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v/vlib/math/big/special_array_ops.v

347 lines
8.1 KiB
V

module big
import math.bits
import strings
[direct_array_access; inline]
fn shrink_tail_zeros(mut a []u32) {
mut alen := a.len
for alen > 0 && a[alen - 1] == 0 {
alen--
}
unsafe {
a.len = alen
}
}
// suppose operand_a bigger than operand_b and both not null.
// Both quotient and remaider are already allocated but of length 0
fn newton_divide_array_by_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut remainder []u32) {
// tranform back to Integers (on the stack without allocation)
a := Integer{
signum: 1
digits: operand_a
}
b := Integer{
signum: 1
digits: operand_b
}
k := a.bit_len() + b.bit_len() // a*b < 2**k
mut x := integer_from_int(2) // 0 < x < 2**(k+1)/b // initial guess for convergence
// https://en.wikipedia.org/wiki/Division_algorithm#Newton%E2%80%93Raphson_division
// use 48/17 - 32/17.D (divisor)
initial_guess := (((integer_from_int(48) - (integer_from_int(32) * b)) * integer_from_i64(0x0f0f0f0f0f0f0f0f)).right_shift(64)).neg() // / 17 == 0x11
if initial_guess > zero_int {
x = initial_guess
}
mut lastx := integer_from_int(0)
pow2_k_plus_1 := pow2(k + 1) // outside of the loop to optimize allocatio
for lastx != x { // main loop
lastx = x
x = (x * (pow2_k_plus_1 - (x * b))).right_shift(u32(k))
}
if x * b < pow2(k) {
x.inc()
}
mut q := (a * x).right_shift(u32(k))
// possible adjustments. see literature
if q * b > a {
q.dec()
}
mut r := a - (q * b)
if r >= b {
q.inc()
r -= b
}
quotient = q.digits.clone()
remainder = r.digits.clone()
shrink_tail_zeros(mut remainder)
}
// bit_length returns the number of bits needed to represent the absolute value of the integer a.
[deprecated: 'use a.bit_len() instead']
[inline]
pub fn bit_length(a Integer) int {
return a.digits.len * 32 - bits.leading_zeros_32(a.digits.last())
}
[direct_array_access; inline]
fn debug_u32_str(a []u32) string {
mut sb := strings.new_builder(30)
sb.write_string('[')
mut first := true
for i in 0 .. a.len {
if !first {
sb.write_string(', ')
}
sb.write_string('0x${a[i].hex()}')
first = false
}
sb.write_string(']')
return sb.str()
}
[direct_array_access; inline]
fn found_multiplication_base_case(operand_a []u32, operand_b []u32, mut storage []u32) bool {
// base case necessary to end recursion
if operand_a.len == 0 || operand_b.len == 0 {
storage.clear()
return true
}
if operand_a.len < operand_b.len {
multiply_digit_array(operand_b, operand_a, mut storage)
return true
}
if operand_b.len == 1 {
multiply_array_by_digit(operand_a, operand_b[0], mut storage)
return true
}
return false
}
// karatsuba algorithm for multiplication
// possible optimisations:
// - transform one or all the recurrences in loops
[direct_array_access]
fn karatsuba_multiply_digit_array(operand_a []u32, operand_b []u32, mut storage []u32) {
if found_multiplication_base_case(operand_a, operand_b, mut storage) {
return
}
// thanks to the base cases we can pass zero-length arrays to the mult func
half := imax(operand_a.len, operand_b.len) / 2
a_l := operand_a[0..half]
a_h := operand_a[half..]
mut b_l := []u32{}
mut b_h := []u32{}
if half <= operand_b.len {
b_l = operand_b[0..half]
b_h = operand_b[half..]
} else {
b_l = unsafe { operand_b }
// b_h = []u32{}
}
// use storage for p_1 to avoid allocation and copy later
multiply_digit_array(a_h, b_h, mut storage)
mut p_3 := []u32{len: a_l.len + b_l.len + 1}
multiply_digit_array(a_l, b_l, mut p_3)
mut tmp_1 := []u32{len: imax(a_h.len, a_l.len) + 1}
mut tmp_2 := []u32{len: imax(b_h.len, b_l.len) + 1}
add_digit_array(a_h, a_l, mut tmp_1)
add_digit_array(b_h, b_l, mut tmp_2)
mut p_2 := []u32{len: operand_a.len + operand_b.len + 1}
multiply_digit_array(tmp_1, tmp_2, mut p_2)
subtract_in_place(mut p_2, storage) // p_1
subtract_in_place(mut p_2, p_3)
// return p_1.left_shift(2 * u32(half * 32)) + p_2.left_shift(u32(half * 32)) + p_3
left_shift_digits_in_place(mut storage, 2 * half)
left_shift_digits_in_place(mut p_2, half)
add_in_place(mut storage, p_2)
add_in_place(mut storage, p_3)
shrink_tail_zeros(mut storage)
}
[direct_array_access]
fn toom3_multiply_digit_array(operand_a []u32, operand_b []u32, mut storage []u32) {
if found_multiplication_base_case(operand_a, operand_b, mut storage) {
return
}
// After the base case, we have operand_a as the larger integer in terms of digit length
// k is the length (in u32 digits) of the lower order slices
k := (operand_a.len + 2) / 3
k2 := 2 * k
// The pieces of the calculation need to be worked on as proper big.Integers
// because the intermediate results can be negative. After recombination, the
// final result will be positive.
// Slices of a and b
a0 := Integer{
digits: operand_a[0..k]
signum: 1
}
a1 := Integer{
digits: operand_a[k..k2]
signum: 1
}
a2 := Integer{
digits: operand_a[k2..]
signum: 1
}
// Zero arrays by default
mut b0 := zero_int.clone()
mut b1 := zero_int.clone()
mut b2 := zero_int.clone()
if operand_b.len < k {
b0 = Integer{
digits: operand_b
signum: 1
}
} else if operand_b.len < k2 {
b0 = Integer{
digits: operand_b[0..k]
signum: 1
}
b1 = Integer{
digits: operand_b[k..]
signum: 1
}
} else {
b0 = Integer{
digits: operand_b[0..k]
signum: 1
}
b1 = Integer{
digits: operand_b[k..k2]
signum: 1
}
b2 = Integer{
digits: operand_b[k2..]
signum: 1
}
}
// https://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication#Details
// DOI: 10.1007/978-3-540-73074-3_10
p0 := a0 * b0
mut ptemp := a2 + a0
mut qtemp := b2 + b0
vm1 := (ptemp - a1) * (qtemp - b1)
ptemp += a1
qtemp += b1
p1 := ptemp * qtemp
p2 := ((ptemp + a2).left_shift(1) - a0) * ((qtemp + b2).left_shift(1) - b0)
pinf := a2 * b2
mut t2 := (p2 - vm1) / three_int
mut tm1 := (p1 - vm1).right_shift(1)
mut t1 := p1 - p0
t2 = (t2 - t1).right_shift(1)
t1 = (t1 - tm1 - pinf)
t2 = t2 - pinf.left_shift(1)
tm1 = tm1 - t2
// shift amount
s := u32(k) << 5
result := (((pinf.left_shift(s) + t2).left_shift(s) + t1).left_shift(s) + tm1).left_shift(s) +
p0
storage = result.digits.clone()
}
[inline]
fn pow2(k int) Integer {
mut ret := []u32{len: (k >> 5) + 1}
bit_set(mut ret, k)
return Integer{
signum: 1
digits: ret
}
}
// optimized left shift in place. amount must be positive
[direct_array_access]
fn left_shift_digits_in_place(mut a []u32, amount int) {
a_len := a.len
// control or allocate capacity
for _ in a_len .. a_len + amount {
a << u32(0)
}
for index := a_len - 1; index >= 0; index-- {
a[index + amount] = a[index]
}
for index in 0 .. amount {
a[index] = u32(0)
}
}
// optimized right shift in place. amount must be positive
[direct_array_access]
fn right_shift_digits_in_place(mut a []u32, amount int) {
for index := 0; index < a.len - amount; index++ {
a[index] = a[index + amount]
}
for index := a.len - amount; index < a.len; index++ {
a[index] = u32(0)
}
shrink_tail_zeros(mut a)
}
// operand b can be greater than operand a
// the capacity of both array is supposed to be sufficient
[direct_array_access; inline]
fn add_in_place(mut a []u32, b []u32) {
len_a := a.len
len_b := b.len
max := imax(len_a, len_b)
min := imin(len_a, len_b)
mut carry := u64(0)
for index in 0 .. min {
partial := carry + a[index] + b[index]
a[index] = u32(partial)
carry = u32(partial >> 32)
}
if len_a >= len_b {
for index in min .. max {
partial := carry + a[index]
a[index] = u32(partial)
carry = u32(partial >> 32)
}
} else {
for index in min .. max {
partial := carry + b[index]
a << u32(partial)
carry = u32(partial >> 32)
}
}
}
// a := a - b supposed a >= b
[direct_array_access]
fn subtract_in_place(mut a []u32, b []u32) {
len_a := a.len
len_b := b.len
max := imax(len_a, len_b)
min := imin(len_a, len_b)
mut carry := u32(0)
mut new_carry := u32(0)
for index in 0 .. min {
new_carry = if a[index] < (b[index] + carry) {
u32(1)
} else {
u32(0)
}
a[index] -= (b[index] + carry)
carry = new_carry
}
if len_a >= len_b {
for index in min .. max {
new_carry = if a[index] < carry {
u32(1)
} else {
u32(0)
}
a[index] -= carry
carry = new_carry
}
} else { // if len.b > len.a return zero
a.clear()
}
}