponents
	}
	if exp2 > 0 {
		z.a.abs = z.a.abs.shl(z.a.abs, uint(exp2))
	} else if exp2 < 0 {
		z.b.abs = z.b.abs.shl(z.b.abs, uint(-exp2))
	}

	z.a.neg = neg && len(z.a.abs) > 0 // 0 has no sign

	return z.norm(), true
}

// scanExponent scans the longest possible prefix of r representing a base 10
// (“e”, “E”) or a base 2 (“p”, “P”) exponent, if any. It returns the
// exponent, the exponent base (10 or 2), or a read or syntax error, if any.
//
// If sepOk is set, an underscore character “_” may appear between successive
// exponent digits; such underscores do not change the value of the exponent.
// Incorrect placement of underscores is reported as an error if there are no
// other errors. If sepOk is not set, underscores are not recognized and thus
// terminate scanning like any other character that is not a valid digit.
//
//	exponent = ( "e" | "E" | "p" | "P" ) [ sign ] digits .
//	sign     = "+" | "-" .
//	digits   = digit { [ '_' ] digit } .
//	digit    = "0" ... "9" .
//
// A base 2 exponent is only permitted if base2ok is set.
func scanExponent(r io.ByteScanner, base2ok, sepOk bool) (exp int64, base int, err error) {
	// one char look-ahead
	ch, err := r.ReadByte()
	if err != nil {
		if err == io.EOF {
			err = nil
		}
		return 0, 10, err
	}

	// exponent char
	switch ch {
	case 'e', 'E':
		base = 10
	case 'p', 'P':
		if base2ok {
			base = 2
			break // ok
		}
		fallthrough // binary exponent not permitted
	default:
		r.UnreadByte() // ch does not belong to exponent anymore
		return 0, 10, nil
	}

	// sign
	var digits []byte
	ch, err = r.ReadByte()
	if err == nil && (ch == '+' || ch == '-') {
		if ch == '-' {
			digits = append(digits, '-')
		}
		ch, err = r.ReadByte()
	}

	// prev encodes the previously seen char: it is one
	// of '_', '0' (a digit), or '.' (anything else). A
	// valid separator '_' may only occur after a digit.
	prev := '.'
	invalSep := false

	// exponent value
	hasDigits := false
	for err == nil {
		if '0' <= ch && ch <= '9' {
			digits = append(digits, ch)
			prev = '0'
			hasDigits = true
		} else if ch == '_' && sepOk {
			if prev != '0' {
				invalSep = true
			}
			prev = '_'
		} else {
			r.UnreadByte() // ch does not belong to number anymore
			break
		}
		ch, err = r.ReadByte()
	}

	if err == io.EOF {
		err = nil
	}
	if err == nil && !hasDigits {
		err = errNoDigits
	}
	if err == nil {
		exp, err = strconv.ParseInt(string(digits), 10, 64)
	}
	// other errors take precedence over invalid separators
	if err == nil && (invalSep || prev == '_') {
		err = errInvalSep
	}

	return
}

// String returns a string representation of x in the form "a/b" (even if b == 1).
func (x *Rat) String() string {
	return string(x.marshal())
}

// marshal implements String returning a slice of bytes
func (x *Rat) marshal() []byte {
	var buf []byte
	buf = x.a.Append(buf, 10)
	buf = append(buf, '/')
	if len(x.b.abs) != 0 {
		buf = x.b.Append(buf, 10)
	} else {
		buf = append(buf, '1')
	}
	return buf
}

// RatString returns a string representation of x in the form "a/b" if b != 1,
// and in the form "a" if b == 1.
func (x *Rat) RatString() string {
	if x.IsInt() {
		return x.a.String()
	}
	return x.String()
}

// FloatString returns a string representation of x in decimal form with prec
// digits of precision after the radix point. The last digit is rounded to
// nearest, with halves rounded away from zero.
func (x *Rat) FloatString(prec int) string {
	var buf []byte

	if x.IsInt() {
		buf = x.a.Append(buf, 10)
		if prec > 0 {
			buf = append(buf, '.')
			for i := prec; i > 0; i-- {
				buf = append(buf, '0')
			}
		}
		return string(buf)
	}
	// x.b.abs != 0

	q, r := nat(nil).div(nat(nil), x.a.abs, x.b.abs)

	p := natOne
	if prec > 0 {
		p = nat(nil).expNN(natTen, nat(nil).setUint64(uint64(prec)), nil, false)
	}

	r = r.mul(r, p)
	r, r2 := r.div(nat(nil), r, x.b.abs)

	// see if we need to round up
	r2 = r2.add(r2, r2)
	if x.b.abs.cmp(r2) <= 0 {
		r = r.add(r, natOne)
		if r.cmp(p) >= 0 {
			q = nat(nil).add(q, natOne)
			r = nat(nil).sub(r, p)
		}
	}

	if x.a.neg {
		buf = append(buf, '-')
	}
	buf = append(buf, q.utoa(10)...) // itoa ignores sign if q == 0

	if prec > 0 {
		buf = append(buf, '.')
		rs := r.utoa(10)
		for i := prec - len(rs); i > 0; i-- {
			buf = append(buf, '0')
		}
		buf = append(buf, rs...)
	}

	return string(buf)
}

// Note: FloatPrec (below) is in this file rather than rat.go because
//       its results are relevant for decimal representation/printing.

// FloatPrec returns the number n of non-repeating digits immediately
// following the decimal point of the decimal representation of x.
// The boolean result indicates whether a decimal representation of x
// with that many fractional digits is exact or rounded.
//
// Examples:
//
//	x      n    exact    decimal representation n fractional digits
//	0      0    true     0
//	1      0    true     1
//	1/2    1    true     0.5
//	1/3    0    false    0       (0.333... rounded)
//	1/4    2    true     0.25
//	1/6    1    false    0.2     (0.166... rounded)
func (x *Rat) FloatPrec() (n int, exact bool) {
	// Determine q and largest p2, p5 such that d = q·2^p2·5^p5.
	// The results n, exact are:
	//
	//     n = max(p2, p5)
	//     exact = q == 1
	//
	// For details see:
	// https://en.wikipedia.org/wiki/Repeating_decimal#Reciprocals_of_integers_not_coprime_to_10
	d := x.Denom().abs // d >= 1

	// Determine p2 by counting factors of 2.
	// p2 corresponds to the trailing zero bits in d.
	// Do this first to reduce q as much as possible.
	var q nat
	p2 := d.trailingZeroBits()
	q = q.shr(d, p2)

	// Determine p5 by counting factors of 5.
	// Build a table starting with an initial power of 5,
	// and use repeated squaring until the factor doesn't
	// divide q anymore. Then use the table to determine
	// the power of 5 in q.
	const fp = 13        // f == 5^fp
	var tab []nat        // tab[i] == (5^fp)^(2^i) == 5^(fp·2^i)
	f := nat{1220703125} // == 5^fp (must fit into a uint32 Word)
	var t, r nat         // temporaries
	for {
		if _, r = t.div(r, q, f); len(r) != 0 {
			break // f doesn't divide q evenly
		}
		tab = append(tab, f)
		f = nat(nil).sqr(f) // nat(nil) to ensure a new f for each table entry
	}

	// Factor q using the table entries, if any.
	// We start with the largest factor f = tab[len(tab)-1]
	// that evenly divides q. It does so at most once because
	// otherwise f·f would also divide q. That can't be true
	// because f·f is the next higher table entry, contradicting
	// how f was chosen in the first place.
	// The same reasoning applies to the subsequent factors.
	var p5 uint
	for i := len(tab) - 1; i >= 0; i-- {
		if t, r = t.div(r, q, tab[i]); len(r) == 0 {
			p5 += fp * (1 << i) // tab[i] == 5^(fp·2^i)
			q = q.set(t)
		}
	}

	// If fp != 1, we may still have multiples of 5 left.
	for {
		if t, r = t.div(r, q, natFive); len(r) != 0 {
			break
		}
		p5++
		q = q.set(t)
	}

	return int(max(p2, p5)), q.cmp(natOne) == 0
}

The pages are generated with Golds v0.7.0-preview. (GOOS=linux GOARCH=amd64)
Golds is a Go 101 project developed by Tapir Liu.
PR and bug reports are welcome and can be submitted to the issue list.
Please follow @zigo_101 (reachable from the left QR code) to get the latest news of Golds.