C# 使用SIMD向量型別加速浮點陣列求和運算（3）：回圈展開-有解無憂

作者： zyl910

一、背景
- 1.1 回圈展開簡介
- 1.2 測驗準備
二、在C#中使用
- 2.1 對基礎演算法做回圈展開
  - 2.1.1 測驗結果:
- 2.2 對 Vector4 版演算法做回圈展開
  - 2.2.1 測驗結果:
- 2.3 對 Vector<T> 版演算法做回圈展開
  - 2.3.1 測驗結果:
- 2.4 對 Avx版演算法做回圈展開
  - 2.4.1 測驗結果:
- 2.5 對 Avx版演算法做回圈展開16次
  - 2.5.1 測驗結果:
- 2.6 嘗試用陣列來存盤回圈展開的臨時變數
  - 2.6.1 測驗結果:
- 2.7 嘗試用堆疊陣列來減少相關性
  - 2.7.1 測驗結果:
- 2.8 測驗結果匯總
三、在C++中使用
- 3.1 修改代碼
- 3.2 測驗結果
四、小結
參考文獻

一、背景

先前的2篇文章，說了向量型別的型別選擇問題，本文討論一個使用方面的問題——回圈展開，

現在的CPU采用了流水線、超標量等機制來提高運算性能，如果完全是順序代碼，那么流水線的效果會非常好，
但是程式中不可避免的需要分支與回圈來處理各種復雜的邏輯，分支與回圈會被編譯為跳轉指令，而跳轉指令會導致CPU流水線失效，對性能的影響很大，雖然現代處理器增加了分支預測技術，但總會有預測失敗的概率，
尤其是在使用向量型別進行SIMD運算時，因向量型別僅盡可能榨干CPU內部的ALU（算術邏輯單元），于是在跳轉時的性能損失更大，

故在使用向量型別處理大規模數學計算時，應盡可能的避免分支與回圈，
對于分支，最好盡量將分支挪到內回圈外，若是內回圈中必須的分支，可盡量用位掩碼等辦法來寫無分支代碼，
對于回圈，一般可使用回圈展開技術，來避免短的回圈，

1.1 回圈展開簡介

摘錄——

回圈展開（Loop unrolling）技術是一種提升程式執行速度的非常有效的優化方法，它可以由程式員手工撰寫，也可由編譯器自動優化，回圈展開的本質是，利用CPU指令級并行，來降低回圈的開銷，當然，同時也有利于指令流水線的高效調度，
……

回圈展開的優點：
第一，減少了分支預測失敗的可能性，
第二，增加了回圈體內陳述句并發執行的可能性，當然，這需要回圈體內各陳述句不存在資料相關性，

回圈展開的缺點：
第一，造成代碼膨脹，導致ELF檔案（或Windows PE檔案）尺寸增大，
第二，代碼可讀性顯著降低，前一個人寫的回圈展開代碼，很可能被不熟悉的后續維護人員改回去，

1.2 測驗準備

注意，回圈展開提高的是流水線性能，對小回圈效果明顯，此時分支造成的延時，大多與內回圈的運算耗時差不多，
對于有些復雜的大回圈，內回圈的運算耗時已經很大了，而分支造成的延時仍是常數值，比例下降了很多，此時回圈展開的收益就少了，

由于回圈展開是程式員手工撰寫的，故必須在編碼前就確定好展開次數，
本文就來探討一下大多數時候的展開次數選擇，

展開2倍的話，性能最多為原來2倍，即大多數情況下只有1倍多的性能提升，提升不大，
展開2倍的話，性能最多為原來4倍，區間大了，很多時候能達到2倍以上的提升，
故一開始可以用展開4倍來測驗，下面將進行測驗，

測驗電腦的配置資訊為：lntel(R) Core(TM) i5-8250U CPU @ 1.60GHz、Windows 10，

二、在C#中使用

為了對比測驗 Avx指令的效果，故可在 BenchmarkVectorCore30 工程里進行測驗，因是64位作業系統，故選取 x64、Release版的測驗結果.

2.1 對基礎演算法做回圈展開

回顧一下基礎演算法:

private static float SumBase(float[] src, int count, int loops) {
    float rt = 0; // Result.
    for (int j=0; j< loops; ++j) {
        for(int i=0; i< count; ++i) {
            rt += src[i];
        }
    }
    return rt;
}

改為回圈展開4倍后，代碼為:

private static float SumBaseU4(float[] src, int count, int loops) {
    float rt = 0; // Result.
    float rt1 = 0;
    float rt2 = 0;
    float rt3 = 0;
    int nBlockWidth = 4; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    int p; // Index for src data.
    int i;
    for (int j = 0; j < loops; ++j) {
        p = 0;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            rt += src[p];
            rt1 += src[p + 1];
            rt2 += src[p + 2];
            rt3 += src[p + 3];
            p += nBlockWidth;
        }
        // Remainder processs.
        //p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    rt = rt + rt1 + rt2 + rt3;
    return rt;
}

之前內回圈只處理1個資料，現在內回圈處理了4個資料，
注意內回圈在處理者4個資料時，并不是直接將結果全部累加到 rt 變數，而是使用新增的 rt1、rt2、rt3 變數來臨時存盤累加值，這是為了消除變數之間的相關性，因為變數之間的相關性會影響流水線性能，故分別使用獨立的變數就好了，
最后在 Reduce 階段，將 rt1、rt2、rt3 的值累加到 rt，

2.1.1 測驗結果:

測驗結果摘錄如下：

SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336

可以發現，基礎演算法使用4倍回圈展開后，性能是原先的 2.6336 倍，

2.2 對 Vector4 版演算法做回圈展開

回顧一下Vector4 版演算法:

private static float SumVector4(float[] src, int count, int loops) {
    float rt = 0; // Result.
    const int VectorWidth = 4;
    int nBlockWidth = VectorWidth; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    Vector4 vrt = Vector4.Zero; // Vector result.
    int p; // Index for src data.
    int i;
    // Load.
    Vector4[] vsrc = https://www.cnblogs.com/zyl910/archive/2022/11/16/new Vector4[cntBlock]; // Vector src.
    p = 0;
    for (i = 0; i < vsrc.Length; ++i) {
        vsrc[i] = new Vector4(src[p], src[p + 1], src[p + 2], src[p + 3]);
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            // Equivalent to scalar model: rt += src[i];
            vrt += vsrc[i]; // Add.
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    rt += vrt.X + vrt.Y + vrt.Z + vrt.W;
    return rt;
}

改為回圈展開4倍后，代碼為:

private static float SumVector4U4(float[] src, int count, int loops) {
    float rt = 0; // Result.
    const int LoopUnrolling = 4;
    const int VectorWidth = 4;
    int nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    Vector4 vrt = Vector4.Zero; // Vector result.
    Vector4 vrt1 = Vector4.Zero;
    Vector4 vrt2 = Vector4.Zero;
    Vector4 vrt3 = Vector4.Zero;
    int p; // Index for src data.
    int i;
    // Load.
    Vector4[] vsrc = https://www.cnblogs.com/zyl910/archive/2022/11/16/new Vector4[count / VectorWidth]; // Vector src.
    p = 0;
    for (i = 0; i < vsrc.Length; ++i) {
        vsrc[i] = new Vector4(src[p], src[p + 1], src[p + 2], src[p + 3]);
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = 0;
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt += vsrc[p]; // Add.
            vrt1 += vsrc[p + 1];
            vrt2 += vsrc[p + 2];
            vrt3 += vsrc[p + 3];
            p += LoopUnrolling;
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    vrt = vrt + vrt1 + vrt2 + vrt3;
    rt += vrt.X + vrt.Y + vrt.Z + vrt.W;
    return rt;
}

跟剛才的辦法一樣，使用新增的 rt1、rt2、rt3 變數來臨時存盤累加值，消除變數之間的相關性，
最后在 Reduce 階段，將 vrt1、vrt2、vrt3 的值累加到 vrt，

2.2.1 測驗結果:

測驗結果摘錄如下：

SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVector4:     2.748779E+11    # msUsed=1218, MFLOPS/s=3362.8899835796387, scale=4.054187192118227
SumVector4U4:   1.0995116E+12   # msUsed=532, MFLOPS/s=7699.248120300752, scale=9.281954887218046

SumVector4U4對比基礎演算法（SumBase），性能倍數是 9.281954887218046，
SumVector4U4對比未回圈展開的演算法（SumVector4），倍數是 9.281954887218046/4.054187192118227=2.2894736842105263092984587836542

2.3 對 `Vector<T>` 版演算法做回圈展開

回顧一下 Vector<T> 版演算法:

private static float SumVectorT(float[] src, int count, int loops) {
    float rt = 0; // Result.
    int VectorWidth = Vector<float>.Count; // Block width.
    int nBlockWidth = VectorWidth; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    Vector<float> vrt = Vector<float>.Zero; // Vector result.
    int p; // Index for src data.
    int i;
    // Load.
    Vector<float>[] vsrc = https://www.cnblogs.com/zyl910/archive/2022/11/16/new Vector[cntBlock]; // Vector src.
    p = 0;
    for (i = 0; i < vsrc.Length; ++i) {
        vsrc[i] = new Vector(src, p);
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt += vsrc[i]; // Add.
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt[i];
    }
    return rt;
}

改為回圈展開4倍后，代碼為:

private static float SumVectorTU4(float[] src, int count, int loops) {
    float rt = 0; // Result.
    const int LoopUnrolling = 4;
    int VectorWidth = Vector<float>.Count; // Block width.
    int nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    Vector<float> vrt = Vector<float>.Zero; // Vector result.
    Vector<float> vrt1 = Vector<float>.Zero;
    Vector<float> vrt2 = Vector<float>.Zero;
    Vector<float> vrt3 = Vector<float>.Zero;
    int p; // Index for src data.
    int i;
    // Load.
    Vector<float>[] vsrc = https://www.cnblogs.com/zyl910/archive/2022/11/16/new Vector[count / VectorWidth]; // Vector src.
    p = 0;
    for (i = 0; i < vsrc.Length; ++i) {
        vsrc[i] = new Vector(src, p);
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = 0;
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt += vsrc[p]; // Add.
            vrt1 += vsrc[p + 1];
            vrt2 += vsrc[p + 1];
            vrt3 += vsrc[p + 1];
            p += LoopUnrolling;
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    vrt = vrt + vrt1 + vrt2 + vrt3;
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt[i];
    }
    return rt;
}

跟剛才的辦法一樣，使用新增的 rt1、rt2、rt3 變數來臨時存盤累加值，消除變量之間的相關性，
最后在 Reduce 階段，將 vrt1、vrt2、vrt3 的值累加到 vrt，

2.3.1 測驗結果:

測驗結果摘錄如下：

SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVectorT:     5.497558E+11    # msUsed=609, MFLOPS/s=6725.7799671592775, scale=8.108374384236454
SumVectorTU4:   2.1990233E+12   # msUsed=203, MFLOPS/s=20177.339901477833, scale=24.32512315270936

SumVectorTU4對比基礎演算法（SumBase），性能倍數是 24.32512315270936，
SumVectorTU4對比未回圈展開的演算法（SumVectorT），倍數是 24.32512315270936/8.108374384236454=2.9999999999999997533414337788579
初步發現 Vector<T>回圈展開（2.9999）帶來的性能提升, 比VectorT回圈展開（2.2894）更高一些，

2.4 對 Avx版演算法做回圈展開

先前分別嘗試用陣列、Span、指標的辦法來操縱資料、使用Avx指令集，現在對這3種辦法，均寫一套回圈展開4次的代碼:

/// <summary>
/// Sum - Vector AVX.
/// </summary>
/// <param name="src">Soure array.</param>
/// <param name="count">Soure array count.</param>
/// <param name="loops">Benchmark loops.</param>
/// <returns>Return the sum value.</returns>
private static float SumVectorAvx(float[] src, int count, int loops) {
#if Allow_Intrinsics
    float rt = 0; // Result.
    //int VectorWidth = 32 / 4; // sizeof(__m256) / sizeof(float);
    int VectorWidth = Vector256<float>.Count; // Block width.
    int nBlockWidth = VectorWidth; // Block width.
    int cntBlock = count / nBlockWidth; // Block count.
    int cntRem = count % nBlockWidth; // Remainder count.
    Vector256<float> vrt = Vector256<float>.Zero; // Vector result.
    int p; // Index for src data.
    int i;
    // Load.
    Vector256<float>[] vsrc = https://www.cnblogs.com/zyl910/archive/2022/11/16/new Vector256[cntBlock]; // Vector src.
    p = 0;
    for (i = 0; i < cntBlock; ++i) {
        vsrc[i] = Vector256.Create(src[p], src[p + 1], src[p + 2], src[p + 3], src[p + 4], src[p + 5], src[p + 6], src[p + 7]); // Load.
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt = Avx.Add(vrt, vsrc[i]);    // Add. vrt += vsrc[i];
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt.GetElement(i);
    }
    return rt;
#else
    throw new NotSupportedException();
#endif
}

/// 
/// Sum - Vector AVX - Loop unrolling *4.
/// 
/// [count / VectorWidth]; // Vector src.
    p = 0;
    for (i = 0; i < vsrc.Length; ++i) {
        vsrc[i] = Vector256.Create(src[p], src[p + 1], src[p + 2], src[p + 3], src[p + 4], src[p + 5], src[p + 6], src[p + 7]); // Load.
        p += VectorWidth;
    }
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = 0;
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt = Avx.Add(vrt, vsrc[p]);    // Add. vrt += vsrc[p];
            vrt1 = Avx.Add(vrt1, vsrc[p + 1]);
            vrt2 = Avx.Add(vrt2, vsrc[p + 2]);
            vrt3 = Avx.Add(vrt3, vsrc[p + 3]);
            p += LoopUnrolling;
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    vrt = Avx.Add(Avx.Add(vrt, vrt1), Avx.Add(vrt2, vrt3)); // vrt = vrt + vrt1 + vrt2 + vrt3;
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt.GetElement(i);
    }
    return rt;
#else
    throw new NotSupportedException();
#endif
}

/// 
/// Sum - Vector AVX - Span.
/// 
///  >(new Span(src)); // Reinterpret cast. `float*` to `Vector256*`.
        for (i = 0; i < cntBlock; ++i) {
            vrt = Avx.Add(vrt, vsrc[i]);    // Add. vrt += vsrc[i];
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt.GetElement(i);
    }
    return rt;
#else
    throw new NotSupportedException();
#endif
}

/// 
/// Sum - Vector AVX - Span - Loop unrolling *4.
/// 
/// >(new Span(src)); // Reinterpret cast. `float*` to `Vector256*`.
        for (i = 0; i < cntBlock; ++i) {
            vrt = Avx.Add(vrt, vsrc[p]);    // Add. vrt += vsrc[p];
            vrt1 = Avx.Add(vrt1, vsrc[p + 1]);
            vrt2 = Avx.Add(vrt2, vsrc[p + 2]);
            vrt3 = Avx.Add(vrt3, vsrc[p + 3]);
            p += LoopUnrolling;
        }
        // Remainder processs.
        p = cntBlock * nBlockWidth;
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    vrt = Avx.Add(Avx.Add(vrt, vrt1), Avx.Add(vrt2, vrt3)); // vrt = vrt + vrt1 + vrt2 + vrt3;
    for (i = 0; i < VectorWidth; ++i) {
        rt += vrt.GetElement(i);
    }
    return rt;
#else
    throw new NotSupportedException();
#endif
}

/// 
/// Sum - Vector AVX - Ptr.
/// 
///


2.4.1 測驗結果:
測驗結果摘錄如下：
SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVectorAvx:   5.497558E+11    # msUsed=609, MFLOPS/s=6725.7799671592775, scale=8.108374384236454
SumVectorAvxSpan:       5.497558E+11    # msUsed=625, MFLOPS/s=6553.6, scale=7.9008
SumVectorAvxPtr:        5.497558E+11    # msUsed=610, MFLOPS/s=6714.754098360656, scale=8.095081967213115
SumVectorAvxU4: 2.1990233E+12   # msUsed=328, MFLOPS/s=12487.80487804878, scale=15.054878048780488
SumVectorAvxSpanU4:     2.1990233E+12   # msUsed=312, MFLOPS/s=13128.205128205129, scale=15.826923076923078
SumVectorAvxPtrU4:      2.1990233E+12   # msUsed=157, MFLOPS/s=26089.171974522294, scale=31.452229299363058

未做回圈展開時，這3鐘辦法的性能拉不開差距，都是8倍左右，

而現在用了回圈展開后，陣列版（SumVectorAvxU4）、Span版（SumVectorAvxSpanU4）只有15倍左右的性能提升，而指標版有 31倍性能提升，是 陣列版、Span版 的2倍，

可能是因為指標更貼近底層硬體、更易于編譯器優化，故當使用內在函式時，推薦優先使用指標，
SumVectorAvxPtrU4 對比基礎演算法（SumBase），性能倍數是 31.452229299363058，

SumVectorAvxPtrU4 對比未回圈展開的演算法（SumVectorAvxPtr），倍數是 31.452229299363058/8.095081967213115=3.8853503184713375449974589366746，
2.5 對 Avx版演算法做回圈展開16次
剛才嘗試了4倍回圈展開，故理論上限是4倍，而SumVectorAvxPtrU4版有約 3.8853 倍性能提升，故可考慮進一步加大，于是可測驗一下 4*4=16 次的回圈展開，
將 SumVectorAvxPtr 改造為回圈展開16次的，代碼如下:
private static float SumVectorAvxPtrU16(float[] src, int count, int loops) {
#if Allow_Intrinsics && UNSAFE
    unsafe {
        float rt = 0; // Result.
        const int LoopUnrolling = 16;
        int VectorWidth = Vector256<float>.Count; // Block width.
        int nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
        int cntBlock = count / nBlockWidth; // Block count.
        int cntRem = count % nBlockWidth; // Remainder count.
        Vector256<float> vrt = Vector256<float>.Zero; // Vector result.
        Vector256<float> vrt1 = Vector256<float>.Zero;
        Vector256<float> vrt2 = Vector256<float>.Zero;
        Vector256<float> vrt3 = Vector256<float>.Zero;
        Vector256<float> vrt4 = Vector256<float>.Zero;
        Vector256<float> vrt5 = Vector256<float>.Zero;
        Vector256<float> vrt6 = Vector256<float>.Zero;
        Vector256<float> vrt7 = Vector256<float>.Zero;
        Vector256<float> vrt8 = Vector256<float>.Zero;
        Vector256<float> vrt9 = Vector256<float>.Zero;
        Vector256<float> vrt10 = Vector256<float>.Zero;
        Vector256<float> vrt11 = Vector256<float>.Zero;
        Vector256<float> vrt12 = Vector256<float>.Zero;
        Vector256<float> vrt13 = Vector256<float>.Zero;
        Vector256<float> vrt14 = Vector256<float>.Zero;
        Vector256<float> vrt15 = Vector256<float>.Zero;
        float* p; // Pointer for src data.
        int i;
        // Body.
        fixed (float* p0 = &src[0]) {
            for (int j = 0; j < loops; ++j) {
                p = p0;
                // Vector processs.
                for (i = 0; i < cntBlock; ++i) {
                    //vload = Avx.LoadVector256(p);    // Load. vload = *(*__m256)p;
                    vrt = Avx.Add(vrt, Avx.LoadVector256(p)); // Add. vrt[k] += *((*__m256)(p)+k);
                    vrt1 = Avx.Add(vrt1, Avx.LoadVector256(p + VectorWidth * 1));
                    vrt2 = Avx.Add(vrt2, Avx.LoadVector256(p + VectorWidth * 2));
                    vrt3 = Avx.Add(vrt3, Avx.LoadVector256(p + VectorWidth * 3));
                    vrt4 = Avx.Add(vrt4, Avx.LoadVector256(p + VectorWidth * 4));
                    vrt5 = Avx.Add(vrt5, Avx.LoadVector256(p + VectorWidth * 5));
                    vrt6 = Avx.Add(vrt6, Avx.LoadVector256(p + VectorWidth * 6));
                    vrt7 = Avx.Add(vrt7, Avx.LoadVector256(p + VectorWidth * 7));
                    vrt8 = Avx.Add(vrt8, Avx.LoadVector256(p + VectorWidth * 8));
                    vrt9 = Avx.Add(vrt9, Avx.LoadVector256(p + VectorWidth * 9));
                    vrt10 = Avx.Add(vrt10, Avx.LoadVector256(p + VectorWidth * 10));
                    vrt11 = Avx.Add(vrt11, Avx.LoadVector256(p + VectorWidth * 11));
                    vrt12 = Avx.Add(vrt12, Avx.LoadVector256(p + VectorWidth * 12));
                    vrt13 = Avx.Add(vrt13, Avx.LoadVector256(p + VectorWidth * 13));
                    vrt14 = Avx.Add(vrt14, Avx.LoadVector256(p + VectorWidth * 14));
                    vrt15 = Avx.Add(vrt15, Avx.LoadVector256(p + VectorWidth * 15));
                    p += nBlockWidth;
                }
                // Remainder processs.
                for (i = 0; i < cntRem; ++i) {
                    rt += p[i];
                }
            }
        }
        // Reduce.
        vrt = Avx.Add( Avx.Add( Avx.Add(Avx.Add(vrt, vrt1), Avx.Add(vrt2, vrt3))
            , Avx.Add(Avx.Add(vrt4, vrt5), Avx.Add(vrt6, vrt7)) )
            , Avx.Add( Avx.Add(Avx.Add(vrt8, vrt9), Avx.Add(vrt10, vrt11))
            , Avx.Add(Avx.Add(vrt12, vrt13), Avx.Add(vrt14, vrt15)) ) )
        ; // vrt = vrt + vrt1 + vrt2 + vrt3 + ... vrt15;
        for (i = 0; i < VectorWidth; ++i) {
            rt += vrt.GetElement(i);
        }
        return rt;
    }
#else
    throw new NotSupportedException();
#endif
}

2.5.1 測驗結果:
測驗結果摘錄如下：
SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVectorAvxPtr:        5.497558E+11    # msUsed=610, MFLOPS/s=6714.754098360656, scale=8.095081967213115
SumVectorAvxPtrU4:      2.1990233E+12   # msUsed=157, MFLOPS/s=26089.171974522294, scale=31.452229299363058
SumVectorAvxPtrU16:     8.386202E+12    # msUsed=125, MFLOPS/s=32768, scale=39.504

SumVectorAvxPtrU16 對比基礎演算法（SumBase），性能倍數是 39.504，

SumVectorAvxPtrU16 對比未回圈展開的演算法（SumVectorAvxPtr），倍數是 39.504/8.095081967213115=4.8799999999999998517618469015796，

SumVectorAvxPtrU16 對比回圈展開4次的演算法（SumVectorAvxPtrU4），倍數是 39.504/31.452229299363058=1.2559999999999999730384771162414，
從回圈展開4次，改為回圈展開16次，性能倍數只是從 31倍多，提升到 39 倍多，僅提升 25% 左右，

性能提升的少，但編碼麻煩多了，看來回圈展開16次的性價比很低，故一般情況下用回圈展開4次就行了，
2.6 嘗試用陣列來存盤回圈展開的臨時變數
使用回圈展開N次時，將會導致臨時變數數量是非回圈展開版的N倍，例如剛才的 SumVectorAvxPtrU16 函式，因回圈展開16次，導致臨時變數是非回圈展開版的16倍，寫起了很啰嗦，

這些變數的型別是一樣的，放到陣列中的話，代碼會清晰不少，但會不會影響性能呢？

于是做了一個測驗，代碼如下：
private static float SumVectorAvxPtrU16A(float[] src, int count, int loops) {
#if Allow_Intrinsics && UNSAFE
    unsafe {
        float rt = 0; // Result.
        const int LoopUnrolling = 16;
        int VectorWidth = Vector256<float>.Count; // Block width.
        int nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
        int cntBlock = count / nBlockWidth; // Block count.
        int cntRem = count % nBlockWidth; // Remainder count.
        int i;
        //Vector256<float>[] vrt = new Vector256<float>[LoopUnrolling]; // Vector result.
        Vector256<float>* vrt = stackalloc Vector256<float>[LoopUnrolling]; ; // Vector result.
        for (i = 0; i< LoopUnrolling; ++i) {
            vrt[i] = Vector256<float>.Zero;
        }
        float* p; // Pointer for src data.
        // Body.
        fixed (float* p0 = &src[0]) {
            for (int j = 0; j < loops; ++j) {
                p = p0;
                // Vector processs.
                for (i = 0; i < cntBlock; ++i) {
                    //vload = Avx.LoadVector256(p);    // Load. vload = *(*__m256)p;
                    vrt[0] = Avx.Add(vrt[0], Avx.LoadVector256(p)); // Add. vrt[k] += *((*__m256)(p)+k);
                    vrt[1] = Avx.Add(vrt[1], Avx.LoadVector256(p + VectorWidth * 1));
                    vrt[2] = Avx.Add(vrt[2], Avx.LoadVector256(p + VectorWidth * 2));
                    vrt[3] = Avx.Add(vrt[3], Avx.LoadVector256(p + VectorWidth * 3));
                    vrt[4] = Avx.Add(vrt[4], Avx.LoadVector256(p + VectorWidth * 4));
                    vrt[5] = Avx.Add(vrt[5], Avx.LoadVector256(p + VectorWidth * 5));
                    vrt[6] = Avx.Add(vrt[6], Avx.LoadVector256(p + VectorWidth * 6));
                    vrt[7] = Avx.Add(vrt[7], Avx.LoadVector256(p + VectorWidth * 7));
                    vrt[8] = Avx.Add(vrt[8], Avx.LoadVector256(p + VectorWidth * 8));
                    vrt[9] = Avx.Add(vrt[9], Avx.LoadVector256(p + VectorWidth * 9));
                    vrt[10] = Avx.Add(vrt[10], Avx.LoadVector256(p + VectorWidth * 10));
                    vrt[11] = Avx.Add(vrt[11], Avx.LoadVector256(p + VectorWidth * 11));
                    vrt[12] = Avx.Add(vrt[12], Avx.LoadVector256(p + VectorWidth * 12));
                    vrt[13] = Avx.Add(vrt[13], Avx.LoadVector256(p + VectorWidth * 13));
                    vrt[14] = Avx.Add(vrt[14], Avx.LoadVector256(p + VectorWidth * 14));
                    vrt[15] = Avx.Add(vrt[15], Avx.LoadVector256(p + VectorWidth * 15));
                    p += nBlockWidth;
                }
                // Remainder processs.
                for (i = 0; i < cntRem; ++i) {
                    rt += p[i];
                }
            }
        }
        // Reduce.
        for (i = 1; i < LoopUnrolling; ++i) {
            vrt[0] = Avx.Add(vrt[0], vrt[i]); // vrt[0] += vrt[i]
        }
        for (i = 0; i < VectorWidth; ++i) {
            rt += vrt[0].GetElement(i);
        }
        return rt;
    }
#else
    throw new NotSupportedException();
#endif
}

2.6.1 測驗結果:
測驗結果摘錄如下：
SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVectorAvxPtr:        5.497558E+11    # msUsed=610, MFLOPS/s=6714.754098360656, scale=8.095081967213115
SumVectorAvxPtrU4:      2.1990233E+12   # msUsed=157, MFLOPS/s=26089.171974522294, scale=31.452229299363058
SumVectorAvxPtrU16:     8.386202E+12    # msUsed=125, MFLOPS/s=32768, scale=39.504
SumVectorAvxPtrU16A:    8.3862026E+12   # msUsed=187, MFLOPS/s=21903.74331550802, scale=26.406417112299465

可以發現 SumVectorAvxPtrU16A 的性能比 SumVectorAvxPtrU16 差，

曾經以為是因為陣列是在堆中分配的（new Vector256）引起的，有堆記憶體分配的開銷，且需要多次尋址才能定位變數，

隨后改為堆疊中分配的陣列（stackalloc Vector256），且用最貼近硬體的指標來操作，可性能幾乎一致，故猜測可能是編譯優化時難以將它們優化為暫存器變數，
故在使用回圈展開時，臨時變數不要用陣列來存，還是逐個定義區域變數比較好，
2.7 嘗試用堆疊陣列來減少相關性
還嘗試了用堆疊陣列來減少相關性，代碼如下：
private static float SumVectorAvxPtrUX(float[] src, int count, int loops, int LoopUnrolling) {
#if Allow_Intrinsics && UNSAFE
    unsafe {
        float rt = 0; // Result.
        //const int LoopUnrolling = 16;
        if (LoopUnrolling <= 0) throw new ArgumentOutOfRangeException("LoopUnrolling", "Argument LoopUnrolling must >0 !");
        int VectorWidth = Vector256<float>.Count; // Block width.
        int nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
        int cntBlock = count / nBlockWidth; // Block count.
        int cntRem = count % nBlockWidth; // Remainder count.
        int i;
        //Vector256<float>[] vrt = new Vector256<float>[LoopUnrolling]; // Vector result.
        Vector256<float>* vrt = stackalloc Vector256<float>[LoopUnrolling]; ; // Vector result.
        for (i = 0; i < LoopUnrolling; ++i) {
            vrt[i] = Vector256<float>.Zero;
        }
        float* p; // Pointer for src data.
        // Body.
        fixed (float* p0 = &src[0]) {
            for (int j = 0; j < loops; ++j) {
                p = p0;
                // Vector processs.
                for (i = 0; i < cntBlock; ++i) {
                    for(int k=0; k< LoopUnrolling; ++k) {
                        vrt[k] = Avx.Add(vrt[k], Avx.LoadVector256(p + VectorWidth * k)); // Add. vrt[k] += *(*__m256)(p+k);
                    }
                    p += nBlockWidth;
                }
                // Remainder processs.
                for (i = 0; i < cntRem; ++i) {
                    rt += p[i];
                }
            }
        }
        // Reduce.
        for (i = 1; i < LoopUnrolling; ++i) {
            vrt[0] = Avx.Add(vrt[0], vrt[i]); // vrt[0] += vrt[i]
        } // vrt = vrt + vrt1 + vrt2 + vrt3 + ... vrt15;
        for (i = 0; i < VectorWidth; ++i) {
            rt += vrt[0].GetElement(i);
        }
        return rt;
    }
#else
    throw new NotSupportedException();
#endif
}

測驗代碼:
// SumVectorAvxPtrUX.
int[] LoopUnrollingArray = { 4, 8, 16 };
foreach (int loopUnrolling in LoopUnrollingArray) {
    tickBegin = Environment.TickCount;
    rt = SumVectorAvxPtrUX(src, count, loops, loopUnrolling);
    msUsed = Environment.TickCount - tickBegin;
    mFlops = countMFlops * 1000 / msUsed;
    scale = mFlops / mFlopsBase;
    tw.WriteLine(indent + string.Format("SumVectorAvxPtrUX[{4}]:\t{0}\t# msUsed={1}, MFLOPS/s={2}, scale={3}", rt, msUsed, mFlops, scale, loopUnrolling));
}

2.7.1 測驗結果:
測驗結果摘錄如下：
SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVectorAvxPtr:        5.497558E+11    # msUsed=610, MFLOPS/s=6714.754098360656, scale=8.095081967213115
SumVectorAvxPtrU4:      2.1990233E+12   # msUsed=157, MFLOPS/s=26089.171974522294, scale=31.452229299363058
SumVectorAvxPtrU16:     8.386202E+12    # msUsed=125, MFLOPS/s=32768, scale=39.504
SumVectorAvxPtrU16A:    8.3862026E+12   # msUsed=187, MFLOPS/s=21903.74331550802, scale=26.406417112299465
SumVectorAvxPtrUX[4]:   2.1990233E+12   # msUsed=547, MFLOPS/s=7488.117001828154, scale=9.027422303473491
SumVectorAvxPtrUX[8]:   4.3980465E+12   # msUsed=500, MFLOPS/s=8192, scale=9.876
SumVectorAvxPtrUX[16]:  8.3862026E+12   # msUsed=500, MFLOPS/s=8192, scale=9.876

可以發現 SumVectorAvxPtrUX 版的性能比 非回圈展開版（SumVectorAvxPtr）的性能要好一些，從8倍左右，達到9倍，

調整堆疊陣列長度，達到8之后，性能幾乎差不多，看來已經達到瓶頸了，
該辦法的性能提升少，性價比不高，故還是推薦用經典的回圈展開辦法，
2.8 測驗結果匯總
測驗結果匯總如下：
BenchmarkVectorCore30

IsRelease:      True
EnvironmentVariable(PROCESSOR_IDENTIFIER):      Intel64 Family 6 Model 142 Stepping 10, GenuineIntel
Environment.ProcessorCount:     8
Environment.Is64BitOperatingSystem:     True
Environment.Is64BitProcess:     True
Environment.OSVersion:  Microsoft Windows NT 6.2.9200.0
Environment.Version:    3.1.26
RuntimeEnvironment.GetRuntimeDirectory: C:\Program Files\dotnet\shared\Microsoft.NETCore.App\3.1.26\
RuntimeInformation.FrameworkDescription:        .NET Core 3.1.26
BitConverter.IsLittleEndian:    True
IntPtr.Size:    8
Vector.IsHardwareAccelerated:   True
Vector<byte>.Count:     32      # 256bit
Vector<float>.Count:    8       # 256bit
Vector<double>.Count:   4       # 256bit
Vector4.Assembly.CodeBase:      file:///C:/Program Files/dotnet/shared/Microsoft.NETCore.App/3.1.26/System.Numerics.Vectors.dll
Vector<T>.Assembly.CodeBase:    file:///C:/Program Files/dotnet/shared/Microsoft.NETCore.App/3.1.26/System.Private.CoreLib.dll

Benchmark:      count=4096, loops=1000000, countMFlops=4096
SumBase:        6.871948E+10    # msUsed=4938, MFLOPS/s=829.485621709194
SumBaseU4:      2.748779E+11    # msUsed=1875, MFLOPS/s=2184.5333333333333, scale=2.6336
SumVector4:     2.748779E+11    # msUsed=1218, MFLOPS/s=3362.8899835796387, scale=4.054187192118227
SumVector4U4:   1.0995116E+12   # msUsed=532, MFLOPS/s=7699.248120300752, scale=9.281954887218046
SumVectorT:     5.497558E+11    # msUsed=609, MFLOPS/s=6725.7799671592775, scale=8.108374384236454
SumVectorTU4:   2.1990233E+12   # msUsed=203, MFLOPS/s=20177.339901477833, scale=24.32512315270936
SumVectorAvx:   5.497558E+11    # msUsed=609, MFLOPS/s=6725.7799671592775, scale=8.108374384236454
SumVectorAvxSpan:       5.497558E+11    # msUsed=625, MFLOPS/s=6553.6, scale=7.9008
SumVectorAvxPtr:        5.497558E+11    # msUsed=610, MFLOPS/s=6714.754098360656, scale=8.095081967213115
SumVectorAvxU4: 2.1990233E+12   # msUsed=328, MFLOPS/s=12487.80487804878, scale=15.054878048780488
SumVectorAvxSpanU4:     2.1990233E+12   # msUsed=312, MFLOPS/s=13128.205128205129, scale=15.826923076923078
SumVectorAvxPtrU4:      2.1990233E+12   # msUsed=157, MFLOPS/s=26089.171974522294, scale=31.452229299363058
SumVectorAvxPtrU16:     8.386202E+12    # msUsed=125, MFLOPS/s=32768, scale=39.504
SumVectorAvxPtrU16A:    8.3862026E+12   # msUsed=187, MFLOPS/s=21903.74331550802, scale=26.406417112299465
SumVectorAvxPtrUX[4]:   2.1990233E+12   # msUsed=547, MFLOPS/s=7488.117001828154, scale=9.027422303473491
SumVectorAvxPtrUX[8]:   4.3980465E+12   # msUsed=500, MFLOPS/s=8192, scale=9.876
SumVectorAvxPtrUX[16]:  8.3862026E+12   # msUsed=500, MFLOPS/s=8192, scale=9.876

三、在C++中使用
3.1 修改代碼
參考上面的經驗，現在來將 C++ 版程式也改為回圈展開的，

BenchmarkVectorCpp.cpp 的全部代碼如下:
// BenchmarkVectorCpp.cpp : This file contains the 'main' function. Program execution begins and ends there.
//

#include <immintrin.h>
#include <malloc.h>
#include <stdio.h>
#include <time.h>

#ifndef EXCEPTION_EXECUTE_HANDLER 
#define EXCEPTION_EXECUTE_HANDLER (1)
#endif // !EXCEPTION_EXECUTE_HANDLER 

// Sum - base.
float SumBase(const float* src, size_t count, int loops) {
    float rt = 0; // Result.
    size_t i;
    for (int j = 0; j < loops; ++j) {
        for (i = 0; i < count; ++i) {
            rt += src[i];
        }
    }
    return rt;
}

// Sum - base - Loop unrolling *4.
float SumBaseU4(const float* src, size_t count, int loops) {
    float rt = 0; // Result.
    float rt1=0;
    float rt2 = 0;
    float rt3 = 0;
    size_t nBlockWidth = 4; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    size_t p; // Index for src data.
    size_t i;
    for (int j = 0; j < loops; ++j) {
        p = 0;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            rt += src[p];
            rt1 += src[p + 1];
            rt2 += src[p + 2];
            rt3 += src[p + 3];
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += src[p + i];
        }
    }
    // Reduce.
    rt = rt + rt1 + rt2 + rt3;
    return rt;
}

// Sum - Vector AVX.
float SumVectorAvx(const float* src, size_t count, int loops) {
    float rt = 0; // Result.
    size_t VectorWidth = sizeof(__m256) / sizeof(float); // Block width.
    size_t nBlockWidth = VectorWidth; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    __m256 vrt = _mm256_setzero_ps(); // Vector result. [AVX] Set zero.
    __m256 vload; // Vector load.
    const float* p; // Pointer for src data.
    size_t i;
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = src;
        // Vector processs.
        for (i = 0; i < cntBlock; ++i) {
            vload = _mm256_load_ps(p);    // Load. vload = *(*__m256)p;
            vrt = _mm256_add_ps(vrt, vload);    // Add. vrt += vload;
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += p[i];
        }
    }
    // Reduce.
    p = (const float*)&vrt;
    for (i = 0; i < VectorWidth; ++i) {
        rt += p[i];
    }
    return rt;
}

// Sum - Vector AVX - Loop unrolling *4.
float SumVectorAvxU4(const float* src, size_t count, int loops) {
    float rt = 0;    // Result.
    const int LoopUnrolling = 4;
    size_t VectorWidth = sizeof(__m256) / sizeof(float); // Block width.
    size_t nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    __m256 vrt = _mm256_setzero_ps(); // Vector result. [AVX] Set zero.
    __m256 vrt1 = _mm256_setzero_ps();
    __m256 vrt2 = _mm256_setzero_ps();
    __m256 vrt3 = _mm256_setzero_ps();
    __m256 vload; // Vector load.
    __m256 vload1, vload2, vload3;
    const float* p; // Pointer for src data.
    size_t i;
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = src;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            vload = _mm256_load_ps(p);    // Load. vload = *(*__m256)p;
            vload1 = _mm256_load_ps(p + VectorWidth * 1);
            vload2 = _mm256_load_ps(p + VectorWidth * 2);
            vload3 = _mm256_load_ps(p + VectorWidth * 3);
            vrt = _mm256_add_ps(vrt, vload);    // Add. vrt += vload;
            vrt1 = _mm256_add_ps(vrt1, vload1);
            vrt2 = _mm256_add_ps(vrt2, vload2);
            vrt3 = _mm256_add_ps(vrt3, vload3);
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += p[i];
        }
    }
    // Reduce.
    vrt = _mm256_add_ps(_mm256_add_ps(vrt, vrt1), _mm256_add_ps(vrt2, vrt3)); // vrt = vrt + vrt1 + vrt2 + vrt3;
    p = (const float*)&vrt;
    for (i = 0; i < VectorWidth; ++i) {
        rt += p[i];
    }
    return rt;
}

// Sum - Vector AVX - Loop unrolling *16.
float SumVectorAvxU16(const float* src, size_t count, int loops) {
    float rt = 0;    // Result.
    const int LoopUnrolling = 16;
    size_t VectorWidth = sizeof(__m256) / sizeof(float); // Block width.
    size_t nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    __m256 vrt = _mm256_setzero_ps(); // Vector result. [AVX] Set zero.
    __m256 vrt1 = _mm256_setzero_ps();
    __m256 vrt2 = _mm256_setzero_ps();
    __m256 vrt3 = _mm256_setzero_ps();
    __m256 vrt4 = _mm256_setzero_ps();
    __m256 vrt5 = _mm256_setzero_ps();
    __m256 vrt6 = _mm256_setzero_ps();
    __m256 vrt7 = _mm256_setzero_ps();
    __m256 vrt8 = _mm256_setzero_ps();
    __m256 vrt9 = _mm256_setzero_ps();
    __m256 vrt10 = _mm256_setzero_ps();
    __m256 vrt11 = _mm256_setzero_ps();
    __m256 vrt12 = _mm256_setzero_ps();
    __m256 vrt13 = _mm256_setzero_ps();
    __m256 vrt14 = _mm256_setzero_ps();
    __m256 vrt15 = _mm256_setzero_ps();
    const float* p; // Pointer for src data.
    size_t i;
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = src;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt = _mm256_add_ps(vrt, _mm256_load_ps(p));    // Add. vrt += *((*__m256)(p)+k);
            vrt1 = _mm256_add_ps(vrt1, _mm256_load_ps(p + VectorWidth * 1));
            vrt2 = _mm256_add_ps(vrt2, _mm256_load_ps(p + VectorWidth * 2));
            vrt3 = _mm256_add_ps(vrt3, _mm256_load_ps(p + VectorWidth * 3));
            vrt4 = _mm256_add_ps(vrt4, _mm256_load_ps(p + VectorWidth * 4));
            vrt5 = _mm256_add_ps(vrt5, _mm256_load_ps(p + VectorWidth * 5));
            vrt6 = _mm256_add_ps(vrt6, _mm256_load_ps(p + VectorWidth * 6));
            vrt7 = _mm256_add_ps(vrt7, _mm256_load_ps(p + VectorWidth * 7));
            vrt8 = _mm256_add_ps(vrt8, _mm256_load_ps(p + VectorWidth * 8));
            vrt9 = _mm256_add_ps(vrt9, _mm256_load_ps(p + VectorWidth * 9));
            vrt10 = _mm256_add_ps(vrt10, _mm256_load_ps(p + VectorWidth * 10));
            vrt11 = _mm256_add_ps(vrt11, _mm256_load_ps(p + VectorWidth * 11));
            vrt12 = _mm256_add_ps(vrt12, _mm256_load_ps(p + VectorWidth * 12));
            vrt13 = _mm256_add_ps(vrt13, _mm256_load_ps(p + VectorWidth * 13));
            vrt14 = _mm256_add_ps(vrt14, _mm256_load_ps(p + VectorWidth * 14));
            vrt15 = _mm256_add_ps(vrt15, _mm256_load_ps(p + VectorWidth * 15));
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += p[i];
        }
    }
    // Reduce.
    vrt = _mm256_add_ps(_mm256_add_ps(_mm256_add_ps(_mm256_add_ps(vrt, vrt1), _mm256_add_ps(vrt2, vrt3))
        , _mm256_add_ps(_mm256_add_ps(vrt4, vrt5), _mm256_add_ps(vrt6, vrt7)))
        , _mm256_add_ps(_mm256_add_ps(_mm256_add_ps(vrt8, vrt9), _mm256_add_ps(vrt10, vrt11))
        , _mm256_add_ps(_mm256_add_ps(vrt12, vrt13), _mm256_add_ps(vrt14, vrt15))))
    ; // vrt = vrt + vrt1 + vrt2 + vrt3 + ... vrt15;
    p = (const float*)&vrt;
    for (i = 0; i < VectorWidth; ++i) {
        rt += p[i];
    }
    return rt;
}

// Sum - Vector AVX - Loop unrolling *16 - Array.
float SumVectorAvxU16A(const float* src, size_t count, int loops) {
    float rt = 0;    // Result.
    const int LoopUnrolling = 16;
    size_t VectorWidth = sizeof(__m256) / sizeof(float); // Block width.
    size_t nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    size_t i;
    __m256 vrt[LoopUnrolling]; // Vector result.
    for (i = 0; i < LoopUnrolling; ++i) {
        vrt[i] = _mm256_setzero_ps(); // [AVX] Set zero.
    }
    const float* p; // Pointer for src data.
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = src;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            vrt[0] = _mm256_add_ps(vrt[0], _mm256_load_ps(p));    // Add. vrt += *((*__m256)(p)+k);
            vrt[1] = _mm256_add_ps(vrt[1], _mm256_load_ps(p + VectorWidth * 1));
            vrt[2] = _mm256_add_ps(vrt[2], _mm256_load_ps(p + VectorWidth * 2));
            vrt[3] = _mm256_add_ps(vrt[3], _mm256_load_ps(p + VectorWidth * 3));
            vrt[4] = _mm256_add_ps(vrt[4], _mm256_load_ps(p + VectorWidth * 4));
            vrt[5] = _mm256_add_ps(vrt[5], _mm256_load_ps(p + VectorWidth * 5));
            vrt[6] = _mm256_add_ps(vrt[6], _mm256_load_ps(p + VectorWidth * 6));
            vrt[7] = _mm256_add_ps(vrt[7], _mm256_load_ps(p + VectorWidth * 7));
            vrt[8] = _mm256_add_ps(vrt[8], _mm256_load_ps(p + VectorWidth * 8));
            vrt[9] = _mm256_add_ps(vrt[9], _mm256_load_ps(p + VectorWidth * 9));
            vrt[10] = _mm256_add_ps(vrt[10], _mm256_load_ps(p + VectorWidth * 10));
            vrt[11] = _mm256_add_ps(vrt[11], _mm256_load_ps(p + VectorWidth * 11));
            vrt[12] = _mm256_add_ps(vrt[12], _mm256_load_ps(p + VectorWidth * 12));
            vrt[13] = _mm256_add_ps(vrt[13], _mm256_load_ps(p + VectorWidth * 13));
            vrt[14] = _mm256_add_ps(vrt[14], _mm256_load_ps(p + VectorWidth * 14));
            vrt[15] = _mm256_add_ps(vrt[15], _mm256_load_ps(p + VectorWidth * 15));
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += p[i];
        }
    }
    // Reduce.
    vrt[0] = _mm256_add_ps(_mm256_add_ps(_mm256_add_ps(_mm256_add_ps(vrt[0], vrt[1]), _mm256_add_ps(vrt[2], vrt[3]))
        , _mm256_add_ps(_mm256_add_ps(vrt[4], vrt[5]), _mm256_add_ps(vrt[6], vrt[7])))
        , _mm256_add_ps(_mm256_add_ps(_mm256_add_ps(vrt[8], vrt[9]), _mm256_add_ps(vrt[10], vrt[11]))
        , _mm256_add_ps(_mm256_add_ps(vrt[12], vrt[13]), _mm256_add_ps(vrt[14], vrt[15]))))
    ; // vrt = vrt + vrt1 + vrt2 + vrt3 + ... vrt15;
    p = (const float*)&vrt;
    for (i = 0; i < VectorWidth; ++i) {
        rt += p[i];
    }
    return rt;
}

// Sum - Vector AVX - Loop unrolling *X - Array.
float SumVectorAvxUX(const float* src, size_t count, int loops, const int LoopUnrolling) {
    float rt = 0;    // Result.
    size_t VectorWidth = sizeof(__m256) / sizeof(float); // Block width.
    size_t nBlockWidth = VectorWidth * LoopUnrolling; // Block width.
    size_t cntBlock = count / nBlockWidth; // Block count.
    size_t cntRem = count % nBlockWidth; // Remainder count.
    size_t i;
    __m256* vrt = new __m256[LoopUnrolling]; // Vector result.
    for (i = 0; i < LoopUnrolling; ++i) {
        vrt[i] = _mm256_setzero_ps(); // [AVX] Set zero.
    }
    const float* p; // Pointer for src data.
    // Body.
    for (int j = 0; j < loops; ++j) {
        p = src;
        // Block processs.
        for (i = 0; i < cntBlock; ++i) {
            for (int k = 0; k < LoopUnrolling; ++k) {
                vrt[k] = _mm256_add_ps(vrt[k], _mm256_load_ps(p + VectorWidth * k));    // Add. vrt += *((*__m256)(p)+k);
            }
            p += nBlockWidth;
        }
        // Remainder processs.
        for (i = 0; i < cntRem; ++i) {
            rt += p[i];
        }
    }
    // Reduce.
    for (i = 1; i < LoopUnrolling; ++i) {
        vrt[0] = _mm256_add_ps(vrt[0], vrt[i]); // vrt[0] += vrt[i]
    } // vrt = vrt + vrt1 + vrt2 + vrt3 + ... vrt15;
    p = (const float*)&vrt[0];
    for (i = 0; i < VectorWidth; ++i) {
        rt += p[i];
    }
    delete[] vrt;
    return rt;
}

// Do Benchmark.
void Benchmark() {
    const size_t alignment = 256 / 8; // sizeof(__m256) / sizeof(BYTE);
    // init.
    clock_t tickBegin, msUsed;
    double mFlops; // MFLOPS/s .
    double scale;
    float rt;
    const int count = 1024 * 4;
    const int loops = 1000 * 1000;
    //const int loops = 1;
    const double countMFlops = count * (double)loops / (1000.0 * 1000);
    float* src = https://www.cnblogs.com/zyl910/archive/2022/11/16/(float*)_aligned_malloc(sizeof(float)*count, alignment); // new float[count];
    if (NULL == src) {
        printf("Memory alloc fail!");
        return;
    }
    for (int i = 0; i < count; ++i) {
        src[i] = (float)i;
    }
    printf("Benchmark: \tcount=%d, loops=%d, countMFlops=%f\n", count, loops, countMFlops);
    // SumBase.
    tickBegin = clock();
    rt = SumBase(src, count, loops);
    msUsed = clock() - tickBegin;
    mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
    printf("SumBase:\t%g\t# msUsed=%d, MFLOPS/s=%f\n", rt, (int)msUsed, mFlops);
    double mFlopsBase = mFlops;
    // SumBaseU4.
    tickBegin = clock();
    rt = SumBaseU4(src, count, loops);
    msUsed = clock() - tickBegin;
    mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
    scale = mFlops / mFlopsBase;
    printf("SumBaseU4:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", rt, (int)msUsed, mFlops, scale);
    // SumVectorAvx.
    __try {
        tickBegin = clock();
        rt = SumVectorAvx(src, count, loops);
        msUsed = clock() - tickBegin;
        mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
        scale = mFlops / mFlopsBase;
        printf("SumVectorAvx:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", rt, (int)msUsed, mFlops, scale);
        // SumVectorAvxU4.
        tickBegin = clock();
        rt = SumVectorAvxU4(src, count, loops);
        msUsed = clock() - tickBegin;
        mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
        scale = mFlops / mFlopsBase;
        printf("SumVectorAvxU4:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", rt, (int)msUsed, mFlops, scale);
        // SumVectorAvxU16.
        tickBegin = clock();
        rt = SumVectorAvxU16(src, count, loops);
        msUsed = clock() - tickBegin;
        mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
        scale = mFlops / mFlopsBase;
        printf("SumVectorAvxU16:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", rt, (int)msUsed, mFlops, scale);
        // SumVectorAvxU16A.
        tickBegin = clock();
        rt = SumVectorAvxU16A(src, count, loops);
        msUsed = clock() - tickBegin;
        mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
        scale = mFlops / mFlopsBase;
        printf("SumVectorAvxU16A:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", rt, (int)msUsed, mFlops, scale);
        // SumVectorAvxUX.
        int LoopUnrollingArray[] = { 4, 8, 16 };
        for (int i = 0; i < sizeof(LoopUnrollingArray) / sizeof(LoopUnrollingArray[0]); ++i) {
            int loopUnrolling = LoopUnrollingArray[i];
            tickBegin = clock();
            rt = SumVectorAvxUX(src, count, loops, loopUnrolling);
            msUsed = clock() - tickBegin;
            mFlops = countMFlops * CLOCKS_PER_SEC / msUsed;
            scale = mFlops / mFlopsBase;
            printf("SumVectorAvxUX[%d]:\t%g\t# msUsed=%d, MFLOPS/s=%f, scale=%f\n", loopUnrolling, rt, (int)msUsed, mFlops, scale);
        }
    }
    __except (EXCEPTION_EXECUTE_HANDLER) {
        printf("Run SumVectorAvx fail!");
    }
    // done.
    _aligned_free(src);
}

int main() {
    printf("BenchmarkVectorCpp\n");
    printf("\n");
    printf("Pointer size:\t%d\n", (int)(sizeof(void*)));
#ifdef _DEBUG
    printf("IsRelease:\tFalse\n");
#else
    printf("IsRelease:\tTrue\n");
#endif // _DEBUG
#ifdef _MSC_VER
    printf("_MSC_VER:\t%d\n", _MSC_VER);
#endif // _MSC_VER
#ifdef __AVX__
    printf("__AVX__:\t%d\n", __AVX__);
#endif // __AVX__
    printf("\n");
    // Benchmark.
    Benchmark();
}

3.2 測驗結果
測驗結果匯總如下：
BenchmarkVectorCpp

Pointer size:   8
IsRelease:      True
_MSC_VER:       1916
__AVX__:        1

Benchmark:      count=4096, loops=1000000, countMFlops=4096.000000
SumBase:        6.87195e+10     # msUsed=4938, MFLOPS/s=829.485622
SumBaseU4:      2.74878e+11     # msUsed=1229, MFLOPS/s=3332.790887, scale=4.017901
SumVectorAvx:   5.49756e+11     # msUsed=616, MFLOPS/s=6649.350649, scale=8.016234
SumVectorAvxU4: 2.19902e+12     # msUsed=247, MFLOPS/s=16582.995951, scale=19.991903
SumVectorAvxU16:        8.3862e+12      # msUsed=89, MFLOPS/s=46022.471910, scale=55.483146
SumVectorAvxU16A:       8.3862e+12      # msUsed=89, MFLOPS/s=46022.471910, scale=55.483146
SumVectorAvxUX[4]:      2.19902e+12     # msUsed=465, MFLOPS/s=8808.602151, scale=10.619355
SumVectorAvxUX[8]:      4.39805e+12     # msUsed=336, MFLOPS/s=12190.476190, scale=14.696429
SumVectorAvxUX[16]:     8.3862e+12      # msUsed=323, MFLOPS/s=12681.114551, scale=15.287926

先前做未回圈展開時，C# 與 Visual C++ 程式的性能差距不大，而現在使用回圈展開后，發現差距拉大了——

SumBaseU4：C++版的MFLOPS/s為 3332.790887，C#版的MFLOPS/s為 2184.5333333333333，3332.790887/2184.5333333333333=1.5256305940246582264042754215188，即大約是 1.5256 倍，
SumVectorAvxU4：C++版的MFLOPS/s為 16582.995951，C#版的MFLOPS/s為 12487.80487804878，16582.995951/12487.80487804878=1.3279352226386719268724696343231，即大約是 1.3279 倍，
SumVectorAvxU16：C++版的MFLOPS/s為 46022.471910，C#版的MFLOPS/s為 32768，46022.471910/32768=1.40449438201904296875，即大約是 1.4045 倍，

而且還發現——

SumVectorAvxU16A 與 SumVectorAvxU16 的性能差不多，表示C++編譯器能很好地優化陣列訪問，能達到區域變數同級別的速度，故C++中可以放心使用陣列來存盤回圈展開的臨時變數，
SumVectorAvxUX 的 C++ 版性能比C#好一些，而且臨時陣列長度大于8時，也能帶來一定的性能提升，只可惜還是存在性價比不高的問題，還是經典的回圈展開更好用，

四、小結
C#中使用回圈展開的心得總結——

使用回圈展開能提高性能，但由于編碼比較麻煩，且會增加代碼維護的成本，故應作為最后手段，應該先嘗試其他優化手段，
使用回圈展開后，指標版代碼比陣列版、Span版要高一些，可能是因為指標更貼近底層硬體、更易于編譯器優化，故推薦優先使用指標，
在C#中做回圈展開，一般展開4次就行，展開16次只有少量提升，性價比不高，
回圈展開會引起臨時變數倍增，應該堅持逐個定義區域變數，若用陣列會造成性能下降，包括用指標操作堆疊分配陣列也會下降，

對于回圈展開這樣的運算量重的情況，Visual C++編譯優化的更好，但性能大多只有1倍多，總體差距不大，

C++還有個優點，是可以用陣列來存盤回圈展開的臨時變數，且性能不會下降，
原始碼地址——

https://github.com/zyl910/BenchmarkVector/tree/main/BenchmarkVector
參考文獻

zyl910《C# 使用SIMD向量型別加速浮點陣列求和運算（1）：使用Vector4、Vector<T>》. https://www.cnblogs.com/zyl910/p/dotnet_simd_BenchmarkVector1.html
zyl910《C# 使用SIMD向量型別加速浮點陣列求和運算（2）：C#通過Intrinsic直接使用AVX指令集操作 Vector256<T>，及C++程式對比》. https://www.cnblogs.com/zyl910/p/dotnet_simd_BenchmarkVector2.html
有色金屬?《C 語言性能優化：回圈展開》. https://zhuanlan.zhihu.com/p/359032198




    作者：zyl910    出處：http://www.cnblogs.com/zyl910/    著作權宣告：自由轉載-非商用-非衍生-保持署名 | Creative Commons BY-NC-ND 3.0.





        
      轉載請註明出處，本文鏈接：https://www.uj5u.com/net/535110.html
      標籤：.NET技术 
      上一篇：aspnetcore兩種上傳圖片(檔案)的方式
下一篇：WPF-CommandBinding到F#庫中定義的命令不起作用

C# 使用SIMD向量型別加速浮點陣列求和運算（3）：回圈展開

一、背景

1.1 回圈展開簡介

1.2 測驗準備

二、在C#中使用

2.1 對基礎演算法做回圈展開

2.1.1 測驗結果:

2.2 對 Vector4 版演算法做回圈展開

2.2.1 測驗結果:

2.3 對 Vector<T> 版演算法做回圈展開

2.3.1 測驗結果:

2.4 對 Avx版演算法做回圈展開

2.4.1 測驗結果:

2.5 對 Avx版演算法做回圈展開16次

2.5.1 測驗結果:

2.6 嘗試用陣列來存盤回圈展開的臨時變數

2.6.1 測驗結果:

2.7 嘗試用堆疊陣列來減少相關性

2.7.1 測驗結果:

2.8 測驗結果匯總

三、在C++中使用

3.1 修改代碼

3.2 測驗結果

四、小結

參考文獻

2.3 對 `Vector<T>` 版演算法做回圈展開