Пример #1
0
char
binetasymptotic(floatnum x,
                int digits)
{
  floatstruct recsqr;
  floatstruct sum;
  floatstruct smd;
  floatstruct pwr;
  int i, workprec;

  if (float_getexponent(x) >= digits)
  {
    /* if x is very big, ln(gamma(x)) is
    dominated by x*ln x and the Binet function
    does not contribute anything substantial to
    the final result */
    float_setzero(x);
    return 1;
  }
  float_create(&recsqr);
  float_create(&sum);
  float_create(&smd);
  float_create(&pwr);

  float_copy(&pwr, &c1, EXACT);
  float_setzero(&sum);
  float_div(&smd, &c1, &c12, digits+1);
  workprec = digits - 2*float_getexponent(x)+3;
  i = 1;
  if (workprec > 0)
  {
    float_mul(&recsqr, x, x, workprec);
    float_reciprocal(&recsqr, workprec);
    while (float_getexponent(&smd) > -digits-1
           && ++i <= MAXBERNOULLIIDX)
    {
      workprec = digits + float_getexponent(&smd) + 3;
      float_add(&sum, &sum, &smd, digits+1);
      float_mul(&pwr, &recsqr, &pwr, workprec);
      float_muli(&smd, &cBernoulliDen[i-1], 2*i*(2*i-1), workprec);
      float_div(&smd, &pwr, &smd, workprec);
      float_mul(&smd, &smd, &cBernoulliNum[i-1], workprec);
    }
  }
  else
    /* sum reduces to the first summand*/
    float_move(&sum, &smd);
  if (i > MAXBERNOULLIIDX)
      /* x was not big enough for the asymptotic
    series to converge sufficiently */
    float_setnan(x);
  else
    float_div(x, &sum, x, digits);
  float_free(&pwr);
  float_free(&smd);
  float_free(&sum);
  float_free(&recsqr);
  return i <= MAXBERNOULLIIDX;
}
Пример #2
0
static Error
_pack2frac(
  floatnum x,
  p_ext_seq_desc n,
  int digits)
{
  floatstruct tmp;
  int exp;
  Error result;

  n->seq.digits -= n->seq.trailing0;
  n->seq.trailing0 = 0;
  switch(n->seq.base)
  {
  case IO_BASE_NAN:
    float_setnan(x);
    break;
  case IO_BASE_ZERO:
    float_setzero(x);
    break;
  default:
    if ((result = _pack2int(x, n)) != Success)
      return result;
    float_create(&tmp);
    float_setinteger(&tmp, n->seq.base);
    _raiseposi(&tmp, &exp, n->seq.digits, digits+2);
    float_div(x, x, &tmp, digits + 2);
    float_setexponent(x, float_getexponent(x) - exp);
    float_free(&tmp);
  }
  n->seq.digits += n->seq.trailing0;
  return Success;
}
Пример #3
0
static Error
_packdec2int(
  floatnum x,
  p_ext_seq_desc n)
{
  int ofs;
  int exp;
  int bufsz;
  int i;
  char buf[DECPRECISION];

  float_setnan(x);
  ofs = n->seq.leadingSignDigits;
  exp = n->seq.trailing0;
  bufsz = n->seq.digits - ofs - exp;
  if (bufsz > DECPRECISION)
    return IOBufferOverflow;
  if (bufsz == 0)
    float_setzero(x);
  else
    for (i = -1; ++i < bufsz;)
      buf[i] = n->getdigit(ofs++, &n->seq) + '0';
  float_setsignificand(x, NULL, buf, bufsz);
  float_setexponent(x, exp + bufsz - 1);
  return Success;
}
Пример #4
0
void
_cos(
  floatnum x,
  int digits)
{
  signed char sgn;

  float_abs(x);
  sgn = 1;
  if (float_cmp(x, &cPiDiv2) > 0)
  {
    sgn = -1;
    float_sub(x, &cPi, x, digits+1);
  }
  if (float_cmp(x, &cPiDiv4) <= 0)
  {
    if (2*float_getexponent(x)+2 < - digits)
      float_setzero(x);
    else
      _cosminus1ltPiDiv4(x, digits);
    float_add(x, x, &c1, digits);
  }
  else
  {
    float_sub(x, &cPiDiv2, x, digits+1);
    _sinltPiDiv4(x, digits);
  }
  float_setsign(x, sgn);
}
Пример #5
0
/* series expansion of cos/cosh - 1 used for small x,
   |x| <= 0.01.
   The function returns 0, if an underflow occurs.
   The relative error seems to be less than 5e-100 for
   a 100-digit calculation with |x| < 0.01 */
char
cosminus1series(
  floatnum x,
  int digits,
  char alternating)
{
  floatstruct sum, smd;
  int expsqrx, pwrsz, addsz, i;

  expsqrx = 2 * float_getexponent(x);
  float_setexponent(x, 0);
  float_mul(x, x, x, digits+1);
  float_mul(x, x, &c1Div2, digits+1);
  float_setsign(x, alternating? -1 : 1);
  expsqrx += float_getexponent(x);
  if (float_iszero(x) || expsqrx < EXPMIN)
  {
    /* underflow */
    float_setzero(x);
    return expsqrx == 0;
  }
  float_setexponent(x, expsqrx);
  pwrsz = digits + expsqrx + 2;
  if (pwrsz <= 0)
    /* for very small x, cos/cosh(x) - 1 = (-/+)0.5*x*x */
    return 1;
  addsz = pwrsz;
  float_create(&sum);
  float_create(&smd);
  float_copy(&smd, x, pwrsz);
  float_setzero(&sum);
  i = 2;
  while (pwrsz > 0)
  {
    float_mul(&smd, &smd, x, pwrsz+1);
    float_divi(&smd, &smd, i*(2*i-1), pwrsz);
    float_add(&sum, &sum, &smd, addsz);
    ++i;
    pwrsz = digits + float_getexponent(&smd);
  }
  float_add(x, x, &sum, digits+1);
  float_free(&sum);
  float_free(&smd);
  return 1;
}
Пример #6
0
Error
pack2floatnum(
  floatnum x,
  p_number_desc n)
{
  floatstruct tmp;
  int digits;
  int saveerr;
  int saverange;
  Error result;
  signed char base;

  if ((result = _pack2int(x, &n->intpart)) != Success)
    return result;
  if (float_isnan(x))
    return Success;
  saveerr = float_geterror();
  saverange = float_setrange(MAXEXP);
  float_create(&tmp);
  float_move(&tmp, x);
  float_setzero(x);
  digits = DECPRECISION - float_getexponent(&tmp);
  if (digits <= 0
      || (result = _pack2frac(x, &n->fracpart, digits)) == Success)
    float_add(x, x, &tmp, DECPRECISION);
  if (result != Success)
    return result;
  if ((!float_getlength(x)) == 0) /* no zero, no NaN? */
  {
    base = n->prefix.base;
    float_setinteger(&tmp, base);
    if (n->exp >= 0)
    {
      _raiseposi_(&tmp, n->exp, DECPRECISION + 2);
      float_mul(x, x, &tmp, DECPRECISION + 2);
    }
    else
    {
      _raiseposi_(&tmp, -n->exp, DECPRECISION + 2);
      float_div(x, x, &tmp, DECPRECISION + 2);
    }
  }
  float_free(&tmp);
  float_setsign(x, n->prefix.sign == IO_SIGN_COMPLEMENT? -1 : n->prefix.sign);
  float_geterror();
  float_seterror(saveerr);
  float_setrange(saverange);
  if (!float_isvalidexp(float_getexponent(x)))
    float_setnan(x);
  return float_isnan(x)? IOExpOverflow : Success;
}
Пример #7
0
char
float_cosh(
  floatnum x,
  int digits)
{
  int expx;

  if (!chckmathparam(x, digits))
    return 0;
  expx = float_getexponent(x);
  if (2*expx+2 <= -digits || !_coshminus1(x, digits+2*expx))
  {
    if (expx > 0)
      return _seterror(x, Overflow);
    float_setzero(x);
  }
  return float_add(x, x, &c1, digits);
}
Пример #8
0
static Error
_pack2int(
  floatnum x,
  p_ext_seq_desc n)
{
  switch(n->seq.base)
  {
  case IO_BASE_NAN:
    float_setnan(x);
    break;
  case IO_BASE_ZERO:
    float_setzero(x);
    break;
  case 10:
    return _packdec2int(x, n);
  default:
    return _packbin2int(x, n);
  }
  return Success;
}
Пример #9
0
void
_longint2floatnum(
  floatnum f,
  t_longint* longint)
{
  floatstruct tmp;
  int idx;

  float_setzero(f);
  if(longint->length == 0)
    return;
  float_create(&tmp);
  idx = longint->length - 1;
  for (; idx >= 0; --idx)
  {
    _setunsigned(&tmp, longint->value[idx]);
    float_mul(f, f, &cUnsignedBound, EXACT);
    float_add(f, f, &tmp, EXACT);
  }
  float_free(&tmp);
}
Пример #10
0
char
erfcasymptotic(
  floatnum x,
  int digits)
{
  floatstruct smd, fct;
  int i, workprec, newprec;

  float_create(&smd);
  float_create(&fct);
  workprec = digits - 2 * float_getexponent(x) + 1;
  if (workprec <= 0)
  {
    float_copy(x, &c1, EXACT);
    return 1;
  }
  float_mul(&fct, x, x, digits + 1);
  float_div(&fct, &c1Div2, &fct, digits);
  float_neg(&fct);
  float_copy(&smd, &c1, EXACT);
  float_setzero(x);
  newprec = digits;
  workprec = newprec;
  i = 1;
  while (newprec > 0 && newprec <= workprec)
  {
    workprec = newprec;
    float_add(x, x, &smd, digits + 4);
    float_muli(&smd, &smd, i, workprec + 1);
    float_mul(&smd, &smd, &fct, workprec + 2);
    newprec = digits + float_getexponent(&smd) + 1;
    i += 2;
  }
  float_free(&fct);
  float_free(&smd);
  return newprec <= workprec;
}
Пример #11
0
static char
_pochhammer_g(
  floatnum x,
  cfloatnum n,
  int digits)
{
  /* this generalizes the rising Pochhammer symbol using the
     formula pochhammer(x,n) = Gamma(x+1)/Gamma(x-n+1) */
  floatstruct tmp, factor1, factor2;
  int inf1, inf2;
  char result;

  float_create(&tmp);
  float_create(&factor1);
  float_create(&factor2);
  inf2 = 0;
  float_add(&tmp, x, n, digits+1);
  result = _lngamma_prim(x, &factor1, &inf1, digits)
           && _lngamma_prim(&tmp, &factor2, &inf2, digits)
           && (inf2 -= inf1) <= 0;
  if (inf2 > 0)
    float_seterror(ZeroDivide);
  if (result && inf2 < 0)
    float_setzero(x);
  if (result && inf2 == 0)
    result = float_div(&factor1, &factor1, &factor2, digits+1)
             && float_sub(x, &tmp, x, digits+1)
             && _exp(x, digits)
             && float_mul(x, x, &factor1, digits+1);
  float_free(&tmp);
  float_free(&factor2);
  float_free(&factor1);
  if (!result)
    float_setnan(x);
  return result;
}
Пример #12
0
/* the Taylor series of arctan/arctanh x at x == 0. For small
   |x| < 0.01 this series converges very fast, yielding 4 or
   more digits of the result with every summand. The working
   precision is adjusted, so that the relative error for
   100-digit arguments is around 5.0e-100. This means, the error
   is 1 in the 100-th place (or less) */
void
arctanseries(
  floatnum x,
  int digits,
  char alternating)
{
  int expx;
  int expsqrx;
  int pwrsz;
  int addsz;
  int i;
  floatstruct xsqr;
  floatstruct pwr;
  floatstruct smd;
  floatstruct sum;

  /* upper limit of log(x) and log(result) */
  expx = float_getexponent(x)+1;

  /* the summands of the series from the second on are
     bounded by x^(2*i-1)/3. So the summation yields a
     result bounded by (x^3/(1-x*x))/3.
     For x < sqrt(1/3) approx.= 0.5, this is less than 0.5*x^3.
     We need to sum up only, if the first <digits> places
     of the result (roughly x) are touched. Ignoring the effect of
     a possile carry, this is only the case, if
     x*x >= 2*10^(-digits) > 10^(-digits)
     Example: for x = 9e-51, a 100-digits result covers
     the decimal places from 1e-51 to 1e-150. x^3/3
     is roughly 3e-151, and so is the sum of the series.
     So we can ignore the sum, but we couldn't for x = 9e-50 */
  if (float_iszero(x) || 2*expx < -digits)
    /* for very tiny arguments arctan/arctanh x is approx.== x */
    return;
  float_create(&xsqr);
  float_create(&pwr);
  float_create(&smd);
  float_create(&sum);

  /* we adapt the working precision to the decreasing
     summands, saving time when multiplying. Unfortunately,
     there is no error bound given for the operations of
     bc_num. Tests show, that the last digit in an incomplete
     multiplication is usually not correct up to 5 ULP's. */
  pwrsz = digits + 2*expx + 1;
  /* the precision of the addition must not decrease, of course */
  addsz = pwrsz;
  i = 3;
  float_mul(&xsqr, x, x, pwrsz);
  float_setsign(&xsqr, alternating? -1 : 1);
  expsqrx = float_getexponent(&xsqr);
  float_copy(&pwr, x, pwrsz);
  float_setzero(&sum);

  for(; pwrsz > 0; )
  {
    /* x^i */
    float_mul(&pwr, &pwr, &xsqr, pwrsz+1);
    /* x^i/i */
    float_divi(&smd, &pwr, i, pwrsz);
    /* The addition virtually does not introduce errors */
    float_add(&sum, &sum, &smd, addsz);
    /* reduce the working precision according to the decreasing powers */
    pwrsz = digits - expx + float_getexponent(&smd) + expsqrx + 3;
    i += 2;
  }
  /* add the first summand */
  float_add(x, x, &sum, digits+1);

  float_free(&xsqr);
  float_free(&pwr);
  float_free(&smd);
  float_free(&sum);
}
Пример #13
0
char
erfcsum(
  floatnum x, /* should be the square of the parameter to erfc */
  int digits)
{
  int i, workprec;
  floatstruct sum, smd;
  floatnum Ei;

  if (digits > erfcdigits)
  {
    /* cannot re-use last evaluation's intermediate results */
    for (i = MAXERFCIDX; --i >= 0;)
      /* clear all exp(-k*k*alpha*alpha) to indicate their absence */
      float_free(&erfccoeff[i]);
    /* current precision */
    erfcdigits = digits;
    /* create new alpha appropriate for the desired precision
       This alpha need not be high precision, any alpha near the
       one evaluated here would do */
    float_muli(&erfcalpha, &cLn10, digits + 4, 3);
    float_sqrt(&erfcalpha, 3);
    float_div(&erfcalpha, &cPi, &erfcalpha, 3);
    float_mul(&erfcalphasqr, &erfcalpha, &erfcalpha, EXACT);
    /* the exp(-k*k*alpha*alpha) are later evaluated iteratively.
       Initiate the iteration here */
    float_copy(&erfct2, &erfcalphasqr, EXACT);
    float_neg(&erfct2);
    _exp(&erfct2, digits + 3); /* exp(-alpha*alpha) */
    float_copy(erfccoeff, &erfct2, EXACT); /* start value */
    float_mul(&erfct3, &erfct2, &erfct2, digits + 3); /* exp(-2*alpha*alpha) */
  }
  float_create(&sum);
  float_create(&smd);
  float_setzero(&sum);
  for (i = 0; ++i < MAXERFCIDX;)
  {
    Ei = &erfccoeff[i-1];
    if (float_isnan(Ei))
    {
      /* if exp(-i*i*alpha*alpha) is not available, evaluate it from
         the coefficient of the last summand */
      float_mul(&erfct2, &erfct2, &erfct3, workprec + 3);
      float_mul(Ei, &erfct2, &erfccoeff[i-2], workprec + 3);
    }
    /* Ei finally decays rapidly. save some time by adjusting the
       working precision */
    workprec = digits + float_getexponent(Ei) + 1;
    if (workprec <= 0)
      break;
    /* evaluate the summand exp(-i*i*alpha*alpha)/(i*i*alpha*alpha+x) */
    float_muli(&smd, &erfcalphasqr, i*i, workprec);
    float_add(&smd, x, &smd, workprec + 2);
    float_div(&smd, Ei, &smd, workprec + 1);
    /* add summand to the series */
    float_add(&sum, &sum, &smd, digits + 3);
  }
  float_move(x, &sum);
  float_free(&smd);
  return 1;
}