Question

我想（超级）优化Heaviside函数的实现。

我正在研究一种速度特别重要的数值算法（在Fortran中）。它多次使用Heaviside函数，目前由signum内部函数实现如下：

heaviside = 0.5*sign(1,x)+1

我主要感兴趣的是x是intel处理器上的双精度实数。

是否有可能开发更高效的Heaviside功能实现？也许使用汇编语言，超优化代码或调用现有的外部库？

Answer 1

您打算heaviside = 0.5*(sign(1,x)+1)吗？在任何情况下使用gcc 4.8.1 fortran进行测试都表明高性能Mark的想法应该是有益的。这有三种可能性：

heavyiside1 - 原创 heavyiside2 - 高性能马克的想法 heavyiside3 - 另一种变化

  function heaviside1 (x)
  double precision heaviside1, x
  heaviside1 = 0.5 * (sign(1d0,x) + 1)
  end

  function heaviside2 (x)
  double precision heaviside2, x
  heaviside2 = sign(0.5d0,x) + 0.5
  end

  function heaviside3 (x)
  double precision heaviside3, x
  heaviside3 = 0
  if (x .ge. 0) heaviside3 = 1
  end

  program demo
  double precision heaviside1, heaviside2, heaviside3, x, a, b, c

  do
     x = 0.5 - RAND(0)
     a = heaviside1(x)
     b = heaviside2(x)
     c = heaviside3(x)
     print *, "x=", x, "heaviside(x)=", a, b, c
  enddo
  end

编译时，gcc会生成以下3个独立函数：

<heaviside1_>:
  vmovsd    xmm0,QWORD PTR [rcx]
  vandpd    xmm0,xmm0,XMMWORD PTR [rip+0x2d824]
  vorpd     xmm0,xmm0,XMMWORD PTR [rip+0x2d80c]
  vaddsd    xmm0,xmm0,QWORD PTR [rip+0x2d7f4]
  vmulsd    xmm0,xmm0,QWORD PTR [rip+0x2d81c]
  ret    

<heaviside2_>:
  vmovsd    xmm0,QWORD PTR [rcx]
  vandpd    xmm0,xmm0,XMMWORD PTR [rip+0x2d844]
  vorpd     xmm0,xmm0,XMMWORD PTR [rip+0x2d85c]
  vaddsd    xmm0,xmm0,QWORD PTR [rip+0x2d844]
  ret    

<heaviside3_>:
  vxorpd    xmm0,xmm0,xmm0
  vmovsd    xmm1,QWORD PTR [rip+0x2d844]
  vcmplesd  xmm0,xmm0,QWORD PTR [rcx]
  vandpd    xmm0,xmm1,xmm0
  ret

使用gcc编译时，heaviside1会生成一个可能会降低执行速度的乘法。 heavyiside2消除了繁殖。 heaviside3与heaviside2具有相同数量的指令，但使用的内存访问次数减少了2次。

对于独立功能：

             instruction   memory reference
             count         count
heaviside1   6             5
heaviside2   5             4
heaviside3   5             2

这些函数的内联代码避免了对返回指令的需要，理想情况下传递寄存器中的参数并使用所需的常量预加载其他寄存器。确切的结果取决于使用的编译器和调用代码。对内联代码的估计：

             instruction   memory reference
             count         count
heaviside1   4             0
heaviside2   3             0
heaviside3   2             0

看起来这个函数可以通过两个编译器生成的指令来处理：vcmplesd + vandpd。如果参数为负，则第一条指令创建全零的掩码，否则创建所有零的掩码。第二条指令将掩码应用于寄存器常量值1，以便产生零或一的结果值。

虽然我没有对这些函数进行基准测试，但看起来重量函数不应该花费太多的执行时间。

--- 09/23/2013：添加x86_64汇编语言版本和C语言基准---

file functions.s

//----------------------------------------------------------------------------
.intel_syntax noprefix
.text

//-----------------------------------------------------------------------------
// this heaviside function generates its own register constants
// double  heaviside_a1 (double arg);
.globl heaviside_a1

heaviside_a1:
   mov     rax,0x3ff0000000000000
   xorpd   xmm1,xmm1                # xmm1: constant 0.0
   cmplesd xmm1,xmm0                # xmm1: mask (all Fs or all 0s)
   movq    xmm0,rax                 # xmm0: constant 1.0
   andpd   xmm0,xmm1
   retq

//-----------------------------------------------------------------------------
// this heaviside function uses register constants passed from caller
// double  heaviside_a2 (double arg, double const0, double const1);
.globl heaviside_a2

heaviside_a2:
   cmplesd xmm1,xmm0                # xmm1: mask (all Fs or all 0s)
   movsd   xmm0,xmm2                # xmm0: constant 1.0
   andpd   xmm0,xmm1
   retq

//-----------------------------------------------------------------------------

file ctest.c

#define __USE_MINGW_ANSI_STDIO 1
#include <windows.h>
#include <stdio.h>
#include <stdint.h>

// functions.s
double heaviside_a1 (double x);
double heaviside_a2 (double arg, double const0, double const1);

//-----------------------------------------------------------------------------

static double heaviside_c1 (double x)
    {
    double result = 0;
    if (x >= 0) result = 1;
    return result;
    }

//-----------------------------------------------------------------------------
//
//  queryPerformanceCounter - similar to QueryPerformanceCounter, but returns
//                            count directly.

uint64_t queryPerformanceCounter (void)
    {
    LARGE_INTEGER int64;
    QueryPerformanceCounter (&int64);
    return int64.QuadPart;
    }

//-----------------------------------------------------------------------------
//
// queryPerformanceFrequency - same as QueryPerformanceFrequency, but returns  count direcly.

uint64_t queryPerformanceFrequency (void)
    {
    LARGE_INTEGER int64;

    QueryPerformanceFrequency (&int64);
    return int64.QuadPart;
    }

//----------------------------------------------------------------------------
//
// lfsr64gpr - left shift galois type lfsr for 64-bit data, general purpose register implementation
//
static uint64_t lfsr64gpr (uint64_t data, uint64_t mask)
   {
   uint64_t carryOut = data >> 63;
   uint64_t maskOrZ = -carryOut; 
   return (data << 1) ^ (maskOrZ & mask);
   }

//---------------------------------------------------------------------------

int runtests (uint64_t pattern, uint64_t mask)
    {
    uint64_t startCount, elapsed, index, loops = 800000000;
    double ns;
    double total = 0;

    startCount = queryPerformanceCounter ();
    for (index = 0; index < loops; index++)
        {
        double x, result;
        pattern = lfsr64gpr (pattern, mask);
        x = (double) (int64_t) pattern;
        result = heaviside_c1 (x);
        total += result;
        }
    elapsed = queryPerformanceCounter () - startCount;
    ns = (double) elapsed / queryPerformanceFrequency () * 1000000000 / loops;
    printf ("heaviside_c1: %7.2f ns\n", ns);

    startCount = queryPerformanceCounter ();
    for (index = 0; index < loops; index++)
        {
        double x, result;
        pattern = lfsr64gpr (pattern, mask);
        x = (double) (int64_t) pattern;
        result = heaviside_a1 (x);
        //printf ("heaviside_a1 (%lf): %lf\n", x, result);
        total += result;
        }
    elapsed = queryPerformanceCounter () - startCount;
    ns = (double) elapsed / queryPerformanceFrequency () * 1000000000 / loops;
    printf ("heaviside_a1: %7.2f ns\n", ns);

    startCount = queryPerformanceCounter ();
    for (index = 0; index < loops; index++)
        {
        double x, result;
        const double const0 = 0.0;
        const double const1 = 1.0;
        pattern = lfsr64gpr (pattern, mask);
        x = (double) (int64_t) pattern;
        result = heaviside_a2 (x, const0, const1);
        //printf ("heaviside_a2 (%lf): %lf\n", x, result);
        total += result;
        }
    elapsed = queryPerformanceCounter () - startCount;
    ns = (double) elapsed / queryPerformanceFrequency () * 1000000000 / loops;
    printf ("heaviside_a2: %7.2f ns\n", ns);
    return total;
    }

//---------------------------------------------------------------------------

int main (void)
    {
    uint64_t mask;

    mask = 0xBEFFFFFFFFFFFFFF;

    // raise our priority to increase measurement accuracy
    SetPriorityClass (GetCurrentProcess (), REALTIME_PRIORITY_CLASS);

    printf ("using pseudo-random data\n");
    runtests (1, mask);
    return 0;
    }

//---------------------------------------------------------------------------

mingw64 build command: gcc -Wall -Wextra -O3 -octest.exe ctest.c functions.s

英特尔酷睿i7-2600K的4.0 GHz程序输出：

using pseudo-random data
heaviside_c1:    2.24 ns
heaviside_a1:    2.00 ns
heaviside_a2:    2.00 ns

这些时序结果包括执行伪随机参数生成和结果总计代码，以防止优化器消除未使用的heaviside_c1本地函数。

heaviside_c1来自原始的fortran建议，移植到C. heaviside_a1是汇编语言实现。 heaviside_a2是汇编语言版本的修改，它使用调用者传递的寄存器常量来避免生成它们的开销。对于我的处理器，基准测试显示传递常量没有优势。

汇编语言函数假设xmm0返回结果，xmm1和xmm2可用作临时寄存器。这适用于Windows使用的x64调用约定。其他调用约定应该确认这个假设。

为了避免内存访问，汇编语言版本要求参数通过寄存器（XMM0）中的值传递。因为这不是fortran默认值，所以需要特殊声明。这个似乎适用于gfortran 64位：

  interface
  real(c_double) function heaviside_a1(x)
  use iso_c_binding, only: c_double
  real(c_double), VALUE :: x
  end function heaviside_a1
  end interface

Heaviside函数的优化实现

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