コード例 #1
0
ファイル: tisqrt.c プロジェクト: BrianGladman/mpfr
static void
tst_icbrt (unsigned long n, unsigned long r)
{
  unsigned long i;

  i = __gmpfr_cuberoot (n);
  if (i != r)
    {
      printf ("Error in __gmpfr_cuberoot (%lu): got %lu instead of %lu\n",
              n, i, r);
      exit (1);
    }
}
コード例 #2
0
/* use Brent's formula exp(x) = (1+r+r^2/2!+r^3/3!+...)^(2^K)*2^n
   where x = n*log(2)+(2^K)*r
   together with Brent-Kung O(t^(1/2)) algorithm for the evaluation of
   power series. The resulting complexity is O(n^(1/3)*M(n)).
*/
int
mpfr_exp_2 (mpfr_ptr y, mpfr_srcptr x, mp_rnd_t rnd_mode)
{
  long n;
  unsigned long K, k, l, err; /* FIXME: Which type ? */
  int error_r;
  mp_exp_t exps;
  mp_prec_t q, precy;
  int inexact;
  mpfr_t r, s, t;
  mpz_t ss;
  TMP_DECL(marker);

  precy = MPFR_PREC(y);
  
  MPFR_TRACE ( printf("Py=%d Px=%d", MPFR_PREC(y), MPFR_PREC(x)) );
  MPFR_TRACE ( MPFR_DUMP (x) );

  n = (long) (mpfr_get_d1 (x) / LOG2);

  /* error bounds the cancelled bits in x - n*log(2) */
  if (MPFR_UNLIKELY(n == 0))
    error_r = 0;
  else
    count_leading_zeros (error_r, (mp_limb_t) (n < 0) ? -n : n);
  error_r = BITS_PER_MP_LIMB - error_r + 2;

  /* for the O(n^(1/2)*M(n)) method, the Taylor series computation of
     n/K terms costs about n/(2K) multiplications when computed in fixed
     point */
  K = (precy < SWITCH) ? __gmpfr_isqrt ((precy + 1) / 2)
    : __gmpfr_cuberoot (4*precy);
  l = (precy - 1) / K + 1;
  err = K + MPFR_INT_CEIL_LOG2 (2 * l + 18);
  /* add K extra bits, i.e. failure probability <= 1/2^K = O(1/precy) */
  q = precy + err + K + 5;
  
  /*q = ( (q-1)/BITS_PER_MP_LIMB + 1) * BITS_PER_MP_LIMB; */

  mpfr_init2 (r, q + error_r);
  mpfr_init2 (s, q + error_r);
  mpfr_init2 (t, q);

  /* the algorithm consists in computing an upper bound of exp(x) using
     a precision of q bits, and see if we can round to MPFR_PREC(y) taking
     into account the maximal error. Otherwise we increase q. */
  for (;;)
    {
      MPFR_TRACE ( printf("n=%d K=%d l=%d q=%d\n",n,K,l,q) );
      
      /* if n<0, we have to get an upper bound of log(2)
	 in order to get an upper bound of r = x-n*log(2) */
      mpfr_const_log2 (s, (n >= 0) ? GMP_RNDZ : GMP_RNDU);
      /* s is within 1 ulp of log(2) */
      
      mpfr_mul_ui (r, s, (n < 0) ? -n : n, (n >= 0) ? GMP_RNDZ : GMP_RNDU);
      /* r is within 3 ulps of n*log(2) */
      if (n < 0)
	mpfr_neg (r, r, GMP_RNDD); /* exact */
      /* r = floor(n*log(2)), within 3 ulps */
      
      MPFR_TRACE ( MPFR_DUMP (x) );
      MPFR_TRACE ( MPFR_DUMP (r) );
      
      mpfr_sub (r, x, r, GMP_RNDU);
      /* possible cancellation here: the error on r is at most
	 3*2^(EXP(old_r)-EXP(new_r)) */
      while (MPFR_IS_NEG (r))
	{ /* initial approximation n was too large */
	  n--;
	  mpfr_add (r, r, s, GMP_RNDU);
	}
      mpfr_prec_round (r, q, GMP_RNDU);
      MPFR_TRACE ( MPFR_DUMP (r) );
      MPFR_ASSERTD (MPFR_IS_POS (r));
      mpfr_div_2ui (r, r, K, GMP_RNDU); /* r = (x-n*log(2))/2^K, exact */
      
      TMP_MARK(marker);
      MY_INIT_MPZ(ss, 3 + 2*((q-1)/BITS_PER_MP_LIMB));
      exps = mpfr_get_z_exp (ss, s);
      /* s <- 1 + r/1! + r^2/2! + ... + r^l/l! */
      l = (precy < SWITCH) ? 
	mpfr_exp2_aux (ss, r, q, &exps)      /* naive method */
	: mpfr_exp2_aux2 (ss, r, q, &exps);  /* Brent/Kung method */
      
      MPFR_TRACE(printf("l=%d q=%d (K+l)*q^2=%1.3e\n", l, q, (K+l)*(double)q*q));
      
      for (k = 0; k < K; k++)
	{
	  mpz_mul (ss, ss, ss);
	  exps <<= 1;
	  exps += mpz_normalize (ss, ss, q);
	}
      mpfr_set_z (s, ss, GMP_RNDN);
      
      MPFR_SET_EXP(s, MPFR_GET_EXP (s) + exps);
      TMP_FREE(marker); /* don't need ss anymore */
      
      if (n>0) 
	mpfr_mul_2ui(s, s, n, GMP_RNDU);
      else 
	mpfr_div_2ui(s, s, -n, GMP_RNDU);
      
      /* error is at most 2^K*(3l*(l+1)) ulp for mpfr_exp2_aux */
      l = (precy < SWITCH) ? 3*l*(l+1) : l*(l+4) ;
      k = MPFR_INT_CEIL_LOG2 (l);
      /* k = 0; while (l) { k++; l >>= 1; } */

      /* now k = ceil(log(error in ulps)/log(2)) */
      K += k;

      MPFR_TRACE ( printf("after mult. by 2^n:\n") );
      MPFR_TRACE ( MPFR_DUMP (s) );
      MPFR_TRACE ( printf("err=%d bits\n", K) );
      
      if (mpfr_can_round (s, q - K, GMP_RNDN, GMP_RNDZ,
			  precy + (rnd_mode == GMP_RNDN)) )
	break;
      MPFR_TRACE (printf("prec++, use %d\n", q+BITS_PER_MP_LIMB) );
      MPFR_TRACE (printf("q=%d q-K=%d precy=%d\n",q,q-K,precy) );
      q += BITS_PER_MP_LIMB;
      mpfr_set_prec (r, q);
      mpfr_set_prec (s, q);
      mpfr_set_prec (t, q);
    }
  
  inexact = mpfr_set (y, s, rnd_mode);

  mpfr_clear (r); 
  mpfr_clear (s); 
  mpfr_clear (t);

  return inexact;
}