Fix folder names in primitives (#4358)

* sr-arithmetic -> arithmetic

* sr-sandbox -> sandbox

* primitives/sr-staking-primitives -> primitives/staking

* primitives/sr-version -> primitives/version

* primitives/block-builder/runtime-api -> primitives/block-builder

substrate/primitives/arithmetic/src/helpers_128bit.rs

@@ -0,0 +1,112 @@
+// Copyright 2019 Parity Technologies (UK) Ltd.
+// This file is part of Substrate.
+
+// Substrate is free software: you can redistribute it and/or modify
+// it under the terms of the GNU General Public License as published by
+// the Free Software Foundation, either version 3 of the License, or
+// (at your option) any later version.
+
+// Substrate is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+// GNU General Public License for more details.
+
+// You should have received a copy of the GNU General Public License
+// along with Substrate.  If not, see <http://www.gnu.org/licenses/>.
+
+//! Some helper functions to work with 128bit numbers. Note that the functionality provided here is
+//! only sensible to use with 128bit numbers because for smaller sizes, you can always rely on
+//! assumptions of a bigger type (u128) being available, or simply create a per-thing and use the
+//! multiplication implementation provided there.
+
+use crate::biguint;
+use num_traits::Zero;
+use sp_std::{cmp::{min, max}, convert::TryInto, mem};
+
+/// Helper gcd function used in Rational128 implementation.
+pub fn gcd(a: u128, b: u128) -> u128 {
+	match ((a, b), (a & 1, b & 1)) {
+		((x, y), _) if x == y => y,
+		((0, x), _) | ((x, 0), _) => x,
+		((x, y), (0, 1)) | ((y, x), (1, 0)) => gcd(x >> 1, y),
+		((x, y), (0, 0)) => gcd(x >> 1, y >> 1) << 1,
+		((x, y), (1, 1)) => {
+			let (x, y) = (min(x, y), max(x, y));
+			gcd((y - x) >> 1, x)
+		},
+		_ => unreachable!(),
+	}
+}
+
+/// split a u128 into two u64 limbs
+pub fn split(a: u128) -> (u64, u64) {
+	let al = a as u64;
+	let ah = (a >> 64) as u64;
+	(ah, al)
+}
+
+/// Convert a u128 to a u32 based biguint.
+pub fn to_big_uint(x: u128) -> biguint::BigUint {
+	let (xh, xl) = split(x);
+	let (xhh, xhl) = biguint::split(xh);
+	let (xlh, xll) = biguint::split(xl);
+	let mut n = biguint::BigUint::from_limbs(&[xhh, xhl, xlh, xll]);
+	n.lstrip();
+	n
+}
+
+/// Safely and accurately compute `a * b / c`. The approach is:
+///   - Simply try `a * b / c`.
+///   - Else, convert them both into big numbers and re-try. `Err` is returned if the result
+///     cannot be safely casted back to u128.
+///
+/// Invariant: c must be greater than or equal to 1.
+pub fn multiply_by_rational(mut a: u128, mut b: u128, mut c: u128) -> Result<u128, &'static str> {
+	if a.is_zero() || b.is_zero() { return Ok(Zero::zero()); }
+	c = c.max(1);
+
+	// a and b are interchangeable by definition in this function. It always helps to assume the
+	// bigger of which is being multiplied by a `0 < b/c < 1`. Hence, a should be the bigger and
+	// b the smaller one.
+	if b > a {
+		mem::swap(&mut a, &mut b);
+	}
+
+	// Attempt to perform the division first
+	if a % c == 0 {
+		a /= c;
+		c = 1;
+	} else if b % c == 0 {
+		b /= c;
+		c = 1;
+	}
+
+	if let Some(x) = a.checked_mul(b) {
+		// This is the safest way to go. Try it.
+		Ok(x / c)
+	} else {
+		let a_num = to_big_uint(a);
+		let b_num = to_big_uint(b);
+		let c_num = to_big_uint(c);
+
+		let mut ab = a_num * b_num;
+		ab.lstrip();
+		let mut q = if c_num.len() == 1 {
+			// PROOF: if `c_num.len() == 1` then `c` fits in one limb.
+			ab.div_unit(c as biguint::Single)
+		} else {
+			// PROOF: both `ab` and `c` cannot have leading zero limbs; if length of `c` is 1,
+			// the previous branch would handle. Also, if ab for sure has a bigger size than
+			// c, because `a.checked_mul(b)` has failed, hence ab must be at least one limb
+			// bigger than c. In this case, returning zero is defensive-only and div should
+			// always return Some.
+			let (mut q, r) = ab.div(&c_num, true).unwrap_or((Zero::zero(), Zero::zero()));
+			let r: u128 = r.try_into()
+				.expect("reminder of div by c is always less than c; qed");
+			if r > (c / 2) { q = q.add(&to_big_uint(1)); }
+			q
+		};
+		q.lstrip();
+		q.try_into().map_err(|_| "result cannot fit in u128")
+	}
+}