multprec.cpp: Simplify with _Unsigned128

[StephanTLavavej]

Followup to #5436, which reimplemented linear_congruential_engine in <random> by using our modern _Unsigned128. This PR shrinks our separately compiled, now-retained-for-bincompat multprec.cpp by also using _Unsigned128. Ordinarily we don't put a lot of effort into simplifying retained-for-bincompat code because the benefit is low and the risk is unnecessary. In this case, it's worth the risk because this fixes #1008 by removing squirrelly code that needed an /analyze suppression. (Being /analyze-clean is very important now, and keeping suppressions around in our codebase is a minor debt.)

My new comments do a hopefully better job than before of explaining _MP_arr's representation, and the assumptions behind the various functions. I am aiming for simplicity here, not micro-optimizations; e.g. _MP_Rem() is computing modulo a 64-bit value, so only the lower 64 bits will be non-zero, but I'm still going through the general assign_mp_from_u128() helper function.

Retained-for-bincompat code isn't something that we expend effort on testing (the idea is that we don't mess with it and it keeps working). In this case, I performed manual randomized testing. I wrote a test case to generate random values, subject to the conditions in linear_congruential_engine, and compared not only the ultimate output but also the intermediate _MP_arr contents. I compared the new implementation (after building the STL and running set_environment.bat, of course) to a copy of the old classic implementation, for 12.8 billion trials, and everything agreed.

Click to expand test case:

C:\Temp>type meow.cpp

// Copyright (c) Microsoft Corporation.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception

#include <atomic>
#include <cassert>
#include <chrono>
#include <cstdint>
#include <cstring>
#include <limits>
#include <print>
#include <random>
#include <thread>
#include <vector>
using namespace std;
using namespace std::chrono;

namespace {
    using _MP_arr         = uint64_t[5];
    constexpr int _MP_len = 5;

    // implements multiprecision math for random number generators

    constexpr int shift                 = numeric_limits<unsigned long long>::digits / 2;
    constexpr unsigned long long mask   = ~(~0ULL << shift);
    constexpr unsigned long long maxVal = mask + 1;

    [[nodiscard]] unsigned long long __CLRCALL_PURE_OR_CDECL old_Get(
        _MP_arr u) noexcept { // convert multi-word value to scalar value
        return (u[1] << shift) + u[0];
    }

    static void add(unsigned long long* u, int ulen, unsigned long long* v,
        int vlen) noexcept { // add multi-word value to multi-word value
        int i;
        unsigned long long k = 0;
        for (i = 0; i < vlen; ++i) { // add multi-word values
            u[i] += v[i] + k;
            k = u[i] >> shift;
            u[i] &= mask;
        }
        for (; k != 0 && i < ulen; ++i) { // propagate carry
            u[i] += k;
            k = u[i] >> shift;
            u[i] &= mask;
        }
    }

    void __CLRCALL_PURE_OR_CDECL old_Add(
        _MP_arr u, unsigned long long v0) noexcept { // add scalar value to multi-word value
        unsigned long long v[2];
        v[0] = v0 & mask;
        v[1] = v0 >> shift;
        add(u, _MP_len, v, 2);
    }

    static void mul(unsigned long long* u, int ulen,
        unsigned long long v0) noexcept { // multiply multi-word value by single-word value
        unsigned long long k = 0;
        for (int i = 0; i < ulen; ++i) { // multiply and propagate carry
            u[i] = u[i] * v0 + k;
            k    = u[i] >> shift;
            u[i] &= mask;
        }
    }

    void __CLRCALL_PURE_OR_CDECL old_Mul(_MP_arr w, unsigned long long u0,
        unsigned long long v0) noexcept { // multiply multi-word value by multi-word value
        constexpr int m = 2;
        constexpr int n = 2;
        unsigned long long u[2];
        unsigned long long v[2];
        u[0] = u0 & mask;
        u[1] = u0 >> shift;
        v[0] = v0 & mask;
        v[1] = v0 >> shift;

        // Knuth, vol. 2, p. 268, Algorithm M
        // M1: [Initialize.]
        for (int i = 0; i < m + n + 1; ++i) {
            w[i] = 0;
        }

        for (int j = 0; j < n; ++j) { // M2: [Zero multiplier?]
            if (v[j] == 0) {
                w[j + m] = 0;
            } else { // multiply by non-zero value
                unsigned long long k = 0;
                int i;
                // M3: [Initialize i.]
                for (i = 0; i < m; ++i) { // M4: [Multiply and add.]
                    w[i + j] = u[i] * v[j] + w[i + j] + k;
                    k        = w[i + j] >> shift;
                    w[i + j] &= mask;
                    // M5: [Loop on i.]
                }
                w[i + j] = k;
            }
            // M6: [Loop on j.]
        }
    }

    static void div(_MP_arr u,
        unsigned long long
            v0) noexcept { // divide multi-word value by scalar value (fits in lower half of unsigned long long)
        unsigned long long k = 0;
        int ulen             = _MP_len;
        while (0 <= --ulen) { // propagate remainder and divide
            unsigned long long tmp = (k << shift) + u[ulen];
            u[ulen]                = tmp / v0;
            k                      = tmp % v0;
        }
    }

    [[nodiscard]] static int limit(const unsigned long long* u, int ulen) noexcept { // get index of last non-zero value
        while (u[ulen - 1] == 0) {
            --ulen;
        }

        return ulen;
    }

    void __CLRCALL_PURE_OR_CDECL old_Rem(
        _MP_arr u, unsigned long long v0) noexcept { // divide multi-word value by value, leaving remainder in u
        unsigned long long v[2];
        v[0]        = v0 & mask;
        v[1]        = v0 >> shift;
        const int n = limit(v, 2);
        _Analysis_assume_(n > 0);
        _Analysis_assume_(n <= 2);
        const int m = limit(u, _MP_len) - n;
        _Analysis_assume_(m > 0);
        _Analysis_assume_(m <= _MP_len - n);

        // Knuth, vol. 2, p. 272, Algorithm D
        // D1: [Normalize.]
        unsigned long long d = maxVal / (v[n - 1] + 1);
        if (d != 1) { // scale numerator and divisor
            mul(u, _MP_len, d);
            mul(v, n, d);
        }
        // D2: [Initialize j.]
        for (int j = m; 0 <= j; --j) { // D3: [Calculate qh.]
            unsigned long long qh = ((u[j + n] << shift) + u[j + n - 1]) / v[n - 1];
            if (qh == 0) {
                continue;
            }

            unsigned long long rh = ((u[j + n] << shift) + u[j + n - 1]) % v[n - 1];
            for (;;) {
#pragma warning(push)
#pragma warning(disable : 6385) // TRANSITION, GH-1008
                if (qh < maxVal && qh * v[n - 2] <= (rh << shift) + u[j + n - 2]) {
#pragma warning(pop)
                    break;
                } else { // reduce tentative value and retry
                    --qh;
                    rh += v[n - 1];
                    if (maxVal <= rh) {
                        break;
                    }
                }
            }

            // D4: [Multiply and subtract.]
            unsigned long long k = 0;
            int i;
            for (i = 0; i < n; ++i) { // multiply and subtract
                u[j + i] -= qh * v[i] + k;
                k = u[j + i] >> shift;
                if (k) {
                    k = maxVal - k;
                }

                u[j + i] &= mask;
            }
            for (; k != 0 && j + i < _MP_len; ++i) { // propagate borrow
                u[j + i] -= k;
                k = u[j + i] >> shift;
                if (k) {
                    k = maxVal - k;
                }

                u[j + i] &= mask;
            }
            // D5: [Test remainder.]
            if (k != 0) { // D6: [Add back.]
                --qh;
                add(u + j, n + 1, v, n);
            }
            // D7: [Loop on j.]
        }
        // D8: [Unnormalize.]
        if (d != 1) {
            div(u, d);
        }
    }
} // unnamed namespace

_STD_BEGIN
[[nodiscard]] _CRTIMP2_PURE uint64_t __CLRCALL_PURE_OR_CDECL _MP_Get(_MP_arr _Wx) noexcept;
_CRTIMP2_PURE void __CLRCALL_PURE_OR_CDECL _MP_Add(_MP_arr _Wx, uint64_t _Cx) noexcept;
_CRTIMP2_PURE void __CLRCALL_PURE_OR_CDECL _MP_Mul(_MP_arr _Wx, uint64_t _Prev, uint64_t _Ax) noexcept;
_CRTIMP2_PURE void __CLRCALL_PURE_OR_CDECL _MP_Rem(_MP_arr _Wx, uint64_t _Mx) noexcept;
_STD_END

int main() {
    atomic<uint64_t> atom_actual_trials{0};

    constexpr uint32_t num_threads           = 32;
    constexpr uint64_t num_trials_per_thread = 400'000'000;

    vector<jthread> worker_threads;
    worker_threads.reserve(num_threads);

    const auto start = steady_clock::now();
    for (uint32_t worker = 0; worker < num_threads; ++worker) {
        worker_threads.emplace_back([worker, &atom_actual_trials] {
            mt19937_64 gen{1729 + worker};

            uint64_t local_actual_trials = 0;

            for (uint64_t trials = 0; trials < num_trials_per_thread; ++trials) {
                // `_Mx == 0` is special-cased; `_Mx == 1` would force `_Ax == 0` (see below)
                uniform_int_distribution<uint64_t> mx_dist{2, UINT64_MAX};
                const uint64_t mx = mx_dist(gen);

                // N5008 [rand.eng.lcong]/3: "If the template parameter m is not 0,
                // the following relations shall hold: a < m and c < m."

                uniform_int_distribution<uint64_t> ax_dist{1, mx - 1}; // `_Ax == 0` is special-cased
                const uint64_t ax = ax_dist(gen);

                uniform_int_distribution<uint64_t> cx_dist{0, mx - 1};
                const uint64_t cx = cx_dist(gen);

                uniform_int_distribution<uint64_t> prev_dist{0, mx - 1}; // the result is always taken modulo `_Mx`
                const uint64_t prev = prev_dist(gen);

                if (cx <= UINT32_MAX && mx - 1 <= (UINT32_MAX - cx) / ax) {
                    continue;
                } else if (mx - 1 <= (UINT64_MAX - cx) / ax) {
                    continue;
                } else {
                    ++local_actual_trials;

                    _MP_arr wx_old;
                    _MP_arr wx_new;

                    old_Mul(wx_old, prev, ax);
                    _MP_Mul(wx_new, prev, ax);
                    assert(memcmp(wx_old, wx_new, sizeof(_MP_arr)) == 0);

                    old_Add(wx_old, cx);
                    _MP_Add(wx_new, cx);
                    assert(memcmp(wx_old, wx_new, sizeof(_MP_arr)) == 0);

                    old_Rem(wx_old, mx);
                    _MP_Rem(wx_new, mx);
                    assert(memcmp(wx_old, wx_new, sizeof(_MP_arr)) == 0);

                    const auto result_old = old_Get(wx_old);
                    const auto result_new = _MP_Get(wx_new);
                    assert(result_old == result_new);
                }
            }

            atom_actual_trials += local_actual_trials;
        });
    }
    worker_threads.clear();
    const auto finish = steady_clock::now();

    println("num_threads: {}", num_threads);
    println("num_trials_per_thread: {}", num_trials_per_thread);
    println("atom_actual_trials: {}", atom_actual_trials.load());
    println("Time: {} ms", duration_cast<milliseconds>(finish - start).count());
}

C:\Temp>cl /EHsc /nologo /W4 /std:c++latest /MT /O2 /GL meow.cpp
meow.cpp
Generating code
Finished generating code

C:\Temp>meow
num_threads: 32
num_trials_per_thread: 400000000
atom_actual_trials: 12799999995
Time: 58741 ms

[StephanTLavavej]

I'm mirroring this to the MSVC-internal repo - please notify me if any further changes are pushed.