Heaviside 函数的优化实现

Question

我想（超级）优化Heaviside函数的实现。

我正在研究一种数值算法（在Fortran中），其中速度特别重要。这多次使用 Heaviside 函数，目前由 signum 内函数实现，如下所示：

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

我主要对 x 是英特尔处理器上的双精度实数的情况感兴趣。

是否有可能开发更有效的Heaviside函数实现？也许使用汇编语言、超级优化代码或调用现有的外部库？

Answer 1

你打算heaviside = 0.5*(sign(1,x)+1)吗？无论如何，使用 gcc 4.8.1 fortran 进行测试表明，高性能 Mark 的想法应该是有益的。这里有3种可能性：

重赛德1 - 原装heaviside2 - 高性能马克的想法Heaviside3 - 另一种变体

  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 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

当使用 gcc 编译时，heaviside1 会生成一个可能会减慢执行速度的乘法。Heaviside2消除了乘法。Heaviside3 的指令数与 Heaviside2 相同，但使用的内存访问次数少 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，在寄存器中传递参数，并使用所需的常量预加载其他寄存器。确切的结果取决于所使用的编译器和调用代码。内联代码的估计值：

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

看起来该函数可以通过两个编译器生成的指令来处理：vcmplesd+vandpd。如果参数为负数，则第一条指令创建所有零的掩码，否则创建所有 1 的掩码。第二条指令将掩码应用于寄存器常量值 1，以产生 0 或 1 的结果值。

虽然我没有对这些函数进行基准测试，但看起来 heaviside 函数应该不会花费太多执行时间。

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

文件函数。

//----------------------------------------------------------------------------
.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
//-----------------------------------------------------------------------------

文件 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 nsn", 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): %lfn", x, result);
        total += result;
        }
    elapsed = queryPerformanceCounter () - startCount;
    ns = (double) elapsed / queryPerformanceFrequency () * 1000000000 / loops;
    printf ("heaviside_a1: %7.2f nsn", 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): %lfn", x, result);
        total += result;
        }
    elapsed = queryPerformanceCounter () - startCount;
    ns = (double) elapsed / queryPerformanceFrequency () * 1000000000 / loops;
    printf ("heaviside_a2: %7.2f nsn", 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 datan");
    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 在 4.0 GHz 时的程序输出：

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

这些计时结果包括执行伪随机参数生成和结果累加代码，以防止优化程序消除否则未使用的heaviside_c1局部函数。

heaviside_c1来自最初的 fortran 建议，移植到 C。heaviside_a1是一种汇编语言实现。heaviside_a2是对汇编语言版本的修改，它使用调用方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任何优势。

汇编语言函数假定 xmm0 返回结果，xmm1 和 xmm2 可用作暂存寄存器。这对 Windows 使用的 x64 调用约定有效。对于其他调用约定，应确认此假设。

为了避免内存访问，汇编语言版本期望参数按寄存器 （XMM0） 中的值传递。由于这不是 fortran 默认值，因此需要特殊声明。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