Faster i256 Division (2-100x) (#4663) (#4672)

* Faster i256 Division (2-100x) (#4663)

* Clippy

* Use inline assembly

* Fix non-x64

* Add repr(C)

* More docs

* Format

arrow-buffer/benches/i256.rs

@@ -21,18 +21,7 @@ use rand::rngs::StdRng;
 use rand::{Rng, SeedableRng};
 use std::str::FromStr;
 
-/// Returns fixed seedable RNG
-fn seedable_rng() -> StdRng {
-    StdRng::seed_from_u64(42)
-}
-
-fn create_i256_vec(size: usize) -> Vec<i256> {
-    let mut rng = seedable_rng();
-
-    (0..size)
-        .map(|_| i256::from_i128(rng.gen::<i128>()))
-        .collect()
-}
+const SIZE: usize = 1024;
 
 fn criterion_benchmark(c: &mut Criterion) {
     let numbers = vec![
@@ -54,24 +43,40 @@ fn criterion_benchmark(c: &mut Criterion) {
         });
     }
 
-    c.bench_function("i256_div", |b| {
+    let mut rng = StdRng::seed_from_u64(42);
+
+    let numerators: Vec<_> = (0..SIZE)
+        .map(|_| {
+            let high = rng.gen_range(1000..i128::MAX);
+            let low = rng.gen();
+            i256::from_parts(low, high)
+        })
+        .collect();
+
+    let divisors: Vec<_> = numerators
+        .iter()
+        .map(|n| {
+            let quotient = rng.gen_range(1..100_i32);
+            n.wrapping_div(i256::from(quotient))
+        })
+        .collect();
+
+    c.bench_function("i256_div_rem small quotient", |b| {
         b.iter(|| {
-            for number_a in create_i256_vec(10) {
-                for number_b in create_i256_vec(5) {
-                    number_a.checked_div(number_b);
-                    number_a.wrapping_div(number_b);
-                }
+            for (n, d) in numerators.iter().zip(&divisors) {
+                black_box(n.wrapping_div(*d));
             }
         });
     });
 
-    c.bench_function("i256_rem", |b| {
+    let divisors: Vec<_> = (0..SIZE)
+        .map(|_| i256::from(rng.gen_range(1..100_i32)))
+        .collect();
+
+    c.bench_function("i256_div_rem small divisor", |b| {
         b.iter(|| {
-            for number_a in create_i256_vec(10) {
-                for number_b in create_i256_vec(5) {
-                    number_a.checked_rem(number_b);
-                    number_a.wrapping_rem(number_b);
-                }
+            for (n, d) in numerators.iter().zip(&divisors) {
+                black_box(n.wrapping_div(*d));
             }
         });
     });

arrow-buffer/src/bigint/div.rs

@@ -0,0 +1,312 @@
+// Licensed to the Apache Software Foundation (ASF) under one
+// or more contributor license agreements.  See the NOTICE file
+// distributed with this work for additional information
+// regarding copyright ownership.  The ASF licenses this file
+// to you 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
+//
+//   http://www.apache.org/licenses/LICENSE-2.0
+//
+// 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.
+
+//! N-digit division
+//!
+//! Implementation heavily inspired by [uint]
+//!
+//! [uint]: https://github.com/paritytech/parity-common/blob/d3a9327124a66e52ca1114bb8640c02c18c134b8/uint/src/uint.rs#L844
+
+/// Unsigned, little-endian, n-digit division with remainder
+///
+/// # Panics
+///
+/// Panics if divisor is zero
+pub fn div_rem<const N: usize>(
+    numerator: &[u64; N],
+    divisor: &[u64; N],
+) -> ([u64; N], [u64; N]) {
+    let numerator_bits = bits(numerator);
+    let divisor_bits = bits(divisor);
+    assert_ne!(divisor_bits, 0, "division by zero");
+
+    if numerator_bits < divisor_bits {
+        return ([0; N], *numerator);
+    }
+
+    if divisor_bits <= 64 {
+        return div_rem_small(numerator, divisor[0]);
+    }
+
+    let numerator_words = (numerator_bits + 63) / 64;
+    let divisor_words = (divisor_bits + 63) / 64;
+    let n = divisor_words;
+    let m = numerator_words - divisor_words;
+
+    div_rem_knuth(numerator, divisor, n, m)
+}
+
+/// Return the least number of bits needed to represent the number
+fn bits(arr: &[u64]) -> usize {
+    for (idx, v) in arr.iter().enumerate().rev() {
+        if *v > 0 {
+            return 64 - v.leading_zeros() as usize + 64 * idx;
+        }
+    }
+    0
+}
+
+/// Division of numerator by a u64 divisor
+fn div_rem_small<const N: usize>(
+    numerator: &[u64; N],
+    divisor: u64,
+) -> ([u64; N], [u64; N]) {
+    let mut rem = 0u64;
+    let mut numerator = *numerator;
+    numerator.iter_mut().rev().for_each(|d| {
+        let (q, r) = div_rem_word(rem, *d, divisor);
+        *d = q;
+        rem = r;
+    });
+
+    let mut rem_padded = [0; N];
+    rem_padded[0] = rem;
+    (numerator, rem_padded)
+}
+
+/// Use Knuth Algorithm D to compute `numerator / divisor` returning the
+/// quotient and remainder
+///
+/// `n` is the number of non-zero 64-bit words in `divisor`
+/// `m` is the number of non-zero 64-bit words present in `numerator` beyond `divisor`, and
+/// therefore the number of words in the quotient
+///
+/// A good explanation of the algorithm can be found [here](https://ridiculousfish.com/blog/posts/labor-of-division-episode-iv.html)
+fn div_rem_knuth<const N: usize>(
+    numerator: &[u64; N],
+    divisor: &[u64; N],
+    n: usize,
+    m: usize,
+) -> ([u64; N], [u64; N]) {
+    assert!(n + m <= N);
+
+    // The algorithm works by incrementally generating guesses `q_hat`, for the next digit
+    // of the quotient, starting from the most significant digit.
+    //
+    // This relies on the property that for any `q_hat` where
+    //
+    //      (q_hat << (j * 64)) * divisor <= numerator`
+    //
+    // We can set
+    //
+    //      q += q_hat << (j * 64)
+    //      numerator -= (q_hat << (j * 64)) * divisor
+    //
+    // And then iterate until `numerator < divisor`
+
+    // We normalize the divisor so that the highest bit in the highest digit of the
+    // divisor is set, this ensures our initial guess of `q_hat` is at most 2 off from
+    // the correct value for q[j]
+    let shift = divisor[n - 1].leading_zeros();
+    // As the shift is computed based on leading zeros, don't need to perform full_shl
+    let divisor = shl_word(divisor, shift);
+    // numerator may have fewer leading zeros than divisor, so must add another digit
+    let mut numerator = full_shl(numerator, shift);
+
+    // The two most significant digits of the divisor
+    let b0 = divisor[n - 1];
+    let b1 = divisor[n - 2];
+
+    let mut q = [0; N];
+
+    for j in (0..=m).rev() {
+        let a0 = numerator[j + n];
+        let a1 = numerator[j + n - 1];
+
+        let mut q_hat = if a0 < b0 {
+            // The first estimate is [a1, a0] / b0, it may be too large by at most 2
+            let (mut q_hat, mut r_hat) = div_rem_word(a0, a1, b0);
+
+            // r_hat = [a1, a0] - q_hat * b0
+            //
+            // Now we want to compute a more precise estimate [a2,a1,a0] / [b1,b0]
+            // which can only be less or equal to the current q_hat
+            //
+            // q_hat is too large if:
+            // [a2,a1,a0] < q_hat * [b1,b0]
+            // [a2,r_hat] < q_hat * b1
+            let a2 = numerator[j + n - 2];
+            loop {
+                let r = u128::from(q_hat) * u128::from(b1);
+                let (lo, hi) = (r as u64, (r >> 64) as u64);
+                if (hi, lo) <= (r_hat, a2) {
+                    break;
+                }
+
+                q_hat -= 1;
+                let (new_r_hat, overflow) = r_hat.overflowing_add(b0);
+                r_hat = new_r_hat;
+
+                if overflow {
+                    break;
+                }
+            }
+            q_hat
+        } else {
+            u64::MAX
+        };
+
+        // q_hat is now either the correct quotient digit, or in rare cases 1 too large
+
+        // Compute numerator -= (q_hat * divisor) << (j * 64)
+        let q_hat_v = full_mul_u64(&divisor, q_hat);
+        let c = sub_assign(&mut numerator[j..], &q_hat_v[..n + 1]);
+
+        // If underflow, q_hat was too large by 1
+        if c {
+            // Reduce q_hat by 1
+            q_hat -= 1;
+
+            // Add back one multiple of divisor
+            let c = add_assign(&mut numerator[j..], &divisor[..n]);
+            numerator[j + n] = numerator[j + n].wrapping_add(u64::from(c));
+        }
+
+        // q_hat is the correct value for q[j]
+        q[j] = q_hat;
+    }
+
+    // The remainder is what is left in numerator, with the initial normalization shl reversed
+    let remainder = full_shr(&numerator, shift);
+    (q, remainder)
+}
+
+/// Perform narrowing division of a u128 by a u64 divisor, returning the quotient and remainder
+///
+/// This method may trap or panic if hi >= divisor, i.e. the quotient would not fit
+/// into a 64-bit integer
+fn div_rem_word(hi: u64, lo: u64, divisor: u64) -> (u64, u64) {
+    debug_assert!(hi < divisor);
+    debug_assert_ne!(divisor, 0);
+
+    // LLVM fails to use the div instruction as it is not able to prove
+    // that hi < divisor, and therefore the result will fit into 64-bits
+    #[cfg(target_arch = "x86_64")]
+    unsafe {
+        let mut quot = lo;
+        let mut rem = hi;
+        std::arch::asm!(
+            "div {divisor}",
+            divisor = in(reg) divisor,
+            inout("rax") quot,
+            inout("rdx") rem,
+            options(pure, nomem, nostack)
+        );
+        (quot, rem)
+    }
+    #[cfg(not(target_arch = "x86_64"))]
+    {
+        let x = (u128::from(hi) << 64) + u128::from(lo);
+        let y = u128::from(divisor);
+        ((x / y) as u64, (x % y) as u64)
+    }
+}
+
+/// Perform `a += b`
+fn add_assign(a: &mut [u64], b: &[u64]) -> bool {
+    binop_slice(a, b, u64::overflowing_add)
+}
+
+/// Perform `a -= b`
+fn sub_assign(a: &mut [u64], b: &[u64]) -> bool {
+    binop_slice(a, b, u64::overflowing_sub)
+}
+
+/// Converts an overflowing binary operation on scalars to one on slices
+fn binop_slice(
+    a: &mut [u64],
+    b: &[u64],
+    binop: impl Fn(u64, u64) -> (u64, bool) + Copy,
+) -> bool {
+    let mut c = false;
+    a.iter_mut().zip(b.iter()).for_each(|(x, y)| {
+        let (res1, overflow1) = y.overflowing_add(u64::from(c));
+        let (res2, overflow2) = binop(*x, res1);
+        *x = res2;
+        c = overflow1 || overflow2;
+    });
+    c
+}
+
+/// Widening multiplication of an N-digit array with a u64
+fn full_mul_u64<const N: usize>(a: &[u64; N], b: u64) -> ArrayPlusOne<u64, N> {
+    let mut carry = 0;
+    let mut out = [0; N];
+    out.iter_mut().zip(a).for_each(|(o, v)| {
+        let r = *v as u128 * b as u128 + carry as u128;
+        *o = r as u64;
+        carry = (r >> 64) as u64;
+    });
+    ArrayPlusOne(out, carry)
+}
+
+/// Left shift of an N-digit array by at most 63 bits
+fn shl_word<const N: usize>(v: &[u64; N], shift: u32) -> [u64; N] {
+    full_shl(v, shift).0
+}
+
+/// Widening left shift of an N-digit array by at most 63 bits
+fn full_shl<const N: usize>(v: &[u64; N], shift: u32) -> ArrayPlusOne<u64, N> {
+    debug_assert!(shift < 64);
+    if shift == 0 {
+        return ArrayPlusOne(*v, 0);
+    }
+    let mut out = [0u64; N];
+    out[0] = v[0] << shift;
+    for i in 1..N {
+        out[i] = v[i - 1] >> (64 - shift) | v[i] << shift
+    }
+    let carry = v[N - 1] >> (64 - shift);
+    ArrayPlusOne(out, carry)
+}
+
+/// Narrowing right shift of an (N+1)-digit array by at most 63 bits
+fn full_shr<const N: usize>(a: &ArrayPlusOne<u64, N>, shift: u32) -> [u64; N] {
+    debug_assert!(shift < 64);
+    if shift == 0 {
+        return a.0;
+    }
+    let mut out = [0; N];
+    for i in 0..N - 1 {
+        out[i] = a[i] >> shift | a[i + 1] << (64 - shift)
+    }
+    out[N - 1] = a[N - 1] >> shift;
+    out
+}
+
+/// An array of N + 1 elements
+///
+/// This is a hack around lack of support for const arithmetic
+#[repr(C)]
+struct ArrayPlusOne<T, const N: usize>([T; N], T);
+
+impl<T, const N: usize> std::ops::Deref for ArrayPlusOne<T, N> {
+    type Target = [T];
+
+    #[inline]
+    fn deref(&self) -> &Self::Target {
+        let x = self as *const Self;
+        unsafe { std::slice::from_raw_parts(x as *const T, N + 1) }
+    }
+}
+
+impl<T, const N: usize> std::ops::DerefMut for ArrayPlusOne<T, N> {
+    fn deref_mut(&mut self) -> &mut Self::Target {
+        let x = self as *mut Self;
+        unsafe { std::slice::from_raw_parts_mut(x as *mut T, N + 1) }
+    }
+}