Fortran/gfortran的幂运算达到高精度 [英] Exponentiation in Fortran/gfortran to high precision

问题描述

gfortran如何处理整数与实数的幂运算?我一直以为是相同的，但请考虑示例:

How does gfortran handle exponentiation with a integer vs a real? I always assumed it was the same, but consider the example:

program main

implicit none

integer, parameter :: dp = selected_real_kind(33,4931)

real(kind=dp) :: x = 82.4754500815524510_dp

print *, x
print *, x**4
print *, x**4.0_dp

end program main

使用gfortran进行编译会得到

Compiling with gfortran gives

82.4754500815524510000000000000000003      
46269923.0191143410452125643548442147      
46269923.0191143410452125643548442211 

现在显然这些数字几乎是一致的-但是如果gfortran以相同的方式处理整数和实数以求幂，我希望它们是相同的.有什么作用?

Now clearly these numbers almost agree - but if gfortran handles integers and reals for exponentiation in the same way I would expect them to be identical. What gives?

推荐答案

略微扩展程序将显示正在发生的事情:

Expanding your program slightly shows what is going on:

ijb@ianbushdesktop ~/work/stack $ cat exp.f90
program main

implicit none

integer, parameter :: dp = selected_real_kind(33,4931)

real(kind=dp) :: x = 82.4754500815524510_dp

print *, x
print *, x**4
print *,(x*x)*(x*x)
print *,Nearest((x*x)*(x*x),+1.0_dp)
print *, x**4.0_dp

end program main

编译和运行可以得到:

ijb@ianbushdesktop ~/work/stack $ gfortran --version
GNU Fortran (GCC) 7.4.0
Copyright (C) 2017 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

ijb@ianbushdesktop ~/work/stack $ gfortran exp.f90 
ijb@ianbushdesktop ~/work/stack $ ./a.out
   82.4754500815524510000000000000000003      
   46269923.0191143410452125643548442147      
   46269923.0191143410452125643548442147      
   46269923.0191143410452125643548442211      
   46269923.0191143410452125643548442211      
ijb@ianbushdesktop ~/work/stack $ 

因此

1. 看起来编译器很聪明，可以算出可以通过乘法来完成整数幂运算，这比一般的幂运算要快得多

1. It looks like the compiler is clever enough to work out that integer powers can be done by multiplies, which is much quicker than a general exponentiation function

使用通用幂函数与乘法答案仅相差1位.鉴于我们不能说本身更准确，因此我们必须接受两者都一样准确

Using the general exponentiation function is only 1 bit different from the multiplication answer. Given we can't say per se which is more accurate we must accepts both as equally accurate

因此，总而言之，编译器会在可能的情况下使用简单的乘法运算，而不是使用完整的幂运算程序，但是即使必须使用更昂贵的路由，编译器也会给出相同的答案，以供仔细考虑相同"一词的含义.

Thus in conclusion the compiler uses simple multiplies where possible instead of a full blown exponentiation routine, but even if it has to use the more expensive route it gives the same answer, for carefully considered meaning of the word "same"

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