使用哪种算法在GNU C ++标准库中计算指数函数？

Question

Please consider std::exp defined in the header cmath in C++ numerics library. Now, please consider an implementation of the C++ Standard Library, say libstdc++ .

Considering there are various algorithms to compute the elementary functions, such as arithmetic-geometric mean iteration algorithm to compute the exponential function and three others shown here ;

Could you please name the particular algorithm being used to compute the exponential function in libstdc++ , if possible?

PS: I could not pinpoint either the correct tarballs containing the std::exp implementation or comprehend the relevant file contents, I'm afraid.

Answer 1

It doesn't use any intricate algorithm at all. Note that std::exp is only defined for a very limited number of types: float , double and long double + any Integral type that is castable to double . That makes it not necessary to implement complicated maths.

Currently, it uses the builtin __builtin_expf as can be verified from the source code . This compiles to a call to expf on my machine which is a call into libm coming from glibc . Let's see what we find in their source code . When we search for expf we find that this internally calls __ieee754_expf which is a system-dependant implementation. Both i686 and x86_64 just include a glibc/sysdeps/ieee754/flt-32/e_expf.c which finally gives us an implementation (reduced for brevity, the look into the sources

It is basically a order 3 polynomial approximation for floats:

static inline uint32_t
top12 (float x)
{
  return asuint (x) >> 20;
}

float
__expf (float x)
{
  uint64_t ki, t;
  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  double_t kd, xd, z, r, r2, y, s;

  xd = (double_t) x;
  // [...] skipping fast under/overflow handling

  /* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k.  */
  z = InvLn2N * xd;

  /* Round and convert z to int, the result is in [-150*N, 128*N] and
     ideally ties-to-even rule is used, otherwise the magnitude of r
     can be bigger which gives larger approximation error.  */
  kd = roundtoint (z);
  ki = converttoint (z);
  r = z - kd;

  /* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
  t = T[ki % N];
  t += ki << (52 - EXP2F_TABLE_BITS);
  s = asdouble (t);
  z = C[0] * r + C[1];
  r2 = r * r;
  y = C[2] * r + 1;
  y = z * r2 + y;
  y = y * s;
  return (float) y;
}

Similarly, for 128-bit long double , it's an order 7 approximation and for double they use more complicated algorithm that I can't make sense of right now.