erf(x) and math.h

问题:

According to this site the error function erf(x) comes from math.h. But actually looking in math.h, it isn't there, and gcc cannot compile the following test program while g++ can:

#include <math.h>
#include <stdio.h>

int main(int argc, char* argv[]) {
  double x;
  double erfX;
  x = 1.0;
  erfX = erf(x);

  printf("erf(%f) = %f", x, erfX);
}

$ gcc mathHTest.c
/tmp/ccWfNox5.o: In function `main':
mathHTest.c:(.text+0x28): undefined reference to `erf'
collect2: ld returned 1 exit status
$ g++ mathHTest.c

What does g++ pull in that gcc doesn't? Looking in /usr/include, the only place I could find erf(x) was in tgmath.h, which I don't include. So g++ must be grabbing different headers than gcc, but which ones?

EDIT: I wasn't linking in libm with gcc, hence the link error. However, I still don't understand why erf() is not in math.h. Where is it coming from?

回答1:

'erf' is actually declared in bits/mathcalls.h, which is #included by math.h. The actual declaration is heavily obscured by macro magic to make it do the right thing for both C and C++

回答2:

I had a similar problem and needed to find the exact definition of erf so let me expand on this. As said by Chris Dodd the function is declared in bits/mathcalls.h which is included by maths.h.

bits/mathcalls.h:

...
#if defined __USE_MISC || defined __USE_XOPEN || defined __USE_ISOC99
__BEGIN_NAMESPACE_C99
/* Error and gamma functions.  */
__MATHCALL (erf,, (_Mdouble_));
__MATHCALL (erfc,, (_Mdouble_));
__MATHCALL (lgamma,, (_Mdouble_));
__END_NAMESPACE_C99
#endif
...

Macro magic expands __MATHCALL (erf,, (_Mdouble_)); to

extern double erf (double) throw (); extern double __erf (double) throw ();

The actual code is in libm.a or libm.so (gcc -lm):

$ nm /usr/lib/libm.a
...
s_erf.o:
00000400 T __erf
00000000 T __erfc
         U __ieee754_exp
00000400 W erf
00000000 W erfc
...

The source can be obtained from the gnu libc webpage. For a rough idea on the actual implementation here a few lines of the source:

sysdeps/ieee754/dbl-64/s_erf.c:

/* double erf(double x)
 * double erfc(double x)
 *                           x
 *                    2      |\
 *     erf(x)  =  ---------  | exp(-t*t)dt
 *                 sqrt(pi) \|
 *                           0
 *
 *     erfc(x) =  1-erf(x)
 *  Note that
 *              erf(-x) = -erf(x)
 *              erfc(-x) = 2 - erfc(x)
 *
 * Method:
 *      1. For |x| in [0, 0.84375]
 *          erf(x)  = x + x*R(x^2)
 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
 *         where R = P/Q where P is an odd poly of degree 8 and
 *         Q is an odd poly of degree 10.
 *                                               -57.90
 *                      | R - (erf(x)-x)/x | <= 2
 *
 *
 *         Remark. The formula is derived by noting
 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
 *         and that
 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
 *         is close to one. The interval is chosen because the fix
 *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
 *         near 0.6174), and by some experiment, 0.84375 is chosen to
 *         guarantee the error is less than one ulp for erf.
 *
 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and     
 ...

回答3:

You need to link the math library (libm) too:

$ gcc mathHTest.c -lm

All of the normal math library functions are actually there, and not in the standard C library (libc).

According to my tests, g++ does include libm automatically, but gcc doesn't.

回答4:

I had the same problem using gcc from cygwin on a x86 processor. The "-lm" library include parameter (after the file list!) worked perfectly.