Back to index

glibc  2.9
Functions
e_pow.c File Reference
#include "endian.h"
#include "upow.h"
#include "dla.h"
#include "mydefs.h"
#include "MathLib.h"
#include "upow.tbl"
#include "math_private.h"

Go to the source code of this file.

Functions

double __exp1 (double x, double xx, double error)
static double log1 (double x, double *delta, double *error)
static double my_log2 (double x, double *delta, double *error)
double __slowpow (double x, double y, double z)
static double power1 (double x, double y)
static int checkint (double x)
double __ieee754_pow (double x, double y)

Function Documentation

double __exp1 ( double  x,
double  xx,
double  error 
)

Definition at line 156 of file e_exp.c.

                                                 {
  double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
  mynumber junk1, junk2, binexp  = {{0,0}};
#if 0
  int4 k;
#endif
  int4 i,j,m,n,ex;

  junk1.x = x;
  m = junk1.i[HIGH_HALF];
  n = m&hugeint;                 /* no sign */

  if (n > smallint && n < bigint) {
    y = x*log2e.x + three51.x;
    bexp = y - three51.x;      /*  multiply the result by 2**bexp        */

    junk1.x = y;

    eps = bexp*ln_two2.x;      /* x = bexp*ln(2) + t - eps               */
    t = x - bexp*ln_two1.x;

    y = t + three33.x;
    base = y - three33.x;      /* t rounded to a multiple of 2**-18      */
    junk2.x = y;
    del = (t - base) + (xx-eps);    /*  x = bexp*ln(2) + base + del      */
    eps = del + del*del*(p3.x*del + p2.x);

    binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+1023)<<20;

    i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356;
    j = (junk2.i[LOW_HALF]&511)<<1;

    al = coar.x[i]*fine.x[j];
    bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1];

    rem=(bet + bet*eps)+al*eps;
    res = al + rem;
    cor = (al - res) + rem;
    if  (res == (res+cor*(1.0+error+err_1))) return res*binexp.x;
    else return -10.0;
  }

  if (n <= smallint) return 1.0; /*  if x->0 e^x=1 */

  if (n >= badint) {
    if (n > infint) return(zero/zero);    /* x is NaN,  return invalid */
    if (n < infint) return ( (x>0) ? (hhuge*hhuge) : (tiny*tiny) );
    /* x is finite,  cause either overflow or underflow  */
    if (junk1.i[LOW_HALF] != 0)  return (zero/zero);        /*  x is NaN  */
    return ((x>0)?inf.x:zero );   /* |x| = inf;  return either inf or 0 */
  }

  y = x*log2e.x + three51.x;
  bexp = y - three51.x;
  junk1.x = y;
  eps = bexp*ln_two2.x;
  t = x - bexp*ln_two1.x;
  y = t + three33.x;
  base = y - three33.x;
  junk2.x = y;
  del = (t - base) + (xx-eps);
  eps = del + del*del*(p3.x*del + p2.x);
  i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356;
  j = (junk2.i[LOW_HALF]&511)<<1;
  al = coar.x[i]*fine.x[j];
  bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1];
  rem=(bet + bet*eps)+al*eps;
  res = al + rem;
  cor = (al - res) + rem;
  if (m>>31) {
    ex=junk1.i[LOW_HALF];
    if (res < 1.0) {res+=res; cor+=cor; ex-=1;}
    if (ex >=-1022) {
      binexp.i[HIGH_HALF] = (1023+ex)<<20;
      if  (res == (res+cor*(1.0+error+err_1))) return res*binexp.x;
      else return -10.0;
    }
    ex = -(1022+ex);
    binexp.i[HIGH_HALF] = (1023-ex)<<20;
    res*=binexp.x;
    cor*=binexp.x;
    eps=1.00000000001+(error+err_1)*binexp.x;
    t=1.0+res;
    y = ((1.0-t)+res)+cor;
    res=t+y;
    cor = (t-res)+y;
    if (res == (res + eps*cor))
      {binexp.i[HIGH_HALF] = 0x00100000; return (res-1.0)*binexp.x;}
    else return -10.0;
  }
  else {
    binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+767)<<20;
    if  (res == (res+cor*(1.0+error+err_1)))
      return res*binexp.x*t256.x;
    else return -10.0;
  }
}
double __ieee754_pow ( double  x,
double  y 
)

Definition at line 58 of file e_pow.c.

                                         {
  double z,a,aa,error, t,a1,a2,y1,y2;
#if 0
  double gor=1.0;
#endif
  mynumber u,v;
  int k;
  int4 qx,qy;
  v.x=y;
  u.x=x;
  if (v.i[LOW_HALF] == 0) { /* of y */
    qx = u.i[HIGH_HALF]&0x7fffffff;
    /* Checking  if x is not too small to compute */
    if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x;
    if (y == 1.0) return x;
    if (y == 2.0) return x*x;
    if (y == -1.0) return 1.0/x;
    if (y == 0) return 1.0;
  }
  /* else */
  if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)||        /* x>0 and not x->0 */
       (u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0))  &&
                                      /*   2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
      (v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) {              /* if y<-1 or y>1   */
    z = log1(x,&aa,&error);                                 /* x^y  =e^(y log (X)) */
    t = y*134217729.0;
    y1 = t - (t-y);
    y2 = y - y1;
    t = z*134217729.0;
    a1 = t - (t-z);
    a2 = (z - a1)+aa;
    a = y1*a1;
    aa = y2*a1 + y*a2;
    a1 = a+aa;
    a2 = (a-a1)+aa;
    error = error*ABS(y);
    t = __exp1(a1,a2,1.9e16*error);     /* return -10 or 0 if wasn't computed exactly */
    return (t>0)?t:power1(x,y);
  }

  if (x == 0) {
    if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
       || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)
      return y;
    if (ABS(y) > 1.0e20) return (y>0)?0:INF.x;
    k = checkint(y);
    if (k == -1)
      return y < 0 ? 1.0/x : x;
    else
      return y < 0 ? 1.0/ABS(x) : 0.0;                               /* return 0 */
  }

  qx = u.i[HIGH_HALF]&0x7fffffff;  /*   no sign   */
  qy = v.i[HIGH_HALF]&0x7fffffff;  /*   no sign   */

  if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) return NaNQ.x;
  if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))
    return x == 1.0 ? 1.0 : NaNQ.x;

  /* if x<0 */
  if (u.i[HIGH_HALF] < 0) {
    k = checkint(y);
    if (k==0) {
      if (qy == 0x7ff00000) {
       if (x == -1.0) return 1.0;
       else if (x > -1.0) return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
       else return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
      }
      else if (qx == 0x7ff00000)
       return y < 0 ? 0.0 : INF.x;
      return NaNQ.x;                              /* y not integer and x<0 */
    }
    else if (qx == 0x7ff00000)
      {
       if (k < 0)
         return y < 0 ? nZERO.x : nINF.x;
       else
         return y < 0 ? 0.0 : INF.x;
      }
    return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */
  }
  /* x>0 */

  if (qx == 0x7ff00000)                              /* x= 2^-0x3ff */
    {if (y == 0) return NaNQ.x;
    return (y>0)?x:0; }

  if (qy > 0x45f00000 && qy < 0x7ff00000) {
    if (x == 1.0) return 1.0;
    if (y>0) return (x>1.0)?INF.x:0;
    if (y<0) return (x<1.0)?INF.x:0;
  }

  if (x == 1.0) return 1.0;
  if (y>0) return (x>1.0)?INF.x:0;
  if (y<0) return (x<1.0)?INF.x:0;
  return 0;     /* unreachable, to make the compiler happy */
}

Here is the call graph for this function:

Here is the caller graph for this function:

double __slowpow ( double  x,
double  y,
double  z 
)

Definition at line 43 of file slowpow.c.

                                               {
  double res,res1;
  mp_no mpx, mpy, mpz,mpw,mpp,mpr,mpr1;
  static const mp_no eps = {-3,{1.0,4.0}};
  int p;

  res = __halfulp(x,y);        /* halfulp() returns -10 or x^y             */
  if (res >= 0) return res;  /* if result was really computed by halfulp */
                  /*  else, if result was not really computed by halfulp */
  p = 10;         /*  p=precision   */
  __dbl_mp(x,&mpx,p);
  __dbl_mp(y,&mpy,p);
  __dbl_mp(z,&mpz,p);
  __mplog(&mpx, &mpz, p);     /* log(x) = z   */
  __mul(&mpy,&mpz,&mpw,p);    /*  y * z =w    */
  __mpexp(&mpw, &mpp, p);     /*  e^w =pp     */
  __add(&mpp,&eps,&mpr,p);    /*  pp+eps =r   */
  __mp_dbl(&mpr, &res, p);
  __sub(&mpp,&eps,&mpr1,p);   /*  pp -eps =r1 */
  __mp_dbl(&mpr1, &res1, p);  /*  converting into double precision */
  if (res == res1) return res;

  p = 32;     /* if we get here result wasn't calculated exactly, continue */
  __dbl_mp(x,&mpx,p);                          /* for more exact calculation */
  __dbl_mp(y,&mpy,p);
  __dbl_mp(z,&mpz,p);
  __mplog(&mpx, &mpz, p);   /* log(c)=z  */
  __mul(&mpy,&mpz,&mpw,p);  /* y*z =w    */
  __mpexp(&mpw, &mpp, p);   /* e^w=pp    */
  __mp_dbl(&mpp, &res, p);  /* converting into double precision */
  return res;
}

Here is the call graph for this function:

static int checkint ( double  x) [static]

Definition at line 366 of file e_pow.c.

                              {
  union {int4 i[2]; double x;} u;
  int k,m,n;
#if 0
  int l;
#endif
  u.x = x;
  m = u.i[HIGH_HALF]&0x7fffffff;    /* no sign */
  if (m >= 0x7ff00000) return 0;    /*  x is +/-inf or NaN  */
  if (m >= 0x43400000) return 1;    /*  |x| >= 2**53   */
  if (m < 0x40000000) return 0;     /* |x| < 2,  can not be 0 or 1  */
  n = u.i[LOW_HALF];
  k = (m>>20)-1023;                 /*  1 <= k <= 52   */
  if (k == 52) return (n&1)? -1:1;  /* odd or even*/
  if (k>20) {
    if (n<<(k-20)) return 0;        /* if not integer */
    return (n<<(k-21))?-1:1;
  }
  if (n) return 0;                  /*if  not integer*/
  if (k == 20) return (m&1)? -1:1;
  if (m<<(k+12)) return 0;
  return (m<<(k+11))?-1:1;
}

Here is the caller graph for this function:

static double log1 ( double  x,
double *  delta,
double *  error 
) [static]

Definition at line 183 of file e_pow.c.

                                                           {
  int i,j,m;
#if 0
  int n;
#endif
  double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
#if 0
  double cor;
#endif
  mynumber u,v;
#ifdef BIG_ENDI
  mynumber
 two52          = {{0x43300000, 0x00000000}}; /* 2**52         */
#else
#ifdef LITTLE_ENDI
  mynumber
 two52          = {{0x00000000, 0x43300000}}; /* 2**52         */
#endif
#endif

  u.x = x;
  m = u.i[HIGH_HALF];
  *error = 0;
  *delta = 0;
  if (m < 0x00100000)             /*  1<x<2^-1007 */
    { x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];}

  if ((m&0x000fffff) < 0x0006a09e)
    {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); }
  else
    {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; }

  v.x = u.x + bigu.x;
  uu = v.x - bigu.x;
  i = (v.i[LOW_HALF]&0x000003ff)<<2;
  if (two52.i[LOW_HALF] == 1023)         /* nx = 0              */
  {
      if (i > 1192 && i < 1208)          /* |x-1| < 1.5*2**-10  */
      {
         t = x - 1.0;
         t1 = (t+5.0e6)-5.0e6;
         t2 = t-t1;
         e1 = t - 0.5*t1*t1;
         e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1);
         res = e1+e2;
         *error = 1.0e-21*ABS(t);
         *delta = (e1-res)+e2;
         return res;
      }                  /* |x-1| < 1.5*2**-10  */
      else
      {
         v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x;
         vv = v.x-bigv.x;
         j = v.i[LOW_HALF]&0x0007ffff;
         j = j+j+j;
         eps = u.x - uu*vv;
         e1 = eps*ui.x[i];
         e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1]));
         e = e1+e2;
         e2 =  ((e1-e)+e2);
         t=ui.x[i+2]+vj.x[j+1];
         t1 = t+e;
         t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4));
         res=t1+t2;
         *error = 1.0e-24;
         *delta = (t1-res)+t2;
         return res;
      }
  }   /* nx = 0 */
  else                            /* nx != 0   */
  {
      eps = u.x - uu;
      nx = (two52.x - two52e.x)+add;
      e1 = eps*ui.x[i];
      e2 = eps*ui.x[i+1];
      e=e1+e2;
      e2 = (e1-e)+e2;
      t=nx*ln2a.x+ui.x[i+2];
      t1=t+e;
      t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6))));
      res = t1+t2;
      *error = 1.0e-21;
      *delta = (t1-res)+t2;
      return res;
  }                                /* nx != 0   */
}

Here is the caller graph for this function:

static double my_log2 ( double  x,
double *  delta,
double *  error 
) [static]

Definition at line 275 of file e_pow.c.

                                                              {
  int i,j,m;
#if 0
  int n;
#endif
  double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
#if 0
  double cor;
#endif
  double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2;
  double y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8;
  mynumber u,v;
#ifdef BIG_ENDI
  mynumber
 two52          = {{0x43300000, 0x00000000}}; /* 2**52         */
#else
#ifdef LITTLE_ENDI
  mynumber
 two52          = {{0x00000000, 0x43300000}}; /* 2**52         */
#endif
#endif

  u.x = x;
  m = u.i[HIGH_HALF];
  *error = 0;
  *delta = 0;
  add=0;
  if (m<0x00100000) {  /* x < 2^-1022 */
    x = x*t52.x;  add = -52.0; u.x = x; m = u.i[HIGH_HALF]; }

  if ((m&0x000fffff) < 0x0006a09e)
    {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); }
  else
    {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; }

  v.x = u.x + bigu.x;
  uu = v.x - bigu.x;
  i = (v.i[LOW_HALF]&0x000003ff)<<2;
  /*------------------------------------- |x-1| < 2**-11-------------------------------  */
  if ((two52.i[LOW_HALF] == 1023)  && (i == 1200))
  {
      t = x - 1.0;
      EMULV(t,s3,y,yy,j1,j2,j3,j4,j5);
      ADD2(-0.5,0,y,yy,z,zz,j1,j2);
      MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8);
      MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8);

      e1 = t+z;
      e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8)))));
      res = e1+e2;
      *error = 1.0e-25*ABS(t);
      *delta = (e1-res)+e2;
      return res;
  }
  /*----------------------------- |x-1| > 2**-11  --------------------------  */
  else
  {          /*Computing log(x) according to log table                        */
      nx = (two52.x - two52e.x)+add;
      ou1 = ui.x[i];
      ou2 = ui.x[i+1];
      lu1 = ui.x[i+2];
      lu2 = ui.x[i+3];
      v.x = u.x*(ou1+ou2)+bigv.x;
      vv = v.x-bigv.x;
      j = v.i[LOW_HALF]&0x0007ffff;
      j = j+j+j;
      eps = u.x - uu*vv;
      ov  = vj.x[j];
      lv1 = vj.x[j+1];
      lv2 = vj.x[j+2];
      a = (ou1+ou2)*(1.0+ov);
      a1 = (a+1.0e10)-1.0e10;
      a2 = a*(1.0-a1*uu*vv);
      e1 = eps*a1;
      e2 = eps*a2;
      e = e1+e2;
      e2 = (e1-e)+e2;
      t=nx*ln2a.x+lu1+lv1;
      t1 = t+e;
      t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4));
      res=t1+t2;
      *error = 1.0e-27;
      *delta = (t1-res)+t2;
      return res;
  }
}

Here is the call graph for this function:

Here is the caller graph for this function:

static double power1 ( double  x,
double  y 
) [static]

Definition at line 160 of file e_pow.c.

                                         {
  double z,a,aa,error, t,a1,a2,y1,y2;
  z = my_log2(x,&aa,&error);
  t = y*134217729.0;
  y1 = t - (t-y);
  y2 = y - y1;
  t = z*134217729.0;
  a1 = t - (t-z);
  a2 = z - a1;
  a = y*z;
  aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y;
  a1 = a+aa;
  a2 = (a-a1)+aa;
  error = error*ABS(y);
  t = __exp1(a1,a2,1.9e16*error);
  return (t >= 0)?t:__slowpow(x,y,z);
}

Here is the call graph for this function:

Here is the caller graph for this function: