c - 数学指数函数的Python源代码？

Question

当我在 Python 中进行数学计算时，我们使用的是哪个库。例如。

>>> 2**0.5
1.4142135623730951

如何找到使用过的源代码？这只是 math.pow() 函数吗？不幸的是，inspect.getsource(pow) 返回了一种错误。

在 Github 上搜索将其缩小到 13可能的文件。而且我不完全理解 cPython 是如何构建的。

/*[clinic input]
math.pow
    x: double
    y: double
    /
Return x**y (x to the power of y).
[clinic start generated code]*/

static PyObject *
math_pow_impl(PyObject *module, double x, double y)
/*[clinic end generated code: output=fff93e65abccd6b0 input=c26f1f6075088bfd]*/
{
    double r;
    int odd_y;

    /* deal directly with IEEE specials, to cope with problems on various
       platforms whose semantics don't exactly match C99 */
    r = 0.; /* silence compiler warning */
    if (!Py_IS_FINITE(x) || !Py_IS_FINITE(y)) {
        errno = 0;
        if (Py_IS_NAN(x))
            r = y == 0. ? 1. : x; /* NaN**0 = 1 */
        else if (Py_IS_NAN(y))
            r = x == 1. ? 1. : y; /* 1**NaN = 1 */
        else if (Py_IS_INFINITY(x)) {
            odd_y = Py_IS_FINITE(y) && fmod(fabs(y), 2.0) == 1.0;
            if (y > 0.)
                r = odd_y ? x : fabs(x);
            else if (y == 0.)
                r = 1.;
            else /* y < 0. */
                r = odd_y ? copysign(0., x) : 0.;
        }
        else if (Py_IS_INFINITY(y)) {
            if (fabs(x) == 1.0)
                r = 1.;
            else if (y > 0. && fabs(x) > 1.0)
                r = y;
            else if (y < 0. && fabs(x) < 1.0) {
                r = -y; /* result is +inf */
                if (x == 0.) /* 0**-inf: divide-by-zero */
                    errno = EDOM;
            }
            else
                r = 0.;
        }
    }
    else {
        /* let libm handle finite**finite */
        errno = 0;
        PyFPE_START_PROTECT("in math_pow", return 0);
        r = pow(x, y);
        PyFPE_END_PROTECT(r);
        /* a NaN result should arise only from (-ve)**(finite
           non-integer); in this case we want to raise ValueError. */
        if (!Py_IS_FINITE(r)) {
            if (Py_IS_NAN(r)) {
                errno = EDOM;
            }
            /*
               an infinite result here arises either from:
               (A) (+/-0.)**negative (-> divide-by-zero)
               (B) overflow of x**y with x and y finite
            */
            else if (Py_IS_INFINITY(r)) {
                if (x == 0.)
                    errno = EDOM;
                else
                    errno = ERANGE;
            }
        }
    }

    if (errno && is_error(r))
        return NULL;
    else
        return PyFloat_FromDouble(r);
}

这是我在 Python 2**0.5 中求 2 的平方根时使用的代码吗？

---

环顾四周，似乎 ** 与 pow() 相同，我们可以在源代码:

• search results for __pow__
• Difference between the built-in pow() and math.pow() for floats, in Python?
• numbers.py对于 Python 如何看待数字

共识似乎是 pow 来自 libm图书馆。可能喜欢this one , e_powf.c。还有e_pow.c

/* e_powf.c -- float version of e_pow.c.
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#include <math.h>
#include <math_private.h>

static const float huge = 1.0e+30, tiny = 1.0e-30;

static const float
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
zero    =  0.0,
one =  1.0,
two =  2.0,
two24   =  16777216.0,  /* 0x4b800000 */
    /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1  =  6.0000002384e-01, /* 0x3f19999a */
L2  =  4.2857143283e-01, /* 0x3edb6db7 */
L3  =  3.3333334327e-01, /* 0x3eaaaaab */
L4  =  2.7272811532e-01, /* 0x3e8ba305 */
L5  =  2.3066075146e-01, /* 0x3e6c3255 */
L6  =  2.0697501302e-01, /* 0x3e53f142 */
P1   =  1.6666667163e-01, /* 0x3e2aaaab */
P2   = -2.7777778450e-03, /* 0xbb360b61 */
P3   =  6.6137559770e-05, /* 0x388ab355 */
P4   = -1.6533901999e-06, /* 0xb5ddea0e */
P5   =  4.1381369442e-08, /* 0x3331bb4c */
lg2  =  6.9314718246e-01, /* 0x3f317218 */
lg2_h  =  6.93145752e-01, /* 0x3f317200 */
lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/

float
__ieee754_powf(float x, float y)
{
    float z,ax,z_h,z_l,p_h,p_l;
    float y1,t1,t2,r,s,t,u,v,w;
    int32_t i,j,k,yisint,n;
    int32_t hx,hy,ix,iy,is;

    GET_FLOAT_WORD(hx,x);
    GET_FLOAT_WORD(hy,y);
    ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

    /* y==zero: x**0 = 1 */
    if(iy==0) return one;

    /* x==+-1 */
    if(x == 1.0) return one;
    if(x == -1.0 && isinf(y)) return one;

    /* +-NaN return x+y */
    if(__builtin_expect(ix > 0x7f800000 ||
                iy > 0x7f800000, 0))
        return x+y;

    /* determine if y is an odd int when x < 0
     * yisint = 0   ... y is not an integer
     * yisint = 1   ... y is an odd int
     * yisint = 2   ... y is an even int
     */
    yisint  = 0;
    if(hx<0) {
        if(iy>=0x4b800000) yisint = 2; /* even integer y */
        else if(iy>=0x3f800000) {
        k = (iy>>23)-0x7f;     /* exponent */
        j = iy>>(23-k);
        if((j<<(23-k))==iy) yisint = 2-(j&1);
        }
    }

    /* special value of y */
    if (__builtin_expect(iy==0x7f800000, 0)) {  /* y is +-inf */
        if (ix==0x3f800000)
        return  y - y;  /* inf**+-1 is NaN */
        else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
        return (hy>=0)? y: zero;
        else            /* (|x|<1)**-,+inf = inf,0 */
        return (hy<0)?-y: zero;
    }
    if(iy==0x3f800000) {    /* y is  +-1 */
        if(hy<0) return one/x; else return x;
    }
    if(hy==0x40000000) return x*x; /* y is  2 */
    if(hy==0x3f000000) {    /* y is  0.5 */
        if(__builtin_expect(hx>=0, 1))  /* x >= +0 */
        return __ieee754_sqrtf(x);
    }

    ax   = fabsf(x);
    /* special value of x */
    if(__builtin_expect(ix==0x7f800000||ix==0||ix==0x3f800000, 0)){
        z = ax;         /*x is +-0,+-inf,+-1*/
        if(hy<0) z = one/z; /* z = (1/|x|) */
        if(hx<0) {
        if(((ix-0x3f800000)|yisint)==0) {
            z = (z-z)/(z-z); /* (-1)**non-int is NaN */
        } else if(yisint==1)
            z = -z;     /* (x<0)**odd = -(|x|**odd) */
        }
        return z;
    }

    /* (x<0)**(non-int) is NaN */
    if(__builtin_expect(((((u_int32_t)hx>>31)-1)|yisint)==0, 0))
        return (x-x)/(x-x);

    /* |y| is huge */
    if(__builtin_expect(iy>0x4d000000, 0)) { /* if |y| > 2**27 */
    /* over/underflow if x is not close to one */
        if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
        if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
    /* now |1-x| is tiny <= 2**-20, suffice to compute
       log(x) by x-x^2/2+x^3/3-x^4/4 */
        t = ax-1;       /* t has 20 trailing zeros */
        w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
        u = ivln2_h*t;  /* ivln2_h has 16 sig. bits */
        v = t*ivln2_l-w*ivln2;
        t1 = u+v;
        GET_FLOAT_WORD(is,t1);
        SET_FLOAT_WORD(t1,is&0xfffff000);
        t2 = v-(t1-u);
    } else {
        float s2,s_h,s_l,t_h,t_l;
        /* Avoid internal underflow for tiny y.  The exact value
           of y does not matter if |y| <= 2**-32.  */
        if (iy < 0x2f800000)
          SET_FLOAT_WORD (y, (hy & 0x80000000) | 0x2f800000);
        n = 0;
    /* take care subnormal number */
        if(ix<0x00800000)
        {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
        n  += ((ix)>>23)-0x7f;
        j  = ix&0x007fffff;
    /* determine interval */
        ix = j|0x3f800000;      /* normalize ix */
        if(j<=0x1cc471) k=0;    /* |x|<sqrt(3/2) */
        else if(j<0x5db3d7) k=1;    /* |x|<sqrt(3)   */
        else {k=0;n+=1;ix -= 0x00800000;}
        SET_FLOAT_WORD(ax,ix);

    /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
        u = ax-bp[k];       /* bp[0]=1.0, bp[1]=1.5 */
        v = one/(ax+bp[k]);
        s = u*v;
        s_h = s;
        GET_FLOAT_WORD(is,s_h);
        SET_FLOAT_WORD(s_h,is&0xfffff000);
    /* t_h=ax+bp[k] High */
        SET_FLOAT_WORD (t_h,
                ((((ix>>1)|0x20000000)+0x00400000+(k<<21))
                 & 0xfffff000));
        t_l = ax - (t_h-bp[k]);
        s_l = v*((u-s_h*t_h)-s_h*t_l);
    /* compute log(ax) */
        s2 = s*s;
        r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
        r += s_l*(s_h+s);
        s2  = s_h*s_h;
        t_h = (float)3.0+s2+r;
        GET_FLOAT_WORD(is,t_h);
        SET_FLOAT_WORD(t_h,is&0xfffff000);
        t_l = r-((t_h-(float)3.0)-s2);
    /* u+v = s*(1+...) */
        u = s_h*t_h;
        v = s_l*t_h+t_l*s;
    /* 2/(3log2)*(s+...) */
        p_h = u+v;
        GET_FLOAT_WORD(is,p_h);
        SET_FLOAT_WORD(p_h,is&0xfffff000);
        p_l = v-(p_h-u);
        z_h = cp_h*p_h;     /* cp_h+cp_l = 2/(3*log2) */
        z_l = cp_l*p_h+p_l*cp+dp_l[k];
    /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
        t = (float)n;
        t1 = (((z_h+z_l)+dp_h[k])+t);
        GET_FLOAT_WORD(is,t1);
        SET_FLOAT_WORD(t1,is&0xfffff000);
        t2 = z_l-(((t1-t)-dp_h[k])-z_h);
    }

    s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
    if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
        s = -one;   /* (-ve)**(odd int) */

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
    GET_FLOAT_WORD(is,y);
    SET_FLOAT_WORD(y1,is&0xfffff000);
    p_l = (y-y1)*t1+y*t2;
    p_h = y1*t1;
    z = p_l+p_h;
    GET_FLOAT_WORD(j,z);
    if (__builtin_expect(j>0x43000000, 0))      /* if z > 128 */
        return s*huge*huge;             /* overflow */
    else if (__builtin_expect(j==0x43000000, 0)) {  /* if z == 128 */
        if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
    }
    else if (__builtin_expect((j&0x7fffffff)>0x43160000, 0))/* z <= -150 */
        return s*tiny*tiny;             /* underflow */
    else if (__builtin_expect((u_int32_t) j==0xc3160000, 0)){/* z == -150*/
        if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
    }
    /*
     * compute 2**(p_h+p_l)
     */
    i = j&0x7fffffff;
    k = (i>>23)-0x7f;
    n = 0;
    if(i>0x3f000000) {      /* if |z| > 0.5, set n = [z+0.5] */
        n = j+(0x00800000>>(k+1));
        k = ((n&0x7fffffff)>>23)-0x7f;  /* new k for n */
        SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
        n = ((n&0x007fffff)|0x00800000)>>(23-k);
        if(j<0) n = -n;
        p_h -= t;
    }
    t = p_l+p_h;
    GET_FLOAT_WORD(is,t);
    SET_FLOAT_WORD(t,is&0xfffff000);
    u = t*lg2_h;
    v = (p_l-(t-p_h))*lg2+t*lg2_l;
    z = u+v;
    w = v-(z-u);
    t  = z*z;
    t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
    r  = (z*t1)/(t1-two)-(w+z*w);
    z  = one-(r-z);
    GET_FLOAT_WORD(j,z);
    j += (n<<23);
    if((j>>23)<=0)  /* subnormal output */
      {
        z = __scalbnf (z, n);
        float force_underflow = z * z;
        math_force_eval (force_underflow);
      }
    else SET_FLOAT_WORD(z,j);
    return s*z;
}
strong_alias (__ieee754_powf, __powf_finite)

Answer 1

正如源代码中的注释 /* let libm handle finite**finite */ 所暗示的那样，实际功能外包给了外部库。 libm 这个名称具有历史意义，是 libc 中进行数学运算的部分。不是每个人都有浮点单元，所以不是每个人都需要一个处理浮点的库，因为当时内存很昂贵，所以它被打包到第二个库中。 (是的，它比那复杂得多，但基本上......)

您正在搜索的代码在您的libc 的源代码中。您可能无法查看您的 libc 的源代码，但其中的函数是标准化的，您可以使用其他库，如 dietlibc、uClibc、newlib (cygwin) , glibc 等等。 (没有给出链接以避免链接失效，但适当的搜索机会找到所有链接)。

其中一些库使用旧的 SunPro 代码(例如:uClibc 和 newlib)，这些代码经过高度优化，接近金属代码但可读和注释，在 uClibc 中查找文件 e_pow.c或新库。

如果您使用 Linux，您可能会想查看 GlibC 的源代码，其中 pow() 的众多实现之一可以在 sysdeps/ieee754/dbl-64 中找到/e_pow.c.

其他库的做法略有不同，但差别不大，例如:dietlibc 对 log() 和 exp() 使用手动 i386 汇编程序。