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```
```// This file is part of Substrate.

// Copyright (C) 2019-2020 Parity Technologies (UK) Ltd.

// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

//! 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")
}
}
```