}
/// Partition of `n` into n > 1e19 and rem <= 1e19
+///
+/// Integer division algorithm is based on the following paper:
+///
+/// T. Granlund and P. Montgomery, “Division by Invariant Integers Using Multiplication”
+/// in Proc. of the SIGPLAN94 Conference on Programming Language Design and
+/// Implementation, 1994, pp. 61–72
+///
fn udiv_1e19(n: u128) -> (u128, u64) {
const DIV: u64 = 1e19 as u64;
- let high = (n >> 64) as u64;
- if high == 0 {
- let low = n as u64;
- return ((low / DIV) as u128, low % DIV);
- }
- let sr = 65 - high.leading_zeros();
- let mut q = n << (128 - sr);
- let mut r = n >> sr;
- let mut carry = 0;
-
- for _ in 0..sr {
- r = (r << 1) | (q >> 127);
- q = (q << 1) | carry as u128;
-
- let s = (DIV as u128).wrapping_sub(r).wrapping_sub(1) as i128 >> 127;
- carry = (s & 1) as u64;
- r -= (DIV as u128) & s as u128;
- }
- ((q << 1) | carry as u128, r as u64)
+ const FACTOR: u128 = 156927543384667019095894735580191660403;
+
+ let quot = if n < 1 << 83 {
+ ((n >> 19) as u64 / (DIV >> 19)) as u128
+ } else {
+ u128_mulhi(n, FACTOR) >> 62
+ };
+
+ let rem = (n - quot * DIV as u128) as u64;
+ (quot, rem)
+}
+
+/// Multiply unsigned 128 bit integers, return upper 128 bits of the result
+#[inline]
+fn u128_mulhi(x: u128, y: u128) -> u128 {
+ let x_lo = x as u64;
+ let x_hi = (x >> 64) as u64;
+ let y_lo = y as u64;
+ let y_hi = (y >> 64) as u64;
+
+ // handle possibility of overflow
+ let carry = (x_lo as u128 * y_lo as u128) >> 64;
+ let m = x_lo as u128 * y_hi as u128 + carry;
+ let high1 = m >> 64;
+
+ let m_lo = m as u64;
+ let high2 = (x_hi as u128 * y_lo as u128 + m_lo as u128) >> 64;
+
+ x_hi as u128 * y_hi as u128 + high1 + high2
}