CPU时间为四倍与双精度

Question

我正在进行一些长期模拟,其中我试图在ODE系统的解决方案中实现尽可能高的准确性.我试图找出四倍(128位)精度计算需要多少时间与双倍(64位)精度.我google了一下,看到了几个意见:有人说它需要4倍,其他60-70倍......所以我决定自己动手,我写了一个简单的Fortran基准程序:

program QUAD_TEST

implicit none

integer,parameter  ::  dp = selected_int_kind(15)
integer,parameter  ::  qp = selected_int_kind(33)

integer   ::  cstart_dp,cend_dp,cstart_qp,cend_qp,crate
real      ::  time_dp,time_qp
real(dp)  ::  sum_dp,sqrt_dp,pi_dp,mone_dp,zero_dp
real(qp)  ::  sum_qp,sqrt_qp,pi_qp,mone_qp,zero_qp
integer   ::  i

! ==============================================================================

! == TEST 1. ELEMENTARY OPERATIONS ==
sum_dp = 1._dp
sum_qp = 1._qp
call SYSTEM_CLOCK(count_rate=crate)

write(*,*) 'Testing elementary operations...'

call SYSTEM_CLOCK(count=cstart_dp)
do i=1,50000000
  sum_dp = sum_dp - 1._dp
  sum_dp = sum_dp + 1._dp
  sum_dp = sum_dp*2._dp
  sum_dp = sum_dp/2._dp
end do
call SYSTEM_CLOCK(count=cend_dp)
time_dp = real(cend_dp - cstart_dp)/real(crate)
write(*,*) 'DP sum: ',sum_dp
write(*,*) 'DP time: ',time_dp,' seconds'

call SYSTEM_CLOCK(count=cstart_qp)
do i=1,50000000
  sum_qp = sum_qp - 1._qp
  sum_qp = sum_qp + 1._qp
  sum_qp = sum_qp*2._qp
  sum_qp = sum_qp/2._qp
end do
call SYSTEM_CLOCK(count=cend_qp)
time_qp = real(cend_qp - cstart_qp)/real(crate)
write(*,*) 'QP sum: ',sum_qp
write(*,*) 'QP time: ',time_qp,' seconds'
write(*,*)
write(*,*) 'DP is ',time_qp/time_dp,' times faster.'
write(*,*)

! == TEST 2. SQUARE ROOT ==
sqrt_dp = 2._dp
sqrt_qp = 2._qp

write(*,*) 'Testing square root ...'

call SYSTEM_CLOCK(count=cstart_dp)
do i = 1,10000000
   sqrt_dp = sqrt(sqrt_dp)
   sqrt_dp = 2._dp
end do
call SYSTEM_CLOCK(count=cend_dp)
time_dp = real(cend_dp - cstart_dp)/real(crate)
write(*,*) 'DP sqrt: ',sqrt_dp
write(*,*) 'DP time: ',time_dp,' seconds'

call SYSTEM_CLOCK(count=cstart_qp)
do i = 1,10000000
   sqrt_qp = sqrt(sqrt_qp)
   sqrt_qp = 2._qp
end do
call SYSTEM_CLOCK(count=cend_qp)
time_qp = real(cend_qp - cstart_qp)/real(crate)
write(*,*) 'QP sqrt: ',sqrt_qp
write(*,*) 'QP time: ',time_qp,' seconds'
write(*,*)
write(*,*) 'DP is ',time_qp/time_dp,' times faster.'
write(*,*)

! == TEST 3. TRIGONOMETRIC FUNCTIONS ==
pi_dp = acos(-1._dp); mone_dp = 1._dp; zero_dp = 0._dp
pi_qp = acos(-1._qp); mone_qp = 1._qp; zero_qp = 0._qp

write(*,*) 'Testing trigonometric functions ...'

call SYSTEM_CLOCK(count=cstart_dp)
do i = 1,10000000
    mone_dp = cos(pi_dp)
    zero_dp = sin(pi_dp)
end do
call SYSTEM_CLOCK(count=cend_dp)
time_dp = real(cend_dp - cstart_dp)/real(crate)
write(*,*) 'DP cos: ',mone_dp
write(*,*) 'DP sin: ',zero_dp
write(*,*) 'DP time: ',time_dp,' seconds'

call SYSTEM_CLOCK(count=cstart_qp)
do i = 1,10000000
    mone_qp = cos(pi_qp)
    zero_qp = sin(pi_qp)
end do
call SYSTEM_CLOCK(count=cend_qp)
time_qp = real(cend_qp - cstart_qp)/real(crate)
write(*,*) 'QP cos: ',mone_qp
write(*,*) 'QP sin: ',zero_qp
write(*,*) 'QP time: ',time_qp,' seconds'
write(*,*)
write(*,*) 'DP is ',time_qp/time_dp,' times faster.'
write(*,*)

end program QUAD_TEST

编译后的典型运行结果gfortran 4.8.4,没有任何优化标志:

 Testing elementary operations...
 DP sum:    1.0000000000000000     
 DP time:   0.572000027      seconds
 QP sum:    1.00000000000000000000000000000000000      
 QP time:    4.32299995      seconds

 DP is    7.55769205      times faster.

 Testing square root ...
 DP sqrt:    2.0000000000000000     
 DP time:    5.20000011E-02  seconds
 QP sqrt:    2.00000000000000000000000000000000000      
 QP time:    2.60700011      seconds

 DP is    50.1346169      times faster.

 Testing trigonometric functions ...
 DP cos:   -1.0000000000000000     
 DP sin:    1.2246467991473532E-016
 DP time:    2.79600000      seconds
 QP cos:   -1.00000000000000000000000000000000000      
 QP sin:    8.67181013012378102479704402604335225E-0035
 QP time:    5.90199995      seconds

 DP is    2.11087275      times faster.

必须要在这里发生一些事情.我的猜测是通过优化算法sqrt计算gfortran出来的,这可能还没有用于四重精度计算.这可能不是这样的sin和cos,偏偏基本操作7.6倍,四精度慢而对于三角函数的事情只有2倍慢下来？如果用于三角函数的算法对于四次和双精度都是相同的,我预计它们的CPU时间也会增加七倍.

与64位相比,使用128位精度时科学计算的平均减速是多少？

我在Intel i7-4771 @ 3.50GHz上运行它.

Answer 1

更多的是评论而不是答案,但......

当前的CPU为双精度浮点运算提供了大量的硬件加速.有些甚至提供扩展精度的设施.除此之外,您仅限于软件实现(正如您所注意到的那样)相当慢.

然而,在一般情况下几乎无法预测这种减速的确切因素.这取决于您的CPU(例如内置的加速度)以及软件堆栈.对于双精度,您通常使用不同的数学库而不是四倍精度,并且这些算法可能使用不同的算法进行基本操作.

对于使用相同算法的给定硬件上的特定操作/算法,您可能可以派生出一个数字,但这肯定不会普遍存在.