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glibc  2.9
Functions | Variables
e_powl.c File Reference
#include "math.h"
#include "math_private.h"

Go to the source code of this file.

Functions

long double __ieee754_powl (long double x, long double y)

Variables

static const long double bp []
static const long double dp_h []
static const long double dp_l []
static const long double zero = 0.0L
static const long double one = 1.0L
static const long double two = 2.0L
static const long double two113 = 1.0384593717069655257060992658440192E34L
static const long double huge = 1.0e3000L
static const long double tiny = 1.0e-3000L
static const long double LN []
static const long double LD []
static const long double PN []
static const long double PD []
static const long double lg2 = 6.9314718055994530941723212145817656807550E-1L
static const long double lg2_h = 6.9314718055994528622676398299518041312695E-1L
static const long double lg2_l = 2.3190468138462996154948554638754786504121E-17L
static const long double ovt = 8.0085662595372944372e-0017L
static const long double cp = 9.6179669392597560490661645400126142495110E-1L
static const long double cp_h = 9.6179669392597555432899980587535537779331E-1L
static const long double cp_l = 5.0577616648125906047157785230014751039424E-17L

Function Documentation

long double __ieee754_powl ( long double  x,
long double  y 
)

Definition at line 152 of file e_powl.c.

{
  long double z, ax, z_h, z_l, p_h, p_l;
  long double y1, t1, t2, r, s, t, u, v, w;
  long double s2, s_h, s_l, t_h, t_l;
  int32_t i, j, k, yisint, n;
  u_int32_t ix, iy;
  int32_t hx, hy;
  ieee854_long_double_shape_type o, p, q;

  p.value = x;
  hx = p.parts32.w0;
  ix = hx & 0x7fffffff;

  q.value = y;
  hy = q.parts32.w0;
  iy = hy & 0x7fffffff;


  /* y==zero: x**0 = 1 */
  if ((iy | q.parts32.w1 | (q.parts32.w2 & 0x7fffffff) | q.parts32.w3) == 0)
    return one;

  /* 1.0**y = 1; -1.0**+-Inf = 1 */
  if (x == one)
    return one;
  if (x == -1.0L && iy == 0x7ff00000
      && (q.parts32.w1 | (q.parts32.w2 & 0x7fffffff) | q.parts32.w3) == 0)
    return one;

  /* +-NaN return x+y */
  if ((ix > 0x7ff00000)
      || ((ix == 0x7ff00000)
         && ((p.parts32.w1 | (p.parts32.w2 & 0x7fffffff) | p.parts32.w3) != 0))
      || (iy > 0x7ff00000)
      || ((iy == 0x7ff00000)
         && ((q.parts32.w1 | (q.parts32.w2 & 0x7fffffff) | q.parts32.w3) != 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 ((q.parts32.w2 & 0x7fffffff) >= 0x43400000)    /* Low part >= 2^53 */
       yisint = 2;          /* even integer y */
      else if (iy >= 0x3ff00000)   /* 1.0 */
       {
         if (__floorl (y) == y)
           {
             z = 0.5 * y;
             if (__floorl (z) == z)
              yisint = 2;
             else
              yisint = 1;
           }
       }
    }

  /* special value of y */
  if ((q.parts32.w1 | (q.parts32.w2 & 0x7fffffff) | q.parts32.w3) == 0)
    {
      if (iy == 0x7ff00000 && q.parts32.w1 == 0) /* y is +-inf */
       {
         if (((ix - 0x3ff00000) | p.parts32.w1
              | (p.parts32.w2 & 0x7fffffff) | p.parts32.w3) == 0)
           return y - y;    /* inf**+-1 is NaN */
         else if (ix > 0x3ff00000 || fabsl (x) > 1.0L)
           /* (|x|>1)**+-inf = inf,0 */
           return (hy >= 0) ? y : zero;
         else
           /* (|x|<1)**-,+inf = inf,0 */
           return (hy < 0) ? -y : zero;
       }
      if (iy == 0x3ff00000)
       {                    /* y is  +-1 */
         if (hy < 0)
           return one / x;
         else
           return x;
       }
      if (hy == 0x40000000)
       return x * x;        /* y is  2 */
      if (hy == 0x3fe00000)
       {                    /* y is  0.5 */
         if (hx >= 0)              /* x >= +0 */
           return __ieee754_sqrtl (x);
       }
    }

  ax = fabsl (x);
  /* special value of x */
  if ((p.parts32.w1 | (p.parts32.w2 & 0x7fffffff) | p.parts32.w3) == 0)
    {
      if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000)
       {
         z = ax;            /*x is +-0,+-inf,+-1 */
         if (hy < 0)
           z = one / z;     /* z = (1/|x|) */
         if (hx < 0)
           {
             if (((ix - 0x3ff00000) | 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 (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
    return (x - x) / (x - x);

  /* |y| is huge.
     2^-16495 = 1/2 of smallest representable value.
     If (1 - 1/131072)^y underflows, y > 1.4986e9 */
  if (iy > 0x41d654b0)
    {
      /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
      if (iy > 0x47d654b0)
       {
         if (ix <= 0x3fefffff)
           return (hy < 0) ? huge * huge : tiny * tiny;
         if (ix >= 0x3ff00000)
           return (hy > 0) ? huge * huge : tiny * tiny;
       }
      /* over/underflow if x is not close to one */
      if (ix < 0x3fefffff)
       return (hy < 0) ? huge * huge : tiny * tiny;
      if (ix > 0x3ff00000)
       return (hy > 0) ? huge * huge : tiny * tiny;
    }

  n = 0;
  /* take care subnormal number */
  if (ix < 0x00100000)
    {
      ax *= two113;
      n -= 113;
      o.value = ax;
      ix = o.parts32.w0;
    }
  n += ((ix) >> 20) - 0x3ff;
  j = ix & 0x000fffff;
  /* determine interval */
  ix = j | 0x3ff00000;             /* normalize ix */
  if (j <= 0x39880)
    k = 0;                  /* |x|<sqrt(3/2) */
  else if (j < 0xbb670)
    k = 1;                  /* |x|<sqrt(3)   */
  else
    {
      k = 0;
      n += 1;
      ix -= 0x00100000;
    }

  o.value = ax;
  o.value = __scalbnl (o.value, ((int) ((ix - o.parts32.w0) * 2)) >> 21);
  ax = o.value;

  /* 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;

  o.value = s_h;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  s_h = o.value;
  /* t_h=ax+bp[k] High */
  t_h = ax + bp[k];
  o.value = t_h;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  t_h = o.value;
  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;
  u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
  v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
  r = s2 * s2 * u / v;
  r += s_l * (s_h + s);
  s2 = s_h * s_h;
  t_h = 3.0 + s2 + r;
  o.value = t_h;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  t_h = o.value;
  t_l = r - ((t_h - 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;
  o.value = p_h;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  p_h = o.value;
  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 = (long double) n;
  t1 = (((z_h + z_l) + dp_h[k]) + t);
  o.value = t1;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  t1 = o.value;
  t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);

  /* s (sign of result -ve**odd) = -1 else = 1 */
  s = one;
  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) */
  y1 = y;
  o.value = y1;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  y1 = o.value;
  p_l = (y - y1) * t1 + y * t2;
  p_h = y1 * t1;
  z = p_l + p_h;
  o.value = z;
  j = o.parts32.w0;
  if (j >= 0x40d00000) /* z >= 16384 */
    {
      /* if z > 16384 */
      if (((j - 0x40d00000) | o.parts32.w1
        | (o.parts32.w2 & 0x7fffffff) | o.parts32.w3) != 0)
       return s * huge * huge;     /* overflow */
      else
       {
         if (p_l + ovt > z - p_h)
           return s * huge * huge; /* overflow */
       }
    }
  else if ((j & 0x7fffffff) >= 0x40d01b90)       /* z <= -16495 */
    {
      /* z < -16495 */
      if (((j - 0xc0d01bc0) | o.parts32.w1
         | (o.parts32.w2 & 0x7fffffff) | o.parts32.w3) != 0)
       return s * tiny * tiny;     /* underflow */
      else
       {
         if (p_l <= z - p_h)
           return s * tiny * tiny; /* underflow */
       }
    }
  /* compute 2**(p_h+p_l) */
  i = j & 0x7fffffff;
  k = (i >> 20) - 0x3ff;
  n = 0;
  if (i > 0x3fe00000)
    {                       /* if |z| > 0.5, set n = [z+0.5] */
      n = __floorl (z + 0.5L);
      t = n;
      p_h -= t;
    }
  t = p_l + p_h;
  o.value = t;
  o.parts32.w3 = 0;
  o.parts32.w2 &= 0xffff8000;
  t = o.value;
  u = t * lg2_h;
  v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
  z = u + v;
  w = v - (z - u);
  /*  exp(z) */
  t = z * z;
  u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
  v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
  t1 = z - t * u / v;
  r = (z * t1) / (t1 - two) - (w + z * w);
  z = one - (r - z);
  z = __scalbnl (z, n);
  return s * z;
}

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Variable Documentation

const long double bp[] [static]
Initial value:
 {
  1.0L,
  1.5L,
}

Definition at line 70 of file e_powl.c.

const long double cp = 9.6179669392597560490661645400126142495110E-1L [static]

Definition at line 143 of file e_powl.c.

const long double cp_h = 9.6179669392597555432899980587535537779331E-1L [static]

Definition at line 144 of file e_powl.c.

const long double cp_l = 5.0577616648125906047157785230014751039424E-17L [static]

Definition at line 145 of file e_powl.c.

const long double dp_h[] [static]
Initial value:
 {
  0.0,
  5.8496250072115607565592654282227158546448E-1L
}

Definition at line 76 of file e_powl.c.

const long double dp_l[] [static]
Initial value:
 {
  0.0,
  1.0579781240112554492329533686862998106046E-16L
}

Definition at line 82 of file e_powl.c.

const long double huge = 1.0e3000L [static]

Definition at line 91 of file e_powl.c.

const long double LD[] [static]
Initial value:
{
 -5.129862866715009066465422805058933131960E1L,
  1.452015077564081884387441590064272782044E2L,
 -1.524043275549860505277434040464085593165E2L,
  7.236063513651544224319663428634139768808E1L,
 -1.494198912340228235853027849917095580053E1L
  
}

Definition at line 106 of file e_powl.c.

const long double lg2 = 6.9314718055994530941723212145817656807550E-1L [static]

Definition at line 138 of file e_powl.c.

const long double lg2_h = 6.9314718055994528622676398299518041312695E-1L [static]

Definition at line 139 of file e_powl.c.

const long double lg2_l = 2.3190468138462996154948554638754786504121E-17L [static]

Definition at line 140 of file e_powl.c.

const long double LN[] [static]
Initial value:
{
 -3.0779177200290054398792536829702930623200E1L,
  6.5135778082209159921251824580292116201640E1L,
 -4.6312921812152436921591152809994014413540E1L,
  1.2510208195629420304615674658258363295208E1L,
 -9.9266909031921425609179910128531667336670E-1L
}

Definition at line 98 of file e_powl.c.

const long double one = 1.0L [static]

Definition at line 88 of file e_powl.c.

const long double ovt = 8.0085662595372944372e-0017L [static]

Definition at line 141 of file e_powl.c.

const long double PD[] [static]
Initial value:
{
  3.049081015149226615468111430031590411682E9L,
  1.069833887183886839966085436512368982758E8L,
  8.259257717868875207333991924545445705394E5L,
  1.872583833284143212651746812884298360922E3L,
  
}

Definition at line 127 of file e_powl.c.

const long double PN[] [static]
Initial value:
{
  5.081801691915377692446852383385968225675E8L,
  9.360895299872484512023336636427675327355E6L,
  4.213701282274196030811629773097579432957E4L,
  5.201006511142748908655720086041570288182E1L,
  9.088368420359444263703202925095675982530E-3L,
}

Definition at line 119 of file e_powl.c.

const long double tiny = 1.0e-3000L [static]

Definition at line 92 of file e_powl.c.

const long double two = 2.0L [static]

Definition at line 89 of file e_powl.c.

const long double two113 = 1.0384593717069655257060992658440192E34L [static]

Definition at line 90 of file e_powl.c.

const long double zero = 0.0L [static]

Definition at line 87 of file e_powl.c.