bigint: division (#11386)
Vincent Laisney
GitHub
parent
56ad5d72ef
commit
43fae5de74
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@ -202,7 +202,7 @@ fn divide_digit_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut
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}
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// Performs division on the non-negative dividend in a by the single digit divisor b. It assumes
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// quotient and remainder are empty zero length arrays with sufficient capacity
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// quotient and remainder are empty zero length arrays without previous allocation
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fn divide_array_by_digit(operand_a []u32, divisor u32, mut quotient []u32, mut remainder []u32) {
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if operand_a.len == 1 {
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// 1 digit for both dividend and divisor
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@ -240,58 +240,6 @@ fn divide_array_by_digit(operand_a []u32, divisor u32, mut quotient []u32, mut r
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}
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}
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// Performs division on the non-negative dividend in a by the multi digit divisor b. It assumes
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// quotient and remainder are empty zero length arrays with sufficient capacity
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// This is different from divide_digit_array because it depends on this very function
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// after making sure that the divisor is indeed multi-digit.
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fn divide_array_by_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut remainder []u32) {
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for index in 0 .. operand_a.len {
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remainder << operand_a[index]
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}
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for _ in 0 .. operand_b.len {
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quotient << 0
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}
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offset := operand_a.len - operand_b.len
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divisor_last_index := operand_b.len - 1
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for index := offset; index >= 0; index-- {
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dividend_last_index := divisor_last_index + index
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value_upper := if remainder.len > dividend_last_index + 1 {
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u64(remainder[dividend_last_index + 1])
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} else {
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u64(0)
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}
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value_lower := if remainder.len > dividend_last_index {
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u64(remainder[dividend_last_index])
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} else {
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u64(0)
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}
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partial := value_lower + (value_upper << 32)
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mut q := u32(partial / operand_b[divisor_last_index])
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if q > 0 {
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mut modified_divisor := []u32{len: operand_b.len + index, init: 0}
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for i in 0 .. operand_b.len {
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modified_divisor[index + i] = operand_b[i]
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}
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mut product := []u32{len: operand_b.len + 1, init: 0}
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multiply_array_by_digit(modified_divisor, q, mut product)
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for q > 0 && compare_digit_array(product, remainder) > 0 {
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q--
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subtract_digit_array(product, modified_divisor, mut product)
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}
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subtract_digit_array(remainder, product, mut remainder)
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}
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quotient[index] = q
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}
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// Remove leading zeros from quotient and remainder
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for quotient.len > 0 && quotient.last() == 0 {
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quotient.delete_last()
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}
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for remainder.len > 0 && remainder.last() == 0 {
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remainder.delete_last()
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}
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}
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// Shifts the contents of the original array by the given amount of bits to the left.
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// This function assumes that the amount is less than 32. The storage is expected to
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// allocated with zeroes.
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@ -0,0 +1,139 @@
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module big
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import math.bits
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// suppose operand_a bigger than operand_b and both not null.
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// Both quotient and remaider are allocated but of length 0
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fn divide_array_by_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut remainder []u32) {
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for index in 0 .. operand_a.len {
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remainder << operand_a[index]
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}
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len_diff := operand_a.len - operand_b.len
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assert len_diff >= 0
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// we must do in place shift and operations.
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mut divisor := []u32{cap: operand_b.len}
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for _ in 0 .. len_diff {
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divisor << u32(0)
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}
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for index in 0 .. operand_b.len {
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divisor << operand_b[index]
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}
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for _ in 0 .. len_diff + 1 {
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quotient << u32(0)
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}
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lead_zer_remainder := u32(bits.leading_zeros_32(remainder.last()))
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lead_zer_divisor := u32(bits.leading_zeros_32(divisor.last()))
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bit_offset := (u32(32) * u32(len_diff)) + (lead_zer_divisor - lead_zer_remainder)
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// align
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if lead_zer_remainder < lead_zer_divisor {
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lshift_in_place(mut divisor, lead_zer_divisor - lead_zer_remainder)
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} else if lead_zer_remainder > lead_zer_divisor {
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lshift_in_place(mut remainder, lead_zer_remainder - lead_zer_divisor)
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}
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assert left_align_p(divisor[divisor.len - 1], remainder[remainder.len - 1])
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for bit_idx := int(bit_offset); bit_idx >= 0; bit_idx-- {
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if greater_equal_from_end(remainder, divisor) {
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bit_set(mut quotient, bit_idx)
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subtract_in_place(mut remainder, divisor)
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}
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rshift_in_place(mut divisor, 1)
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}
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// ajust
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if lead_zer_remainder > lead_zer_divisor {
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// rshift_in_place(mut quotient, lead_zer_remainder - lead_zer_divisor)
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rshift_in_place(mut remainder, lead_zer_remainder - lead_zer_divisor)
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}
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for remainder.len > 0 && remainder.last() == 0 {
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remainder.delete_last()
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}
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for quotient.len > 0 && quotient.last() == 0 {
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quotient.delete_last()
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}
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}
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// help routines for cleaner code but inline for performance
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// quicker than BitField.set_bit
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[inline]
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fn bit_set(mut a []u32, n int) {
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byte_offset := n / 32
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mask := u32(1) << u32(n % 32)
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assert a.len >= byte_offset
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a[byte_offset] |= mask
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}
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// a.len is greater or equal to b.len
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// returns true if a >= b (completed with zeroes)
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[inline]
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fn greater_equal_from_end(a []u32, b []u32) bool {
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assert a.len >= b.len
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offset := a.len - b.len
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for index := a.len - 1; index >= offset; index-- {
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if a[index] > b[index - offset] {
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return true
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} else if a[index] < b[index - offset] {
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return false
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}
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}
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return true
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}
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// logical left shift
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// there is no overflow. We know that the last bits are zero
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// and that n <= 32
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[inline]
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fn lshift_in_place(mut a []u32, n u32) {
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mut carry := u32(0)
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mut prec_carry := u32(0)
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mask := ((u32(1) << n) - 1) << (32 - n)
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for index in 0 .. a.len {
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prec_carry = carry >> (32 - n)
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carry = a[index] & mask
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a[index] <<= n
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a[index] |= prec_carry
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}
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}
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// logical right shift without control because these digits have already been
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// shift left before
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[inline]
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fn rshift_in_place(mut a []u32, n u32) {
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mut carry := u32(0)
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mut prec_carry := u32(0)
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mask := u32((1 << n) - 1)
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for index := a.len - 1; index >= 0; index-- {
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carry = a[index] & mask
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a[index] >>= n
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a[index] |= prec_carry << (32 - n)
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prec_carry = carry
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}
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}
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// a := a - b supposed a >= b
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[inline]
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fn subtract_in_place(mut a []u32, b []u32) {
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mut carry := u32(0)
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mut new_carry := u32(0)
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offset := a.len - b.len
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for index := a.len - b.len; index < a.len; index++ {
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if a[index] < (b[index - offset] + carry) {
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new_carry = 1
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} else {
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new_carry = 0
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}
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a[index] -= (b[index - offset] + carry)
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carry = new_carry
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}
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assert carry == 0
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}
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// for assert
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[inline]
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fn left_align_p(a u32, b u32) bool {
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return bits.leading_zeros_32(a) == bits.leading_zeros_32(b)
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}
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@ -0,0 +1,162 @@
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module big
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import rand
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fn test_lshift_in_place() {
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mut a := [u32(1), 1, 1, 1, 1]
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lshift_in_place(mut a, 1)
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assert a == [u32(2), 2, 2, 2, 2]
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lshift_in_place(mut a, 7)
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assert a == [u32(256), 256, 256, 256, 256]
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mut b := [u32(0x80000001), 0xc0000000, 0x80000000, 0x7fffffff]
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lshift_in_place(mut b, 1)
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assert b == [u32(2), 0x80000001, 1, 0xffffffff]
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mut c := [u32(0x00ffffff), 0xf0f0f0f0, 1, 0x3fffffff, 1]
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lshift_in_place(mut c, 2)
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assert c == [u32(0x3fffffc), 0xc3c3c3c0, 7, 0xfffffffc, 4]
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}
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fn test_rshift_in_place() {
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mut a := [u32(2), 2, 2, 2, 2]
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rshift_in_place(mut a, 1)
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assert a == [u32(1), 1, 1, 1, 1]
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a = [u32(256), 256, 256, 256, 256]
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rshift_in_place(mut a, 7)
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assert a == [u32(2), 2, 2, 2, 2]
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a = [u32(0), 0, 1]
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rshift_in_place(mut a, 1)
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assert a == [u32(0), 0x80000000, 0]
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mut b := [u32(3), 0x80000001, 1, 0xffffffff]
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rshift_in_place(mut b, 1)
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assert b == [u32(0x80000001), 0xc0000000, 0x80000000, 0x7fffffff]
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mut c := [u32(0x03ffffff), 0xc3c3c3c0, 7, 0xfffffffc, 4]
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rshift_in_place(mut c, 2)
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assert c == [u32(0x00ffffff), 0xf0f0f0f0, 1, 0x3fffffff, 1]
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}
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fn test_subtract_in_place() {
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mut a := [u32(2), 2, 2, 2, 2]
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mut b := [u32(1), 1, 2, 1, 1]
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subtract_in_place(mut a, b)
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assert a == [u32(1), 1, 0, 1, 1]
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a = [u32(0), 0, 0, 0, 1]
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b = [u32(0), 0, 1]
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subtract_in_place(mut a, b)
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assert a == [u32(0), 0, 0, 0, 0]
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a = [u32(0), 0, 0, 0, 1, 13]
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b = [u32(1), 0, 1]
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mut c := []u32{len: a.len}
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mut d := [u32(0), 0, 0]
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d << b // to have same length
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subtract_digit_array(a, d, mut c)
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subtract_in_place(mut a, b)
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assert a == [u32(0), 0, 0, u32(-1), 0, 12]
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assert c == a
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}
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fn test_greater_equal_from_end() {
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mut a := [u32(1), 2, 3, 4, 5, 6]
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mut b := [u32(3), 4, 5, 6]
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assert greater_equal_from_end(a, b) == true
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a = [u32(1), 2, 3, 4, 5, 6]
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b = [u32(1), 2, 3, 4, 5, 6]
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assert greater_equal_from_end(a, b) == true
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a = [u32(1), 2, 3, 4, 5, 6]
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b = [u32(2), 2, 3, 4, 5, 6]
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assert greater_equal_from_end(a, b) == false
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a = [u32(0), 0, 0, 4, 5, 6]
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b = [u32(4), 5, 6]
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assert greater_equal_from_end(a, b) == true
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a = [u32(0), 0, 0, 4, 5, 6]
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b = [u32(4), 6, 6]
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assert greater_equal_from_end(a, b) == false
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a = [u32(0), 0, 0, 4, 5, 5]
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b = [u32(4), 5, 6]
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assert greater_equal_from_end(a, b) == false
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}
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fn test_divide_digit_array_03() {
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a := [u32(0), 4]
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b := [u32(0), 1]
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mut q := []u32{cap: a.len - b.len + 1}
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mut r := []u32{cap: a.len}
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divide_digit_array(a, b, mut q, mut r)
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assert q == [u32(4)]
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assert r == []u32{len: 0}
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}
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fn test_divide_digit_array_04() {
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a := [u32(2), 4]
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b := [u32(0), 1]
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mut q := []u32{cap: a.len - b.len + 1}
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mut r := []u32{cap: a.len}
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divide_digit_array(a, b, mut q, mut r)
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assert q == [u32(4)]
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assert r == [u32(2)]
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}
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fn test_divide_digit_array_05() {
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a := [u32(2), 4, 5]
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b := [u32(0), 1]
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mut q := []u32{cap: a.len - b.len + 1}
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mut r := []u32{cap: a.len}
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divide_digit_array(a, b, mut q, mut r)
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assert q == [u32(4), 5]
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assert r == [u32(2)]
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}
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fn test_divide_digit_array_06() {
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a := [u32(2), 4, 5, 3]
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b := [u32(0), 0x8000]
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mut q := []u32{cap: a.len - b.len + 1}
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mut r := []u32{cap: a.len}
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divide_digit_array(a, b, mut q, mut r)
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assert q == [u32(0xa0000), 0x60000]
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assert r == [u32(2), 4]
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}
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// For debugging
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fn integer_from_u32_array(a []u32) Integer {
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mut res := integer_from_i64(0)
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mut multiplicand := integer_from_u32(1)
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for i in 0 .. a.len {
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res += integer_from_u32(a[i]) * multiplicand
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multiplicand = multiplicand.lshift(32)
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}
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return res
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}
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fn test_many_divisions() {
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for _ in 0 .. 100 {
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a := random_number(30)
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b := random_number(30)
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c := a * b
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assert c / a == b
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assert c / b == a
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q, r := a.div_mod(b)
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assert (q * b) + r == a
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}
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}
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fn random_number(length int) Integer {
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numbers := '0123456789'
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mut stri := ''
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for _ in 0 .. length {
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i := rand.intn(10)
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nr := numbers[i]
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stri = stri + nr.ascii_str()
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}
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res := integer_from_string(stri) or { panic('error in random_number') }
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return res
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}
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