Omit low precision digits in calculations

Author: SakiTakamachi

ext/bcmath/libbcmath/src/sqrt.c

@@ -180,10 +180,12 @@ static inline void bc_standard_sqrt(bc_num *num, size_t rscale, size_t num_calc_
 	}
 	guess_vector[guess_arr_size - 1] = 0;
 
-	size_t quot_size = n_arr_size - (guess_arr_size - 1) + 1;
-
 	BC_VECTOR two[1] = { 2 };
 
+	/* The precision (number of vectors) used for the calculation.
+	 * Since the initial value uses two vectors, the initial precision is set to 2. */
+	size_t guess_precision = 2;
+
 	/**
 	 * Newton's algorithm. Iterative expression is `x_{n+1} = (x_n + a / x_n) / 2`
 	 * If break down the calculation into detailed steps, it looks like this:
@@ -194,13 +196,23 @@ static inline void bc_standard_sqrt(bc_num *num, size_t rscale, size_t num_calc_
 	 */
 	bool done = false;
 	do {
+		size_t guess_offset = guess_arr_size - 1 - guess_precision;
+		size_t n_offset = guess_offset * 2;
+
 		/* Since the value changes during division by successive approximation, use a copied version of it. */
-		for (size_t i = 0; i < n_arr_size; i++) {
+		for (size_t i = n_offset; i < n_arr_size; i++) {
 			n_vector_copy[i] = n_vector[i];
 		}
 
+		size_t n_precision = n_arr_size - n_offset;
+		size_t quot_size = n_precision - (guess_precision) + 1;
+
 		/* 1. quot = a / x_n */
-		bool div_ret = bc_divide_vector(n_vector_copy, n_arr_size, guess_vector, guess_arr_size - 1, tmp_div_ret_vector, quot_size);
+		bool div_ret = bc_divide_vector(
+			n_vector_copy + n_offset, n_precision,
+			guess_vector + guess_offset, guess_precision,
+			tmp_div_ret_vector + guess_offset, quot_size
+		);
 		ZEND_ASSERT(div_ret);
 
 		BC_VECTOR *tmp_vptr = guess1_vector;
@@ -209,7 +221,7 @@ static inline void bc_standard_sqrt(bc_num *num, size_t rscale, size_t num_calc_
 
 		/* 2. add = x_n + quot1 */
 		int carry = 0;
-		for (size_t i = 0; i < guess_arr_size - 1; i++) {
+		for (size_t i = guess_offset; i < guess_arr_size; i++) {
 			guess_vector[i] = guess1_vector[i] + tmp_div_ret_vector[i] + carry;
 			if (guess_vector[i] >= BC_VECTOR_BOUNDARY_NUM) {
 				guess_vector[i] -= BC_VECTOR_BOUNDARY_NUM;
@@ -221,24 +233,37 @@ static inline void bc_standard_sqrt(bc_num *num, size_t rscale, size_t num_calc_
 		guess_vector[guess_arr_size - 1] = carry;
 
 		/* 3. x_{n+1} = add / 2 */
-		div_ret = bc_divide_vector(guess_vector, guess_arr_size, two, 1, tmp_div_ret_vector, guess_arr_size);
+		div_ret = bc_divide_vector(
+			guess_vector + guess_offset, guess_precision + 1,
+			two, 1,
+			tmp_div_ret_vector + guess_offset, guess_precision + 1
+		);
 		ZEND_ASSERT(div_ret);
 
-		for (size_t i = 0; i < guess_arr_size; i++) {
+		for (size_t i = guess_offset; i < guess_arr_size; i++) {
 			guess_vector[i] = tmp_div_ret_vector[i];
 		}
 
 		/* 4. repeat until the difference between the `x_n` and `x_{n+1}` is less than or equal to 1. */
-		size_t diff = guess_vector[0] > guess1_vector[0] ? guess_vector[0] - guess1_vector[0] : guess1_vector[0] - guess_vector[0];
+		size_t diff = guess_vector[guess_offset] > guess1_vector[guess_offset]
+			? guess_vector[guess_offset] - guess1_vector[guess_offset]
+			: guess1_vector[guess_offset] - guess_vector[guess_offset];
 		if (diff <= 1) {
 			bool is_same = true;
-			for (size_t i = 1; i < guess_arr_size - 1; i++) {
+			for (size_t i = guess_offset + 1; i < guess_arr_size - 1; i++) {
 				if (guess_vector[i] != guess1_vector[i]) {
 					is_same = false;
 					break;
 				}
 			}
-			done = is_same;
+			if (is_same) {
+				if (guess_precision < guess_arr_size - 1) {
+					/* If the precision has not yet reached the maximum number of digits, it will be increased. */
+					guess_precision = MIN(guess_precision * 3, guess_arr_size - 1);
+				} else {
+					done = is_same;
+				}
+			}
 		}
 	} while (!done);