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zig/lib/std/math/big/rational.zig

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// SPDX-License-Identifier: MIT
// Copyright (c) 2015-2020 Zig Contributors
// This file is part of [zig](https://ziglang.org/), which is MIT licensed.
// The MIT license requires this copyright notice to be included in all copies
// and substantial portions of the software.
const std = @import("../../std.zig");
const debug = std.debug;
const math = std.math;
const mem = std.mem;
const testing = std.testing;
const Allocator = mem.Allocator;
const Limb = std.math.big.Limb;
const DoubleLimb = std.math.big.DoubleLimb;
const Int = std.math.big.int.Managed;
const IntConst = std.math.big.int.Const;
/// An arbitrary-precision rational number.
///
/// Memory is allocated as needed for operations to ensure full precision is kept. The precision
/// of a Rational is only bounded by memory.
///
/// Rational's are always normalized. That is, for a Rational r = p/q where p and q are integers,
/// gcd(p, q) = 1 always.
///
/// TODO rework this to store its own allocator and use a non-managed big int, to avoid double
/// allocator storage.
pub const Rational = struct {
/// Numerator. Determines the sign of the Rational.
p: Int,
/// Denominator. Sign is ignored.
q: Int,
/// Create a new Rational. A small amount of memory will be allocated on initialization.
/// This will be 2 * Int.default_capacity.
pub fn init(a: *Allocator) !Rational {
return Rational{
.p = try Int.init(a),
.q = try Int.initSet(a, 1),
};
}
/// Frees all memory associated with a Rational.
pub fn deinit(self: *Rational) void {
self.p.deinit();
self.q.deinit();
}
/// Set a Rational from a primitive integer type.
pub fn setInt(self: *Rational, a: anytype) !void {
try self.p.set(a);
try self.q.set(1);
}
/// Set a Rational from a string of the form `A/B` where A and B are base-10 integers.
pub fn setFloatString(self: *Rational, str: []const u8) !void {
// TODO: Accept a/b fractions and exponent form
if (str.len == 0) {
return error.InvalidFloatString;
}
const State = enum {
Integer,
Fractional,
};
var state = State.Integer;
var point: ?usize = null;
var start: usize = 0;
if (str[0] == '-') {
start += 1;
}
for (str) |c, i| {
switch (state) {
State.Integer => {
switch (c) {
'.' => {
state = State.Fractional;
point = i;
},
'0'...'9' => {
// okay
},
else => {
return error.InvalidFloatString;
},
}
},
State.Fractional => {
switch (c) {
'0'...'9' => {
// okay
},
else => {
return error.InvalidFloatString;
},
}
},
}
}
// TODO: batch the multiplies by 10
if (point) |i| {
try self.p.setString(10, str[0..i]);
const base = IntConst{ .limbs = &[_]Limb{10}, .positive = true };
var j: usize = start;
while (j < str.len - i - 1) : (j += 1) {
try self.p.ensureMulCapacity(self.p.toConst(), base);
try self.p.mul(self.p.toConst(), base);
}
try self.q.setString(10, str[i + 1 ..]);
try self.p.add(self.p.toConst(), self.q.toConst());
try self.q.set(1);
var k: usize = i + 1;
while (k < str.len) : (k += 1) {
try self.q.mul(self.q.toConst(), base);
}
try self.reduce();
} else {
try self.p.setString(10, str[0..]);
try self.q.set(1);
}
}
/// Set a Rational from a floating-point value. The rational will have enough precision to
/// completely represent the provided float.
pub fn setFloat(self: *Rational, comptime T: type, f: T) !void {
// Translated from golang.go/src/math/big/rat.go.
debug.assert(@typeInfo(T) == .Float);
const UnsignedInt = std.meta.Int(.unsigned, @typeInfo(T).Float.bits);
const f_bits = @bitCast(UnsignedInt, f);
const exponent_bits = math.floatExponentBits(T);
const exponent_bias = (1 << (exponent_bits - 1)) - 1;
const mantissa_bits = math.floatMantissaBits(T);
const exponent_mask = (1 << exponent_bits) - 1;
const mantissa_mask = (1 << mantissa_bits) - 1;
var exponent = @intCast(i16, (f_bits >> mantissa_bits) & exponent_mask);
var mantissa = f_bits & mantissa_mask;
switch (exponent) {
exponent_mask => {
return error.NonFiniteFloat;
},
0 => {
// denormal
exponent -= exponent_bias - 1;
},
else => {
// normal
mantissa |= 1 << mantissa_bits;
exponent -= exponent_bias;
},
}
var shift: i16 = mantissa_bits - exponent;
// factor out powers of two early from rational
while (mantissa & 1 == 0 and shift > 0) {
mantissa >>= 1;
shift -= 1;
}
try self.p.set(mantissa);
self.p.setSign(f >= 0);
try self.q.set(1);
if (shift >= 0) {
try self.q.shiftLeft(self.q, @intCast(usize, shift));
} else {
try self.p.shiftLeft(self.p, @intCast(usize, -shift));
}
try self.reduce();
}
/// Return a floating-point value that is the closest value to a Rational.
///
/// The result may not be exact if the Rational is too precise or too large for the
/// target type.
pub fn toFloat(self: Rational, comptime T: type) !T {
// Translated from golang.go/src/math/big/rat.go.
// TODO: Indicate whether the result is not exact.
debug.assert(@typeInfo(T) == .Float);
const fsize = @typeInfo(T).Float.bits;
const BitReprType = std.meta.Int(.unsigned, fsize);
const msize = math.floatMantissaBits(T);
const msize1 = msize + 1;
const msize2 = msize1 + 1;
const esize = math.floatExponentBits(T);
const ebias = (1 << (esize - 1)) - 1;
const emin = 1 - ebias;
const emax = ebias;
if (self.p.eqZero()) {
return 0;
}
// 1. left-shift a or sub so that a/b is in [1 << msize1, 1 << (msize2 + 1)]
var exp = @intCast(isize, self.p.bitCountTwosComp()) - @intCast(isize, self.q.bitCountTwosComp());
var a2 = try self.p.clone();
defer a2.deinit();
var b2 = try self.q.clone();
defer b2.deinit();
const shift = msize2 - exp;
if (shift >= 0) {
try a2.shiftLeft(a2, @intCast(usize, shift));
} else {
try b2.shiftLeft(b2, @intCast(usize, -shift));
}
// 2. compute quotient and remainder
var q = try Int.init(self.p.allocator);
defer q.deinit();
// unused
var r = try Int.init(self.p.allocator);
defer r.deinit();
try Int.divTrunc(&q, &r, a2.toConst(), b2.toConst());
var mantissa = extractLowBits(q, BitReprType);
var have_rem = r.len() > 0;
// 3. q didn't fit in msize2 bits, redo division b2 << 1
if (mantissa >> msize2 == 1) {
if (mantissa & 1 == 1) {
have_rem = true;
}
mantissa >>= 1;
exp += 1;
}
if (mantissa >> msize1 != 1) {
// NOTE: This can be hit if the limb size is small (u8/16).
@panic("unexpected bits in result");
}
// 4. Rounding
if (emin - msize <= exp and exp <= emin) {
// denormal
const shift1 = @intCast(math.Log2Int(BitReprType), emin - (exp - 1));
const lost_bits = mantissa & ((@intCast(BitReprType, 1) << shift1) - 1);
have_rem = have_rem or lost_bits != 0;
mantissa >>= shift1;
exp = 2 - ebias;
}
// round q using round-half-to-even
var exact = !have_rem;
if (mantissa & 1 != 0) {
exact = false;
if (have_rem or (mantissa & 2 != 0)) {
mantissa += 1;
if (mantissa >= 1 << msize2) {
// 11...1 => 100...0
mantissa >>= 1;
exp += 1;
}
}
}
mantissa >>= 1;
const f = math.scalbn(@intToFloat(T, mantissa), @intCast(i32, exp - msize1));
if (math.isInf(f)) {
exact = false;
}
return if (self.p.isPositive()) f else -f;
}
/// Set a rational from an integer ratio.
pub fn setRatio(self: *Rational, p: anytype, q: anytype) !void {
try self.p.set(p);
try self.q.set(q);
self.p.setSign(@boolToInt(self.p.isPositive()) ^ @boolToInt(self.q.isPositive()) == 0);
self.q.setSign(true);
try self.reduce();
if (self.q.eqZero()) {
@panic("cannot set rational with denominator = 0");
}
}
/// Set a Rational directly from an Int.
pub fn copyInt(self: *Rational, a: Int) !void {
try self.p.copy(a.toConst());
try self.q.set(1);
}
/// Set a Rational directly from a ratio of two Int's.
pub fn copyRatio(self: *Rational, a: Int, b: Int) !void {
try self.p.copy(a.toConst());
try self.q.copy(b.toConst());
self.p.setSign(@boolToInt(self.p.isPositive()) ^ @boolToInt(self.q.isPositive()) == 0);
self.q.setSign(true);
try self.reduce();
}
/// Make a Rational positive.
pub fn abs(r: *Rational) void {
r.p.abs();
}
/// Negate the sign of a Rational.
pub fn negate(r: *Rational) void {
r.p.negate();
}
/// Efficiently swap a Rational with another. This swaps the limb pointers and a full copy is not
/// performed. The address of the limbs field will not be the same after this function.
pub fn swap(r: *Rational, other: *Rational) void {
r.p.swap(&other.p);
r.q.swap(&other.q);
}
/// Returns math.Order.lt, math.Order.eq, math.Order.gt if a < b, a == b or a
/// > b respectively.
pub fn order(a: Rational, b: Rational) !math.Order {
return cmpInternal(a, b, true);
}
/// Returns math.Order.lt, math.Order.eq, math.Order.gt if |a| < |b|, |a| ==
/// |b| or |a| > |b| respectively.
pub fn orderAbs(a: Rational, b: Rational) !math.Order {
return cmpInternal(a, b, false);
}
// p/q > x/y iff p*y > x*q
fn cmpInternal(a: Rational, b: Rational, is_abs: bool) !math.Order {
// TODO: Would a div compare algorithm of sorts be viable and quicker? Can we avoid
// the memory allocations here?
var q = try Int.init(a.p.allocator);
defer q.deinit();
var p = try Int.init(b.p.allocator);
defer p.deinit();
try q.mul(a.p.toConst(), b.q.toConst());
try p.mul(b.p.toConst(), a.q.toConst());
return if (is_abs) q.orderAbs(p) else q.order(p);
}
/// rma = a + b.
///
/// rma, a and b may be aliases. However, it is more efficient if rma does not alias a or b.
///
/// Returns an error if memory could not be allocated.
pub fn add(rma: *Rational, a: Rational, b: Rational) !void {
var r = rma;
var aliased = rma.p.limbs.ptr == a.p.limbs.ptr or rma.p.limbs.ptr == b.p.limbs.ptr;
var sr: Rational = undefined;
if (aliased) {
sr = try Rational.init(rma.p.allocator);
r = &sr;
aliased = true;
}
defer if (aliased) {
rma.swap(r);
r.deinit();
};
try r.p.mul(a.p.toConst(), b.q.toConst());
try r.q.mul(b.p.toConst(), a.q.toConst());
try r.p.add(r.p.toConst(), r.q.toConst());
try r.q.mul(a.q.toConst(), b.q.toConst());
try r.reduce();
}
/// rma = a - b.
///
/// rma, a and b may be aliases. However, it is more efficient if rma does not alias a or b.
///
/// Returns an error if memory could not be allocated.
pub fn sub(rma: *Rational, a: Rational, b: Rational) !void {
var r = rma;
var aliased = rma.p.limbs.ptr == a.p.limbs.ptr or rma.p.limbs.ptr == b.p.limbs.ptr;
var sr: Rational = undefined;
if (aliased) {
sr = try Rational.init(rma.p.allocator);
r = &sr;
aliased = true;
}
defer if (aliased) {
rma.swap(r);
r.deinit();
};
try r.p.mul(a.p.toConst(), b.q.toConst());
try r.q.mul(b.p.toConst(), a.q.toConst());
try r.p.sub(r.p.toConst(), r.q.toConst());
try r.q.mul(a.q.toConst(), b.q.toConst());
try r.reduce();
}
/// rma = a * b.
///
/// rma, a and b may be aliases. However, it is more efficient if rma does not alias a or b.
///
/// Returns an error if memory could not be allocated.
pub fn mul(r: *Rational, a: Rational, b: Rational) !void {
try r.p.mul(a.p.toConst(), b.p.toConst());
try r.q.mul(a.q.toConst(), b.q.toConst());
try r.reduce();
}
/// rma = a / b.
///
/// rma, a and b may be aliases. However, it is more efficient if rma does not alias a or b.
///
/// Returns an error if memory could not be allocated.
pub fn div(r: *Rational, a: Rational, b: Rational) !void {
if (b.p.eqZero()) {
@panic("division by zero");
}
try r.p.mul(a.p.toConst(), b.q.toConst());
try r.q.mul(b.p.toConst(), a.q.toConst());
try r.reduce();
}
/// Invert the numerator and denominator fields of a Rational. p/q => q/p.
pub fn invert(r: *Rational) void {
Int.swap(&r.p, &r.q);
}
// reduce r/q such that gcd(r, q) = 1
fn reduce(r: *Rational) !void {
var a = try Int.init(r.p.allocator);
defer a.deinit();
const sign = r.p.isPositive();
r.p.abs();
try a.gcd(r.p, r.q);
r.p.setSign(sign);
const one = IntConst{ .limbs = &[_]Limb{1}, .positive = true };
if (a.toConst().order(one) != .eq) {
var unused = try Int.init(r.p.allocator);
defer unused.deinit();
// TODO: divexact would be useful here
// TODO: don't copy r.q for div
try Int.divTrunc(&r.p, &unused, r.p.toConst(), a.toConst());
try Int.divTrunc(&r.q, &unused, r.q.toConst(), a.toConst());
}
}
};
fn extractLowBits(a: Int, comptime T: type) T {
testing.expect(@typeInfo(T) == .Int);
const t_bits = @typeInfo(T).Int.bits;
const limb_bits = @typeInfo(Limb).Int.bits;
if (t_bits <= limb_bits) {
return @truncate(T, a.limbs[0]);
} else {
var r: T = 0;
comptime var i: usize = 0;
// Remainder is always 0 since if t_bits >= limb_bits -> Limb | T and both
// are powers of two.
inline while (i < t_bits / limb_bits) : (i += 1) {
r |= math.shl(T, a.limbs[i], i * limb_bits);
}
return r;
}
}
test "big.rational extractLowBits" {
var a = try Int.initSet(testing.allocator, 0x11112222333344441234567887654321);
defer a.deinit();
const a1 = extractLowBits(a, u8);
testing.expect(a1 == 0x21);
const a2 = extractLowBits(a, u16);
testing.expect(a2 == 0x4321);
const a3 = extractLowBits(a, u32);
testing.expect(a3 == 0x87654321);
const a4 = extractLowBits(a, u64);
testing.expect(a4 == 0x1234567887654321);
const a5 = extractLowBits(a, u128);
testing.expect(a5 == 0x11112222333344441234567887654321);
}
test "big.rational set" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
try a.setInt(5);
testing.expect((try a.p.to(u32)) == 5);
testing.expect((try a.q.to(u32)) == 1);
try a.setRatio(7, 3);
testing.expect((try a.p.to(u32)) == 7);
testing.expect((try a.q.to(u32)) == 3);
try a.setRatio(9, 3);
testing.expect((try a.p.to(i32)) == 3);
testing.expect((try a.q.to(i32)) == 1);
try a.setRatio(-9, 3);
testing.expect((try a.p.to(i32)) == -3);
testing.expect((try a.q.to(i32)) == 1);
try a.setRatio(9, -3);
testing.expect((try a.p.to(i32)) == -3);
testing.expect((try a.q.to(i32)) == 1);
try a.setRatio(-9, -3);
testing.expect((try a.p.to(i32)) == 3);
testing.expect((try a.q.to(i32)) == 1);
}
test "big.rational setFloat" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
try a.setFloat(f64, 2.5);
testing.expect((try a.p.to(i32)) == 5);
testing.expect((try a.q.to(i32)) == 2);
try a.setFloat(f32, -2.5);
testing.expect((try a.p.to(i32)) == -5);
testing.expect((try a.q.to(i32)) == 2);
try a.setFloat(f32, 3.141593);
// = 3.14159297943115234375
testing.expect((try a.p.to(u32)) == 3294199);
testing.expect((try a.q.to(u32)) == 1048576);
try a.setFloat(f64, 72.141593120712409172417410926841290461290467124);
// = 72.1415931207124145885245525278151035308837890625
testing.expect((try a.p.to(u128)) == 5076513310880537);
testing.expect((try a.q.to(u128)) == 70368744177664);
}
test "big.rational setFloatString" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
try a.setFloatString("72.14159312071241458852455252781510353");
// = 72.1415931207124145885245525278151035308837890625
testing.expect((try a.p.to(u128)) == 7214159312071241458852455252781510353);
testing.expect((try a.q.to(u128)) == 100000000000000000000000000000000000);
}
test "big.rational toFloat" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
// = 3.14159297943115234375
try a.setRatio(3294199, 1048576);
testing.expect((try a.toFloat(f64)) == 3.14159297943115234375);
// = 72.1415931207124145885245525278151035308837890625
try a.setRatio(5076513310880537, 70368744177664);
testing.expect((try a.toFloat(f64)) == 72.141593120712409172417410926841290461290467124);
}
test "big.rational set/to Float round-trip" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var prng = std.rand.DefaultPrng.init(0x5EED);
var i: usize = 0;
while (i < 512) : (i += 1) {
const r = prng.random.float(f64);
try a.setFloat(f64, r);
testing.expect((try a.toFloat(f64)) == r);
}
}
test "big.rational copy" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Int.initSet(testing.allocator, 5);
defer b.deinit();
try a.copyInt(b);
testing.expect((try a.p.to(u32)) == 5);
testing.expect((try a.q.to(u32)) == 1);
var c = try Int.initSet(testing.allocator, 7);
defer c.deinit();
var d = try Int.initSet(testing.allocator, 3);
defer d.deinit();
try a.copyRatio(c, d);
testing.expect((try a.p.to(u32)) == 7);
testing.expect((try a.q.to(u32)) == 3);
var e = try Int.initSet(testing.allocator, 9);
defer e.deinit();
var f = try Int.initSet(testing.allocator, 3);
defer f.deinit();
try a.copyRatio(e, f);
testing.expect((try a.p.to(u32)) == 3);
testing.expect((try a.q.to(u32)) == 1);
}
test "big.rational negate" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
try a.setInt(-50);
testing.expect((try a.p.to(i32)) == -50);
testing.expect((try a.q.to(i32)) == 1);
a.negate();
testing.expect((try a.p.to(i32)) == 50);
testing.expect((try a.q.to(i32)) == 1);
a.negate();
testing.expect((try a.p.to(i32)) == -50);
testing.expect((try a.q.to(i32)) == 1);
}
test "big.rational abs" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
try a.setInt(-50);
testing.expect((try a.p.to(i32)) == -50);
testing.expect((try a.q.to(i32)) == 1);
a.abs();
testing.expect((try a.p.to(i32)) == 50);
testing.expect((try a.q.to(i32)) == 1);
a.abs();
testing.expect((try a.p.to(i32)) == 50);
testing.expect((try a.q.to(i32)) == 1);
}
test "big.rational swap" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
try a.setRatio(50, 23);
try b.setRatio(17, 3);
testing.expect((try a.p.to(u32)) == 50);
testing.expect((try a.q.to(u32)) == 23);
testing.expect((try b.p.to(u32)) == 17);
testing.expect((try b.q.to(u32)) == 3);
a.swap(&b);
testing.expect((try a.p.to(u32)) == 17);
testing.expect((try a.q.to(u32)) == 3);
testing.expect((try b.p.to(u32)) == 50);
testing.expect((try b.q.to(u32)) == 23);
}
test "big.rational order" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
try a.setRatio(500, 231);
try b.setRatio(18903, 8584);
testing.expect((try a.order(b)) == .lt);
try a.setRatio(890, 10);
try b.setRatio(89, 1);
testing.expect((try a.order(b)) == .eq);
}
test "big.rational add single-limb" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
try a.setRatio(500, 231);
try b.setRatio(18903, 8584);
testing.expect((try a.order(b)) == .lt);
try a.setRatio(890, 10);
try b.setRatio(89, 1);
testing.expect((try a.order(b)) == .eq);
}
test "big.rational add" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
var r = try Rational.init(testing.allocator);
defer r.deinit();
try a.setRatio(78923, 23341);
try b.setRatio(123097, 12441414);
try a.add(a, b);
try r.setRatio(984786924199, 290395044174);
testing.expect((try a.order(r)) == .eq);
}
test "big.rational sub" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
var r = try Rational.init(testing.allocator);
defer r.deinit();
try a.setRatio(78923, 23341);
try b.setRatio(123097, 12441414);
try a.sub(a, b);
try r.setRatio(979040510045, 290395044174);
testing.expect((try a.order(r)) == .eq);
}
test "big.rational mul" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
var r = try Rational.init(testing.allocator);
defer r.deinit();
try a.setRatio(78923, 23341);
try b.setRatio(123097, 12441414);
try a.mul(a, b);
try r.setRatio(571481443, 17082061422);
testing.expect((try a.order(r)) == .eq);
}
test "big.rational div" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var b = try Rational.init(testing.allocator);
defer b.deinit();
var r = try Rational.init(testing.allocator);
defer r.deinit();
try a.setRatio(78923, 23341);
try b.setRatio(123097, 12441414);
try a.div(a, b);
try r.setRatio(75531824394, 221015929);
testing.expect((try a.order(r)) == .eq);
}
test "big.rational div" {
var a = try Rational.init(testing.allocator);
defer a.deinit();
var r = try Rational.init(testing.allocator);
defer r.deinit();
try a.setRatio(78923, 23341);
a.invert();
try r.setRatio(23341, 78923);
testing.expect((try a.order(r)) == .eq);
try a.setRatio(-78923, 23341);
a.invert();
try r.setRatio(-23341, 78923);
testing.expect((try a.order(r)) == .eq);
}
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