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;
}
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;
}
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;
}
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));
}
}
/* 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;
}
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;
}
/* 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;
}
double nan(const char *unused)
{
double x;
#if __GNUC_PREREQ (3, 3)
x = __builtin_nan("");
#else
INSERT_WORDS(x,0x7ff80000,0);
#endif
return x;
}
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;
}
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);
}
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 {
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);
}
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;
}