我遇到了类似的问题,需要找到确切的定义,erf
所以让我对此进行扩展。正如 Chris Dodd 所说,该函数声明在bits/mathcalls.h
其中包含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
...
宏魔法扩展__MATHCALL (erf,, (_Mdouble_));
为
extern double erf (double) throw (); extern double __erf (double) throw ();
实际代码在libm.a
或libm.so
(gcc -lm
)中:
$ nm /usr/lib/libm.a
...
s_erf.o:
00000400 T __erf
00000000 T __erfc
U __ieee754_exp
00000400 W erf
00000000 W erfc
...
源码可以从gnu libc网页获取。对于实际实现的粗略想法,这里有几行源代码:
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
...