bigint: division (#11386)

vlib/math/big/array_ops.v

@@ -202,7 +202,7 @@ fn divide_digit_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut
 }
 
 // Performs division on the non-negative dividend in a by the single digit divisor b. It assumes
-// quotient and remainder are empty zero length arrays with sufficient capacity
+// quotient and remainder are empty zero length arrays without previous allocation
 fn divide_array_by_digit(operand_a []u32, divisor u32, mut quotient []u32, mut remainder []u32) {
 	if operand_a.len == 1 {
 		// 1 digit for both dividend and divisor
@@ -240,58 +240,6 @@ fn divide_array_by_digit(operand_a []u32, divisor u32, mut quotient []u32, mut r
 	}
 }
 
-// Performs division on the non-negative dividend in a by the multi digit divisor b. It assumes
-// quotient and remainder are empty zero length arrays with sufficient capacity
-// This is different from divide_digit_array because it depends on this very function
-// after making sure that the divisor is indeed multi-digit.
-fn divide_array_by_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut remainder []u32) {
-	for index in 0 .. operand_a.len {
-		remainder << operand_a[index]
-	}
-	for _ in 0 .. operand_b.len {
-		quotient << 0
-	}
-	offset := operand_a.len - operand_b.len
-	divisor_last_index := operand_b.len - 1
-	for index := offset; index >= 0; index-- {
-		dividend_last_index := divisor_last_index + index
-		value_upper := if remainder.len > dividend_last_index + 1 {
-			u64(remainder[dividend_last_index + 1])
-		} else {
-			u64(0)
-		}
-		value_lower := if remainder.len > dividend_last_index {
-			u64(remainder[dividend_last_index])
-		} else {
-			u64(0)
-		}
-		partial := value_lower + (value_upper << 32)
-		mut q := u32(partial / operand_b[divisor_last_index])
-		if q > 0 {
-			mut modified_divisor := []u32{len: operand_b.len + index, init: 0}
-			for i in 0 .. operand_b.len {
-				modified_divisor[index + i] = operand_b[i]
-			}
-
-			mut product := []u32{len: operand_b.len + 1, init: 0}
-			multiply_array_by_digit(modified_divisor, q, mut product)
-			for q > 0 && compare_digit_array(product, remainder) > 0 {
-				q--
-				subtract_digit_array(product, modified_divisor, mut product)
-			}
-			subtract_digit_array(remainder, product, mut remainder)
-		}
-		quotient[index] = q
-	}
-	// Remove leading zeros from quotient and remainder
-	for quotient.len > 0 && quotient.last() == 0 {
-		quotient.delete_last()
-	}
-	for remainder.len > 0 && remainder.last() == 0 {
-		remainder.delete_last()
-	}
-}
-
 // Shifts the contents of the original array by the given amount of bits to the left.
 // This function assumes that the amount is less than 32. The storage is expected to
 // allocated with zeroes.

vlib/math/big/division_array_ops.v

@@ -0,0 +1,139 @@
+module big
+
+import math.bits
+
+// suppose operand_a bigger than operand_b and both not null.
+// Both quotient and remaider are allocated but of length 0
+fn divide_array_by_array(operand_a []u32, operand_b []u32, mut quotient []u32, mut remainder []u32) {
+	for index in 0 .. operand_a.len {
+		remainder << operand_a[index]
+	}
+
+	len_diff := operand_a.len - operand_b.len
+	assert len_diff >= 0
+
+	// we must do in place shift and operations.
+	mut divisor := []u32{cap: operand_b.len}
+	for _ in 0 .. len_diff {
+		divisor << u32(0)
+	}
+	for index in 0 .. operand_b.len {
+		divisor << operand_b[index]
+	}
+	for _ in 0 .. len_diff + 1 {
+		quotient << u32(0)
+	}
+
+	lead_zer_remainder := u32(bits.leading_zeros_32(remainder.last()))
+	lead_zer_divisor := u32(bits.leading_zeros_32(divisor.last()))
+	bit_offset := (u32(32) * u32(len_diff)) + (lead_zer_divisor - lead_zer_remainder)
+
+	// align
+	if lead_zer_remainder < lead_zer_divisor {
+		lshift_in_place(mut divisor, lead_zer_divisor - lead_zer_remainder)
+	} else if lead_zer_remainder > lead_zer_divisor {
+		lshift_in_place(mut remainder, lead_zer_remainder - lead_zer_divisor)
+	}
+
+	assert left_align_p(divisor[divisor.len - 1], remainder[remainder.len - 1])
+	for bit_idx := int(bit_offset); bit_idx >= 0; bit_idx-- {
+		if greater_equal_from_end(remainder, divisor) {
+			bit_set(mut quotient, bit_idx)
+			subtract_in_place(mut remainder, divisor)
+		}
+		rshift_in_place(mut divisor, 1)
+	}
+
+	// ajust
+	if lead_zer_remainder > lead_zer_divisor {
+		// rshift_in_place(mut quotient, lead_zer_remainder - lead_zer_divisor)
+		rshift_in_place(mut remainder, lead_zer_remainder - lead_zer_divisor)
+	}
+	for remainder.len > 0 && remainder.last() == 0 {
+		remainder.delete_last()
+	}
+	for quotient.len > 0 && quotient.last() == 0 {
+		quotient.delete_last()
+	}
+}
+
+// help routines for cleaner code but inline for performance
+// quicker than BitField.set_bit
+[inline]
+fn bit_set(mut a []u32, n int) {
+	byte_offset := n / 32
+	mask := u32(1) << u32(n % 32)
+	assert a.len >= byte_offset
+	a[byte_offset] |= mask
+}
+
+// a.len is greater or equal to b.len
+// returns true if a >= b (completed with zeroes)
+[inline]
+fn greater_equal_from_end(a []u32, b []u32) bool {
+	assert a.len >= b.len
+	offset := a.len - b.len
+	for index := a.len - 1; index >= offset; index-- {
+		if a[index] > b[index - offset] {
+			return true
+		} else if a[index] < b[index - offset] {
+			return false
+		}
+	}
+	return true
+}
+
+// logical left shift
+// there is no overflow. We know that the last bits are zero
+// and that n <= 32
+[inline]
+fn lshift_in_place(mut a []u32, n u32) {
+	mut carry := u32(0)
+	mut prec_carry := u32(0)
+	mask := ((u32(1) << n) - 1) << (32 - n)
+	for index in 0 .. a.len {
+		prec_carry = carry >> (32 - n)
+		carry = a[index] & mask
+		a[index] <<= n
+		a[index] |= prec_carry
+	}
+}
+
+// logical right shift without control because these digits have already been
+// shift left before
+[inline]
+fn rshift_in_place(mut a []u32, n u32) {
+	mut carry := u32(0)
+	mut prec_carry := u32(0)
+	mask := u32((1 << n) - 1)
+	for index := a.len - 1; index >= 0; index-- {
+		carry = a[index] & mask
+		a[index] >>= n
+		a[index] |= prec_carry << (32 - n)
+		prec_carry = carry
+	}
+}
+
+// a := a - b supposed a >= b
+[inline]
+fn subtract_in_place(mut a []u32, b []u32) {
+	mut carry := u32(0)
+	mut new_carry := u32(0)
+	offset := a.len - b.len
+	for index := a.len - b.len; index < a.len; index++ {
+		if a[index] < (b[index - offset] + carry) {
+			new_carry = 1
+		} else {
+			new_carry = 0
+		}
+		a[index] -= (b[index - offset] + carry)
+		carry = new_carry
+	}
+	assert carry == 0
+}
+
+// for assert
+[inline]
+fn left_align_p(a u32, b u32) bool {
+	return bits.leading_zeros_32(a) == bits.leading_zeros_32(b)
+}

vlib/math/big/division_array_ops_test.v

@@ -0,0 +1,162 @@
+module big
+
+import rand
+
+fn test_lshift_in_place() {
+	mut a := [u32(1), 1, 1, 1, 1]
+	lshift_in_place(mut a, 1)
+	assert a == [u32(2), 2, 2, 2, 2]
+	lshift_in_place(mut a, 7)
+	assert a == [u32(256), 256, 256, 256, 256]
+	mut b := [u32(0x80000001), 0xc0000000, 0x80000000, 0x7fffffff]
+	lshift_in_place(mut b, 1)
+	assert b == [u32(2), 0x80000001, 1, 0xffffffff]
+	mut c := [u32(0x00ffffff), 0xf0f0f0f0, 1, 0x3fffffff, 1]
+	lshift_in_place(mut c, 2)
+	assert c == [u32(0x3fffffc), 0xc3c3c3c0, 7, 0xfffffffc, 4]
+}
+
+fn test_rshift_in_place() {
+	mut a := [u32(2), 2, 2, 2, 2]
+	rshift_in_place(mut a, 1)
+	assert a == [u32(1), 1, 1, 1, 1]
+	a = [u32(256), 256, 256, 256, 256]
+	rshift_in_place(mut a, 7)
+	assert a == [u32(2), 2, 2, 2, 2]
+	a = [u32(0), 0, 1]
+	rshift_in_place(mut a, 1)
+	assert a == [u32(0), 0x80000000, 0]
+	mut b := [u32(3), 0x80000001, 1, 0xffffffff]
+	rshift_in_place(mut b, 1)
+	assert b == [u32(0x80000001), 0xc0000000, 0x80000000, 0x7fffffff]
+	mut c := [u32(0x03ffffff), 0xc3c3c3c0, 7, 0xfffffffc, 4]
+	rshift_in_place(mut c, 2)
+	assert c == [u32(0x00ffffff), 0xf0f0f0f0, 1, 0x3fffffff, 1]
+}
+
+fn test_subtract_in_place() {
+	mut a := [u32(2), 2, 2, 2, 2]
+	mut b := [u32(1), 1, 2, 1, 1]
+	subtract_in_place(mut a, b)
+	assert a == [u32(1), 1, 0, 1, 1]
+
+	a = [u32(0), 0, 0, 0, 1]
+	b = [u32(0), 0, 1]
+	subtract_in_place(mut a, b)
+	assert a == [u32(0), 0, 0, 0, 0]
+
+	a = [u32(0), 0, 0, 0, 1, 13]
+	b = [u32(1), 0, 1]
+	mut c := []u32{len: a.len}
+	mut d := [u32(0), 0, 0]
+	d << b // to have same length
+	subtract_digit_array(a, d, mut c)
+	subtract_in_place(mut a, b)
+	assert a == [u32(0), 0, 0, u32(-1), 0, 12]
+	assert c == a
+}
+
+fn test_greater_equal_from_end() {
+	mut a := [u32(1), 2, 3, 4, 5, 6]
+	mut b := [u32(3), 4, 5, 6]
+	assert greater_equal_from_end(a, b) == true
+
+	a = [u32(1), 2, 3, 4, 5, 6]
+	b = [u32(1), 2, 3, 4, 5, 6]
+	assert greater_equal_from_end(a, b) == true
+
+	a = [u32(1), 2, 3, 4, 5, 6]
+	b = [u32(2), 2, 3, 4, 5, 6]
+	assert greater_equal_from_end(a, b) == false
+
+	a = [u32(0), 0, 0, 4, 5, 6]
+	b = [u32(4), 5, 6]
+	assert greater_equal_from_end(a, b) == true
+
+	a = [u32(0), 0, 0, 4, 5, 6]
+	b = [u32(4), 6, 6]
+	assert greater_equal_from_end(a, b) == false
+
+	a = [u32(0), 0, 0, 4, 5, 5]
+	b = [u32(4), 5, 6]
+	assert greater_equal_from_end(a, b) == false
+}
+
+fn test_divide_digit_array_03() {
+	a := [u32(0), 4]
+	b := [u32(0), 1]
+	mut q := []u32{cap: a.len - b.len + 1}
+	mut r := []u32{cap: a.len}
+
+	divide_digit_array(a, b, mut q, mut r)
+	assert q == [u32(4)]
+	assert r == []u32{len: 0}
+}
+
+fn test_divide_digit_array_04() {
+	a := [u32(2), 4]
+	b := [u32(0), 1]
+	mut q := []u32{cap: a.len - b.len + 1}
+	mut r := []u32{cap: a.len}
+
+	divide_digit_array(a, b, mut q, mut r)
+	assert q == [u32(4)]
+	assert r == [u32(2)]
+}
+
+fn test_divide_digit_array_05() {
+	a := [u32(2), 4, 5]
+	b := [u32(0), 1]
+	mut q := []u32{cap: a.len - b.len + 1}
+	mut r := []u32{cap: a.len}
+
+	divide_digit_array(a, b, mut q, mut r)
+	assert q == [u32(4), 5]
+	assert r == [u32(2)]
+}
+
+fn test_divide_digit_array_06() {
+	a := [u32(2), 4, 5, 3]
+	b := [u32(0), 0x8000]
+	mut q := []u32{cap: a.len - b.len + 1}
+	mut r := []u32{cap: a.len}
+
+	divide_digit_array(a, b, mut q, mut r)
+	assert q == [u32(0xa0000), 0x60000]
+	assert r == [u32(2), 4]
+}
+
+// For debugging
+fn integer_from_u32_array(a []u32) Integer {
+	mut res := integer_from_i64(0)
+	mut multiplicand := integer_from_u32(1)
+	for i in 0 .. a.len {
+		res += integer_from_u32(a[i]) * multiplicand
+		multiplicand = multiplicand.lshift(32)
+	}
+	return res
+}
+
+fn test_many_divisions() {
+	for _ in 0 .. 100 {
+		a := random_number(30)
+		b := random_number(30)
+		c := a * b
+		assert c / a == b
+		assert c / b == a
+		q, r := a.div_mod(b)
+		assert (q * b) + r == a
+	}
+}
+
+fn random_number(length int) Integer {
+	numbers := '0123456789'
+	mut stri := ''
+	for _ in 0 .. length {
+		i := rand.intn(10)
+		nr := numbers[i]
+		stri = stri + nr.ascii_str()
+	}
+	res := integer_from_string(stri) or { panic('error in random_number') }
+	return res
+}