double npy_nextafter(double x, double y)
{
    volatile double t;
    npy_int32 hx, hy, ix, iy;
    npy_uint32 lx, ly;

    EXTRACT_WORDS(hx, lx, x);
    EXTRACT_WORDS(hy, ly, y);
    ix = hx & 0x7fffffff;       /* |x| */
    iy = hy & 0x7fffffff;       /* |y| */

    if (((ix >= 0x7ff00000) && ((ix - 0x7ff00000) | lx) != 0) ||        /* x is nan */
        ((iy >= 0x7ff00000) && ((iy - 0x7ff00000) | ly) != 0))  /* y is nan */
        return x + y;
    if (x == y)
        return y;               /* x=y, return y */
    if ((ix | lx) == 0) {       /* x == 0 */
        INSERT_WORDS(x, hy & 0x80000000, 1);    /* return +-minsubnormal */
        t = x * x;
        if (t == x)
            return t;
        else
            return x;           /* raise underflow flag */
    }
    if (hx >= 0) {              /* x > 0 */
        if (hx > hy || ((hx == hy) && (lx > ly))) {     /* x > y, x -= ulp */
            if (lx == 0)
                hx -= 1;
            lx -= 1;
        } else {                /* x < y, x += ulp */
            lx += 1;
            if (lx == 0)
                hx += 1;
        }
    } else {                    /* x < 0 */
        if (hy >= 0 || hx > hy || ((hx == hy) && (lx > ly))) {  /* x < y, x -= ulp */
            if (lx == 0)
                hx -= 1;
            lx -= 1;
        } else {                /* x > y, x += ulp */
            lx += 1;
            if (lx == 0)
                hx += 1;
        }
    }
    hy = hx & 0x7ff00000;
    if (hy >= 0x7ff00000)
        return x + x;           /* overflow  */
    if (hy < 0x00100000) {      /* underflow */
        t = x * x;
        if (t != x) {           /* raise underflow flag */
            INSERT_WORDS(y, hx, lx);
            return y;
        }
    }
    INSERT_WORDS(x, hx, lx);
    return x;
}
Beispiel #2
0
double __nexttoward(double x, long double y)
{
	int32_t hx,ix,iy;
	u_int32_t lx,hy,ly,esy;

	EXTRACT_WORDS(hx,lx,x);
	GET_LDOUBLE_WORDS(esy,hy,ly,y);
	ix = hx&0x7fffffff;		/* |x| */
	iy = esy&0x7fff;		/* |y| */

	/* Intel's extended format has the normally implicit 1 explicit
	   present.  Sigh!  */
	if(((ix>=0x7ff00000)&&((ix-0x7ff00000)|lx)!=0) ||   /* x is nan */
	   ((iy>=0x7fff)&&((hy&0x7fffffff)|ly)!=0))        /* y is nan */
	   return x+y;
	if((long double) x==y) return y;	/* x=y, return y */
	if((ix|lx)==0) {			/* x == 0 */
	    double u;
	    INSERT_WORDS(x,(esy&0x8000)<<16,1); /* return +-minsub */
	    u = math_opt_barrier (x);
	    u = u * u;
	    math_force_eval (u);		/* raise underflow flag */
	    return x;
	}
	if(hx>=0) {				/* x > 0 */
	    if (x > y) {			/* x -= ulp */
		if(lx==0) hx -= 1;
		lx -= 1;
	    } else {				/* x < y, x += ulp */
		lx += 1;
		if(lx==0) hx += 1;
	    }
	} else {				/* x < 0 */
	    if (x < y) {			/* x -= ulp */
		if(lx==0) hx -= 1;
		lx -= 1;
	    } else {				/* x > y, x += ulp */
		lx += 1;
		if(lx==0) hx += 1;
	    }
	}
	hy = hx&0x7ff00000;
	if(hy>=0x7ff00000) {
	  double u = x+x;			/* overflow  */
	  math_force_eval (u);
	}
	if(hy<0x00100000) {
	    double u = x*x;			/* underflow */
	    math_force_eval (u);		/* raise underflow flag */
	}
	INSERT_WORDS(x,hx,lx);
	return x;
}
double nextafter(double x, double y) {
	int32_t hx,hy,ix,iy;
	u_int32_t lx,ly;

	EXTRACT_WORDS(hx,lx,x);
	EXTRACT_WORDS(hy,ly,y);
	ix = hx&0x7fffffff;		/* |x| */
	iy = hy&0x7fffffff;		/* |y| */

	if(((ix>=0x7ff00000)&&((ix-0x7ff00000)|lx)!=0) ||   /* x is nan */
	   ((iy>=0x7ff00000)&&((iy-0x7ff00000)|ly)!=0))     /* y is nan */
    return x+y;
	if(x==y) return y;		/* x=y, return y */
	if((ix|lx)==0) {			/* x == 0 */
    double u;
    INSERT_WORDS(x,hy&0x80000000,1);	/* return +-minsubnormal */
    //u = math_opt_barrier (x);
    //u = u*u;
    //math_force_eval (u);		/* raise underflow flag */
    return x;
	}
	if(hx>=0) {				/* x > 0 */
    if(hx>hy||((hx==hy)&&(lx>ly))) {	/* x > y, x -= ulp */
      if(lx==0) hx -= 1;
      lx -= 1;
    } else {				/* x < y, x += ulp */
      lx += 1;
      if(lx==0) hx += 1;
    }
	} else {				/* x < 0 */
    if(hy>=0||hx>hy||((hx==hy)&&(lx>ly))){/* x < y, x -= ulp */
      if(lx==0) hx -= 1;
      lx -= 1;
    } else {				/* x > y, x += ulp */
      lx += 1;
      if(lx==0) hx += 1;
    }
	}
	hy = hx&0x7ff00000;
	if(hy>=0x7ff00000) {
	  x = x+x;	/* overflow  */
	  //if (FLT_EVAL_METHOD != 0 && FLT_EVAL_METHOD != 1)
	  //  asm ("" : "+m"(x));
	  return x;	/* overflow  */
	}
	//if(hy<0x00100000) {
    //double u = x*x;			/* underflow */
    //math_force_eval (u);		/* raise underflow flag */
	//}
	INSERT_WORDS(x,hx,lx);
	return x;
}
Beispiel #4
0
double scalbn(double x, int n)
{
	double scale;

	if (n > 1023) {
		x *= 0x1p1023;
		n -= 1023;
		if (n > 1023) {
			x *= 0x1p1023;
			n -= 1023;
			if (n > 1023)
				return x * 0x1p1023;
		}
	} else if (n < -1022) {
		x *= 0x1p-1022;
		n += 1022;
		if (n < -1022) {
			x *= 0x1p-1022;
			n += 1022;
			if (n < -1022)
				return x * 0x1p-1022;
		}
	}
	INSERT_WORDS(scale, (uint32_t)(0x3ff+n)<<20, 0);
	return x * scale;
}
Beispiel #5
0
double
__kernel_cos(double x, double y)
{
    double a,hz,z,r,qx;
    int32_t ix;
    GET_HIGH_WORD(ix,x);
    ix &= 0x7fffffff;     /* ix = |x|'s high word*/
    if(ix<0x3e400000) {     /* if x < 2**27 */
        if(((int)x)==0) return one;   /* generate inexact */
    }
    z  = x*x;
    r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
    if(ix < 0x3FD33333)       /* if |x| < 0.3 */
        return one - (0.5*z - (z*r - x*y));
    else {
        if(ix > 0x3fe90000) {   /* x > 0.78125 */
            qx = 0.28125;
        } else {
            INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
        }
        hz = 0.5*z-qx;
        a  = one-qx;
        return a - (hz - (z*r-x*y));
    }
}
Beispiel #6
0
/* Return the least floating-point number greater than X.  */
double
__nextup (double x)
{
  int32_t hx, ix;
  uint32_t lx;

  EXTRACT_WORDS (hx, lx, x);
  ix = hx & 0x7fffffff;

  if (((ix >= 0x7ff00000) && ((ix - 0x7ff00000) | lx) != 0))  /* x is nan.  */
    return x + x;
  if ((ix | lx) == 0)
    return DBL_TRUE_MIN;
  if (hx >= 0)
    {				/* x > 0.  */
      if (isinf (x))
        return x;
      lx += 1;
      if (lx == 0)
        hx += 1;
    }
  else
    {				/* x < 0.  */
      if (lx == 0)
        hx -= 1;
      lx -= 1;
    }
  INSERT_WORDS (x, hx, lx);
  return x;
}
Beispiel #7
0
	double infinity()
{
	double x;

	INSERT_WORDS(x,0x7ff00000,0);
	return x;
}
/*
 * FIXME: There is a lot of redundancy between _next* and npy_nextafter*.
 * refactor this at some point
 *
 * p >= 0, returnx x + nulp
 * p < 0, returnx x - nulp
 */
double _next(double x, int p)
{
    volatile double t;
    npy_int32 hx, hy, ix;
    npy_uint32 lx;

    EXTRACT_WORDS(hx, lx, x);
    ix = hx & 0x7fffffff;       /* |x| */

    if (((ix >= 0x7ff00000) && ((ix - 0x7ff00000) | lx) != 0))        /* x is nan */
        return x;
    if ((ix | lx) == 0) {       /* x == 0 */
        if (p >= 0) {
            INSERT_WORDS(x, 0x0, 1);    /* return +minsubnormal */
        } else {
            INSERT_WORDS(x, 0x80000000, 1);    /* return -minsubnormal */
        }
        t = x * x;
        if (t == x)
            return t;
        else
            return x;           /* raise underflow flag */
    }
    if (p < 0) {     /* x -= ulp */
        if (lx == 0)
            hx -= 1;
        lx -= 1;
    } else {         /* x += ulp */
        lx += 1;
        if (lx == 0)
            hx += 1;
    }
    hy = hx & 0x7ff00000;
    if (hy >= 0x7ff00000)
        return x + x;           /* overflow  */
    if (hy < 0x00100000) {      /* underflow */
        t = x * x;
        if (t != x) {           /* raise underflow flag */
            INSERT_WORDS(x, hx, lx);
            return x;
        }
    }
    INSERT_WORDS(x, hx, lx);
    return x;
}
Beispiel #9
0
/* exp(x)/2 for x >= log(DBL_MAX), slightly better than 0.5*exp(x/2)*exp(x/2) */
double __expo2(double x)
{
    double scale;

    /* note that k is odd and scale*scale overflows */
    INSERT_WORDS(scale, (uint32_t)(0x3ff + k/2) << 20, 0);
    /* exp(x - k ln2) * 2**(k-1) */
    return exp(x - kln2) * scale * scale;
}
Beispiel #10
0
	double nan(const char *unused)
{
	double x;

#if __GNUC_PREREQ (3, 3)
	x = __builtin_nan("");
#else
	INSERT_WORDS(x,0x7ff80000,0);
#endif
	return x;
}
Beispiel #11
0
double
nexttoward(double x, long double y)
{
	union IEEEl2bits uy;
	volatile double t;
	int32_t hx,ix;
	u_int32_t lx;

	EXTRACT_WORDS(hx,lx,x);
	ix = hx&0x7fffffff;		/* |x| */
	uy.e = y;

	if(((ix>=0x7ff00000)&&((ix-0x7ff00000)|lx)!=0) ||
	    (uy.bits.exp == 0x7fff &&
	     ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl) != 0))
	   return x+y;	/* x or y is nan */
	if(x==y) return (double)y;		/* x=y, return y */
	if(x==0.0) {
	    INSERT_WORDS(x,uy.bits.sign<<31,1);	/* return +-minsubnormal */
	    t = x*x;
	    if(t==x) return t; else return x;	/* raise underflow flag */
	}
	if(hx>0.0 ^ x < y) {			/* x -= ulp */
	    if(lx==0) hx -= 1;
	    lx -= 1;
	} else {				/* x += ulp */
	    lx += 1;
	    if(lx==0) hx += 1;
	}
	ix = hx&0x7ff00000;
	if(ix>=0x7ff00000) return x+x;	/* overflow  */
	if(ix<0x00100000) {		/* underflow */
	    t = x*x;
	    if(t!=x) {		/* raise underflow flag */
	        INSERT_WORDS(x,hx,lx);
		return x;
	    }
	}
	INSERT_WORDS(x,hx,lx);
	return x;
}
Beispiel #12
0
double
cbrt(double x) 
{
	int32_t	hx;
	double r,s,t=0.0,w;
	u_int32_t sign;
	u_int32_t high,low;

	GET_HIGH_WORD(hx,x);
	sign=hx&0x80000000; 		/* sign= sign(x) */
	hx  ^=sign;
	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
	GET_LOW_WORD(low,x);
	if((hx|low)==0) 
	    return(x);		/* cbrt(0) is itself */

	SET_HIGH_WORD(x,hx);	/* x <- |x| */
    /* rough cbrt to 5 bits */
	if(hx<0x00100000) 		/* subnormal number */
	  {SET_HIGH_WORD(t,0x43500000);	/* set t= 2**54 */
	   t*=x; GET_HIGH_WORD(high,t); SET_HIGH_WORD(t,high/3+B2);
	  }
	else
	  SET_HIGH_WORD(t,hx/3+B1);


    /* new cbrt to 23 bits, may be implemented in single precision */
	r=t*t/x;
	s=C+r*t;
	t*=G+F/(s+E+D/s);	

    /* chopped to 20 bits and make it larger than cbrt(x) */ 
	GET_HIGH_WORD(high,t);
	INSERT_WORDS(t,high+0x00000001,0);


    /* one step newton iteration to 53 bits with error less than 0.667 ulps */
	s=t*t;		/* t*t is exact */
	r=x/s;
	w=t+t;
	r=(r-t)/(w+r);	/* r-s is exact */
	t=t+t*r;

    /* retore the sign bit */
	GET_HIGH_WORD(high,t);
	SET_HIGH_WORD(t,high|sign);
	return(t);
}
Beispiel #13
0
double
expm1(double x)
{
	double y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
	int32_t k,xsb;
	uint32_t hx;

	GET_HIGH_WORD(hx,x);
	xsb = hx&0x80000000;		/* sign bit of x */
	if(xsb==0) y=x; else y= -x;	/* y = |x| */
	hx &= 0x7fffffff;		/* high word of |x| */

    /* filter out huge and non-finite argument */
	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
                if(hx>=0x7ff00000) {
		    uint32_t low;
		    GET_LOW_WORD(low,x);
		    if(((hx&0xfffff)|low)!=0)
		         return x+x; 	 /* NaN */
		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
	        }
	        if(x > o_threshold) return huge*huge; /* overflow */
	    }
	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
		if(x+tiny<0.0)		/* raise inexact */
		return tiny-one;	/* return -1 */
	    }
	}

    /* argument reduction */
	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
		if(xsb==0)
		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
		else
		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
	    } else {
		k  = invln2*x+((xsb==0)?0.5:-0.5);
		t  = k;
		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
		lo = t*ln2_lo;
	    }
	    x  = hi - lo;
	    c  = (hi-x)-lo;
	}
	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
	    t = huge+x;	/* return x with inexact flags when x!=0 */
	    return x - (t-(huge+x));
	}
	else k = 0;

    /* x is now in primary range */
	hfx = 0.5*x;
	hxs = x*hfx;
	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
	t  = 3.0-r1*hfx;
	e  = hxs*((r1-t)/(6.0 - x*t));
	if(k==0) return x - (x*e-hxs);		/* c is 0 */
	else {
	    INSERT_WORDS(twopk,0x3ff00000+(k<<20),0);	/* 2^k */
	    e  = (x*(e-c)-c);
	    e -= hxs;
	    if(k== -1) return 0.5*(x-e)-0.5;
	    if(k==1) {
	       	if(x < -0.25) return -2.0*(e-(x+0.5));
	       	else 	      return  one+2.0*(x-e);
	    }
	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
	        y = one-(e-x);
		if (k == 1024) y = y*2.0*0x1p1023;
		else y = y*twopk;
	        return y-one;
	    }
	    t = one;
	    if(k<20) {
	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
	       	y = t-(e-x);
		y = y*twopk;
	   } else {
Beispiel #14
0
double
cbrt(double x)
{
	int32_t	hx;
	union {
	    double value;
	    u_int64_t bits;
	} u;
	double r,s,t=0.0,w;
	u_int32_t sign;
	u_int32_t high,low;

	EXTRACT_WORDS(hx,low,x);
	sign=hx&0x80000000; 		/* sign= sign(x) */
	hx  ^=sign;
	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */

    /*
     * Rough cbrt to 5 bits:
     *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
     * where e is integral and >= 0, m is real and in [0, 1), and "/" and
     * "%" are integer division and modulus with rounding towards minus
     * infinity.  The RHS is always >= the LHS and has a maximum relative
     * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
     * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
     * floating point representation, for finite positive normal values,
     * ordinary integer divison of the value in bits magically gives
     * almost exactly the RHS of the above provided we first subtract the
     * exponent bias (1023 for doubles) and later add it back.  We do the
     * subtraction virtually to keep e >= 0 so that ordinary integer
     * division rounds towards minus infinity; this is also efficient.
     */
	if(hx<0x00100000) { 		/* zero or subnormal? */
	    if((hx|low)==0)
		return(x);		/* cbrt(0) is itself */
	    SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
	    t*=x;
	    GET_HIGH_WORD(high,t);
	    INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
	} else
	    INSERT_WORDS(t,sign|(hx/3+B1),0);

    /*
     * New cbrt to 23 bits:
     *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
     * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
     * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
     * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
     * gives us bounds for r = t**3/x.
     *
     * Try to optimize for parallel evaluation as in k_tanf.c.
     */
	r=(t*t)*(t/x);
	t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));

    /*
     * Round t away from zero to 23 bits (sloppily except for ensuring that
     * the result is larger in magnitude than cbrt(x) but not much more than
     * 2 23-bit ulps larger).  With rounding towards zero, the error bound
     * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
     * in the rounded t, the infinite-precision error in the Newton
     * approximation barely affects third digit in the final error
     * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
     * before the final error is larger than 0.667 ulps.
     */
	u.value=t;
	u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
	t=u.value;

    /* one step Newton iteration to 53 bits with error < 0.667 ulps */
	s=t*t;				/* t*t is exact */
	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
	w=t+t;				/* t+t is exact */
	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */

	return(t);
}
Beispiel #15
0
double
roundeven (double x)
{
  uint32_t hx, lx, uhx;
  EXTRACT_WORDS (hx, lx, x);
  uhx = hx & 0x7fffffff;
  int exponent = uhx >> (MANT_DIG - 1 - 32);
  if (exponent >= BIAS + MANT_DIG - 1)
    {
      /* Integer, infinity or NaN.  */
      if (exponent == MAX_EXP)
	/* Infinity or NaN; quiet signaling NaNs.  */
	return x + x;
      else
	return x;
    }
  else if (exponent >= BIAS + MANT_DIG - 32)
    {
      /* Not necessarily an integer; integer bit is in low word.
	 Locate the bits with exponents 0 and -1.  */
      int int_pos = (BIAS + MANT_DIG - 1) - exponent;
      int half_pos = int_pos - 1;
      uint32_t half_bit = 1U << half_pos;
      uint32_t int_bit = 1U << int_pos;
      if ((lx & (int_bit | (half_bit - 1))) != 0)
	{
	  /* Carry into the exponent works correctly.  No need to test
	     whether HALF_BIT is set.  */
	  lx += half_bit;
	  hx += lx < half_bit;
	}
      lx &= ~(int_bit - 1);
    }
  else if (exponent == BIAS + MANT_DIG - 33)
    {
      /* Not necessarily an integer; integer bit is bottom of high
	 word, half bit is top of low word.  */
      if (((hx & 1) | (lx & 0x7fffffff)) != 0)
	{
	  lx += 0x80000000;
	  hx += lx < 0x80000000;
	}
      lx = 0;
    }
  else if (exponent >= BIAS)
    {
      /* At least 1; not necessarily an integer, integer bit and half
	 bit are in the high word.  Locate the bits with exponents 0
	 and -1 (when the unbiased exponent is 0, the bit with
	 exponent 0 is implicit, but as the bias is odd it is OK to
	 take it from the low bit of the exponent).  */
      int int_pos = (BIAS + MANT_DIG - 33) - exponent;
      int half_pos = int_pos - 1;
      uint32_t half_bit = 1U << half_pos;
      uint32_t int_bit = 1U << int_pos;
      if (((hx & (int_bit | (half_bit - 1))) | lx) != 0)
	hx += half_bit;
      hx &= ~(int_bit - 1);
      lx = 0;
    }
  else if (exponent == BIAS - 1 && (uhx > 0x3fe00000 || lx != 0))
    {
      /* Interval (0.5, 1).  */
      hx = (hx & 0x80000000) | 0x3ff00000;
      lx = 0;
    }
  else
    {
      /* Rounds to 0.  */
      hx &= 0x80000000;
      lx = 0;
    }
  INSERT_WORDS (x, hx, lx);
  return x;
}