double
__ieee754_exp10 (double arg)
{
int32_t lx;
double arg_high, arg_low;
double exp_high, exp_low;
if (!isfinite (arg))
return __ieee754_exp (arg);
if (arg < DBL_MIN_10_EXP - DBL_DIG - 10)
return DBL_MIN * DBL_MIN;
else if (arg > DBL_MAX_10_EXP + 1)
return DBL_MAX * DBL_MAX;
else if (fabs (arg) < 0x1p-56)
return 1.0;
GET_LOW_WORD (lx, arg);
lx &= 0xf8000000;
arg_high = arg;
SET_LOW_WORD (arg_high, lx);
arg_low = arg - arg_high;
exp_high = arg_high * log10_high;
exp_low = arg_high * log10_low + arg_low * M_LN10;
return __ieee754_exp (exp_high) * __ieee754_exp (exp_low);
}
int
__ieee754_ilogb (double x)
{
int32_t hx, lx, ix;
GET_HIGH_WORD (hx, x);
hx &= 0x7fffffff;
if (hx < 0x00100000)
{
GET_LOW_WORD (lx, x);
if ((hx | lx) == 0)
return FP_ILOGB0; /* ilogb(0) = FP_ILOGB0 */
else /* subnormal x */
if (hx == 0)
{
for (ix = -1043; lx > 0; lx <<= 1)
ix -= 1;
}
else
{
for (ix = -1022, hx <<= 11; hx > 0; hx <<= 1)
ix -= 1;
}
return ix;
}
else if (hx < 0x7ff00000)
return (hx >> 20) - 1023;
else if (FP_ILOGBNAN != INT_MAX)
double atan(double x)
{
double w,s1,s2,z;
int32_t ix,hx,id;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x44100000) { /* if |x| >= 2^66 */
uint32_t low;
GET_LOW_WORD(low, x);
if (ix > 0x7ff00000 ||
(ix == 0x7ff00000 && low != 0)) /* NaN */
return x+x;
if (hx > 0)
return atanhi[3] + *(volatile double *)&atanlo[3];
else
return -atanhi[3] - *(volatile double *)&atanlo[3];
}
if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
if (ix < 0x3e400000) { /* |x| < 2^-27 */
/* raise inexact */
if (huge+x > 1.0)
return x;
}
id = -1;
} else {
x = fabs(x);
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
if (ix < 0x3fe60000) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0*x-1.0)/(2.0+x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x-1.0)/(x+1.0);
}
} else {
if (ix < 0x40038000) { /* |x| < 2.4375 */
id = 2;
x = (x-1.5)/(1.0+1.5*x);
} else { /* 2.4375 <= |x| < 2^66 */
id = 3;
x = -1.0/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - (x*(s1+s2) - atanlo[id] - x);
return hx < 0 ? -z : z;
}
double acos(double x)
{
double z, p, q, r, w, s, c, df;
s32_t hx, ix;
GET_HIGH_WORD(hx,x);
ix = hx & 0x7fffffff;
if (ix >= 0x3ff00000)
{
u32_t lx;
GET_LOW_WORD(lx,x);
if (((ix - 0x3ff00000) | lx) == 0)
{
if (hx > 0)
return 0.0;
else
return pi + 2.0 * pio2_lo;
}
return (x - x) / (x - x);
}
if (ix < 0x3fe00000)
{
if (ix <= 0x3c600000)
return pio2_hi + pio2_lo;
z = x * x;
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
r = p / q;
return pio2_hi - (x - (pio2_lo - x * r));
}
else if (hx < 0)
{
z = (one + x) * 0.5;
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
s = sqrt(z);
r = p / q;
w = r * s - pio2_lo;
return pi - 2.0 * (s + w);
}
else
{
z = (one - x) * 0.5;
s = sqrt(z);
df = s;
SET_LOW_WORD(df,0);
c = (z - df * df) / (s + df);
p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
r = p / q;
w = r * s + c;
return 2.0 * (df + w);
}
}
double
__ieee754_cosh (double x)
{
double t, w;
int32_t ix;
u_int32_t lx;
/* High word of |x|. */
GET_HIGH_WORD (ix, x);
ix &= 0x7fffffff;
/* |x| in [0,22] */
if (ix < 0x40360000)
{
/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
if (ix < 0x3fd62e43)
{
if (ix < 0x3c800000)
return one; /* cosh(tiny) = 1 */
t = __expm1 (fabs (x));
w = one + t;
return one + (t * t) / (w + w);
}
/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
t = __ieee754_exp (fabs (x));
return half * t + half / t;
}
/* |x| in [22, log(maxdouble)] return half*exp(|x|) */
if (ix < 0x40862e42)
return half * __ieee754_exp (fabs (x));
/* |x| in [log(maxdouble), overflowthresold] */
GET_LOW_WORD (lx, x);
if (ix < 0x408633ce || ((ix == 0x408633ce) && (lx <= (u_int32_t) 0x8fb9f87d)))
{
w = __ieee754_exp (half * fabs (x));
t = half * w;
return t * w;
}
/* x is INF or NaN */
if (ix >= 0x7ff00000)
return x * x;
/* |x| > overflowthresold, cosh(x) overflow */
return huge * huge;
}
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 asin(double x)
{
double t=0.0,w,p,q,c,r,s;
int32_t hx,ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x3ff00000) { /* |x|>= 1 */
uint32_t lx;
GET_LOW_WORD(lx, x);
if ((ix-0x3ff00000 | lx) == 0)
/* asin(1) = +-pi/2 with inexact */
return x*pio2_hi + x*pio2_lo;
return (x-x)/(x-x); /* asin(|x|>1) is NaN */
} else if (ix < 0x3fe00000) { /* |x|<0.5 */
if (ix < 0x3e500000) { /* if |x| < 2**-26 */
if (huge+x > 1.0)
return x; /* return x with inexact if x!=0*/
}
t = x*x;
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
q = 1.0+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
w = p/q;
return x + x*w;
}
/* 1 > |x| >= 0.5 */
w = 1.0 - fabs(x);
t = w*0.5;
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
q = 1.0+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
s = sqrt(t);
if (ix >= 0x3FEF3333) { /* if |x| > 0.975 */
w = p/q;
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
} else {
w = s;
SET_LOW_WORD(w,0);
c = (t-w*w)/(s+w);
r = p/q;
p = 2.0*s*r-(pio2_lo-2.0*c);
q = pio4_hi - 2.0*w;
t = pio4_hi - (p-q);
}
if (hx > 0)
return t;
return -t;
}
static double sin_pi(double x)
{
double y,z;
int n,ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
if (ix < 0x3fd00000)
return __sin(pi*x, 0.0, 0);
y = -x; /* negative x is assumed */
/*
* argument reduction, make sure inexact flag not raised if input
* is an integer
*/
z = floor(y);
if (z != y) { /* inexact anyway */
y *= 0.5;
y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
n = (int)(y*4.0);
} else {
if (ix >= 0x43400000) {
y = 0.0; /* y must be even */
n = 0;
} else {
if (ix < 0x43300000)
z = y + two52; /* exact */
GET_LOW_WORD(n, z);
n &= 1;
y = n;
n <<= 2;
}
}
switch (n) {
case 0: y = __sin(pi*y, 0.0, 0); break;
case 1:
case 2: y = __cos(pi*(0.5-y), 0.0); break;
case 3:
case 4: y = __sin(pi*(1.0-y), 0.0, 0); break;
case 5:
case 6: y = -__cos(pi*(y-1.5), 0.0); break;
default: y = __sin(pi*(y-2.0), 0.0, 0); break;
}
return -y;
}
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 */
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;
}
STRICT_ASSIGN(double, x, hi - lo);
c = (hi-x)-lo;
}
else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
double
__ieee754_acos(double x)
{
double z, p, q, r, w, s, c, df;
int32_t hx, ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x3ff00000) { /* |x| >= 1 */
uint32_t lx;
GET_LOW_WORD(lx, x);
if (((ix - 0x3ff00000) | lx) == 0) { /* |x|==1 */
if (hx>0) return 0.0; /* acos(1) = 0 */
else return pi + 2.0*pio2_lo; /* acos(-1)= pi */
}
return (x - x) / (x - x); /* acos(|x|>1) is NaN */
}
if (ix<0x3fe00000) { /* |x| < 0.5 */
if (ix <= 0x3c600000) return pio2_hi + pio2_lo;/*if|x|<2**-57*/
z = x*x;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = one + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p / q;
return pio2_hi - (x - (pio2_lo - x*r));
}
else if (hx<0) { /* x < -0.5 */
z = (one + x)*0.5;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = one + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
s = sqrt(z);
r = p / q;
w = r*s - pio2_lo;
return pi - 2.0*(s + w);
}
else { /* x > 0.5 */
z = (one - x)*0.5;
s = sqrt(z);
df = s;
SET_LOW_WORD(df, 0);
c = (z - df*df) / (s + df);
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = one + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p / q;
w = r*s + c;
return 2.0*(df + w);
}
}
int ilogb(double x)
{
int32_t hx,lx,ix;
GET_HIGH_WORD(hx,x);
hx &= 0x7fffffff;
if(hx<0x00100000) {
GET_LOW_WORD(lx,x);
if((hx|lx)==0)
return 0x80000001; /* ilogb(0) = 0x80000001 */
else /* subnormal x */
if(hx==0) {
for (ix = -1043; lx>0; lx<<=1) ix -=1;
} else {
for (ix = -1022,hx<<=11; hx>0; hx<<=1) ix -=1;
}
return ix;
}
else if (hx<0x7ff00000) return (hx>>20)-1023;
else return 0x7fffffff;
double cosh(double x)
{
double t, w;
s32_t ix;
u32_t lx;
GET_HIGH_WORD(ix,x);
ix &= 0x7fffffff;
if (ix >= 0x7ff00000)
return x * x;
if (ix < 0x3fd62e43)
{
t = expm1(fabs(x));
w = one + t;
if (ix < 0x3c800000)
return w;
return one + (t * t) / (w + w);
}
if (ix < 0x40360000)
{
t = exp(fabs(x));
return half * t + half / t;
}
if (ix < 0x40862E42)
return half * exp(fabs(x));
GET_LOW_WORD(lx,x);
if (ix < 0x408633CE || ((ix == 0x408633ce)
&& (lx <= (u32_t) 0x8fb9f87d)))
{
w = exp(half * fabs(x));
t = half * w;
return t * w;
}
return huge * huge;
}
double
__ieee754_sinh(double x)
{
double t,w,h;
int32_t ix,jx;
uint32_t lx;
/* High word of |x|. */
GET_HIGH_WORD(jx,x);
ix = jx&0x7fffffff;
/* x is INF or NaN */
if(ix>=0x7ff00000) return x+x;
h = 0.5;
if (jx<0) h = -h;
/* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
if (ix < 0x40360000) { /* |x|<22 */
if (ix<0x3e300000) /* |x|<2**-28 */
if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
t = expm1(fabs(x));
if(ix<0x3ff00000) return h*(2.0*t-t*t/(t+one));
return h*(t+t/(t+one));
}
/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
if (ix < 0x40862E42) return h*__ieee754_exp(fabs(x));
/* |x| in [log(maxdouble), overflowthresold] */
GET_LOW_WORD(lx,x);
if (ix<0x408633CE || ((ix==0x408633ce)&&(lx<=(uint32_t)0x8fb9f87d))) {
w = __ieee754_exp(0.5*fabs(x));
t = h*w;
return t*w;
}
/* |x| > overflowthresold, sinh(x) overflow */
return x*shuge;
}
double __kernel_tan(double x, double y, int iy)
{
double z,r,v,w,s;
int32_t ix,hx;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff; /* high word of |x| */
if(ix<0x3e300000) /* x < 2**-28 */
{if((int)x==0) { /* generate inexact */
u_int32_t low;
GET_LOW_WORD(low,x);
if(((ix|low)|(iy+1))==0) return one/fabs(x);
else return (iy==1)? x: -one/x;
}
}
if(ix>=0x3FE59428) { /* |x|>=0.6744 */
if(hx<0) {x = -x; y = -y;}
z = pio4-x;
w = pio4lo-y;
x = z+w; y = 0.0;
}
z = x*x;
w = z*z;
/* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
s = z*x;
r = y + z*(s*(r+v)+y);
r += T[0]*s;
w = x+r;
if(ix>=0x3FE59428) {
v = (double)iy;
return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
}
/*
* exp2(x): compute the base 2 exponential of x
*
* Accuracy: Peak error < 0.503 ulp for normalized results.
*
* Method: (accurate tables)
*
* Reduce x:
* x = 2**k + y, for integer k and |y| <= 1/2.
* Thus we have exp2(x) = 2**k * exp2(y).
*
* Reduce y:
* y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
* with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
*
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
* a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
* The values in exp2t[] and eps[] are chosen such that
* exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
* that exp2t[i] is accurate to 2**-64.
*
* Note that the range of i is +-TBLSIZE/2, so we actually index the tables
* by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are
* virtual tables, interleaved in the real table tbl[].
*
* This method is due to Gal, with many details due to Gal and Bachelis:
*
* Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
* for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
*/
double exp2(double x)
{
double r, t, z;
uint32_t hx, ix, i0;
union {uint32_t u; int32_t i;} k;
/* Filter out exceptional cases. */
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x40900000) { /* |x| >= 1024 */
if (ix >= 0x7ff00000) {
GET_LOW_WORD(ix, x);
if (hx == 0xfff00000 && ix == 0) /* -inf */
return 0;
return x;
}
if (x >= 1024) {
STRICT_ASSIGN(double, x, x * 0x1p1023);
return x;
}
if (x <= -1075) {
STRICT_ASSIGN(double, x, 0x1p-1000*0x1p-1000);
return x;
}
double
__ieee754_hypot(double x, double y)
{
double a, b, t1, t2, y1, y2, w;
int32_t j, k, ha, hb;
GET_HIGH_WORD(ha, x);
ha &= 0x7fffffff;
GET_HIGH_WORD(hb, y);
hb &= 0x7fffffff;
if (hb > ha) { a = y; b = x; j = ha; ha = hb; hb = j; }
else { a = x; b = y; }
a = fabs(a);
b = fabs(b);
if ((ha - hb)>0x3c00000) { return a + b; } /* x/y > 2**60 */
k = 0;
if (ha > 0x5f300000) { /* a>2**500 */
if (ha >= 0x7ff00000) { /* Inf or NaN */
uint32_t low;
/* Use original arg order iff result is NaN; quieten sNaNs. */
w = fabs(x + 0.0) - fabs(y + 0.0);
GET_LOW_WORD(low, a);
if (((ha & 0xfffff) | low) == 0) w = a;
GET_LOW_WORD(low, b);
if (((hb ^ 0x7ff00000) | low) == 0) w = b;
return w;
}
/* scale a and b by 2**-600 */
ha -= 0x25800000; hb -= 0x25800000; k += 600;
SET_HIGH_WORD(a, ha);
SET_HIGH_WORD(b, hb);
}
if (hb < 0x20b00000) { /* b < 2**-500 */
if (hb <= 0x000fffff) { /* subnormal b or 0 */
uint32_t low;
GET_LOW_WORD(low, b);
if ((hb | low) == 0) return a;
t1 = 0;
SET_HIGH_WORD(t1, 0x7fd00000); /* t1=2^1022 */
b *= t1;
a *= t1;
k -= 1022;
}
else { /* scale a and b by 2^600 */
ha += 0x25800000; /* a *= 2^600 */
hb += 0x25800000; /* b *= 2^600 */
k -= 600;
SET_HIGH_WORD(a, ha);
SET_HIGH_WORD(b, hb);
}
}
/* medium size a and b */
w = a - b;
if (w>b) {
t1 = 0;
SET_HIGH_WORD(t1, ha);
t2 = a - t1;
w = sqrt(t1*t1 - (b*(-b) - t2*(a + t1)));
}
else {
a = a + a;
y1 = 0;
SET_HIGH_WORD(y1, hb);
y2 = b - y1;
t1 = 0;
SET_HIGH_WORD(t1, ha + 0x00100000);
t2 = a - t1;
w = sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
}
if (k != 0) {
uint32_t high;
t1 = 1.0;
GET_HIGH_WORD(high, t1);
SET_HIGH_WORD(t1, high + (k << 20));
return t1*w;
}
else return w;
}
double
expm1(double x)
{
double y,hi,lo,c,t,e,hxs,hfx,r1;
int32_t k,xsb;
uint32_t hx;
c = 0;
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 {
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 */
uint32_t high;
y = one-(e-x);
GET_HIGH_WORD(high,y);
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
return y-one;
}
t = one;
if(k<20) {
uint32_t high;
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
y = t-(e-x);
GET_HIGH_WORD(high,y);
SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */
} else {
double
__ieee754_hypot (double x, double y)
{
double a, b, t1, t2, y1, y2, w;
int32_t j, k, ha, hb;
GET_HIGH_WORD (ha, x);
ha &= 0x7fffffff;
GET_HIGH_WORD (hb, y);
hb &= 0x7fffffff;
if (hb > ha)
{
a = y; b = x; j = ha; ha = hb; hb = j;
}
else
{
a = x; b = y;
}
SET_HIGH_WORD (a, ha); /* a <- |a| */
SET_HIGH_WORD (b, hb); /* b <- |b| */
if ((ha - hb) > 0x3c00000)
{
return a + b;
} /* x/y > 2**60 */
k = 0;
if (__glibc_unlikely (ha > 0x5f300000)) /* a>2**500 */
{
if (ha >= 0x7ff00000) /* Inf or NaN */
{
u_int32_t low;
w = a + b; /* for sNaN */
GET_LOW_WORD (low, a);
if (((ha & 0xfffff) | low) == 0)
w = a;
GET_LOW_WORD (low, b);
if (((hb ^ 0x7ff00000) | low) == 0)
w = b;
return w;
}
/* scale a and b by 2**-600 */
ha -= 0x25800000; hb -= 0x25800000; k += 600;
SET_HIGH_WORD (a, ha);
SET_HIGH_WORD (b, hb);
}
if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */
{
if (hb <= 0x000fffff) /* subnormal b or 0 */
{
u_int32_t low;
GET_LOW_WORD (low, b);
if ((hb | low) == 0)
return a;
t1 = 0;
SET_HIGH_WORD (t1, 0x7fd00000); /* t1=2^1022 */
b *= t1;
a *= t1;
k -= 1022;
GET_HIGH_WORD (ha, a);
GET_HIGH_WORD (hb, b);
if (hb > ha)
{
t1 = a;
a = b;
b = t1;
j = ha;
ha = hb;
hb = j;
}
}
else /* scale a and b by 2^600 */
{
ha += 0x25800000; /* a *= 2^600 */
hb += 0x25800000; /* b *= 2^600 */
k -= 600;
SET_HIGH_WORD (a, ha);
SET_HIGH_WORD (b, hb);
}
}
/* medium size a and b */
w = a - b;
if (w > b)
{
t1 = 0;
SET_HIGH_WORD (t1, ha);
t2 = a - t1;
w = __ieee754_sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1)));
}
else
{
a = a + a;
y1 = 0;
SET_HIGH_WORD (y1, hb);
y2 = b - y1;
t1 = 0;
SET_HIGH_WORD (t1, ha + 0x00100000);
t2 = a - t1;
w = __ieee754_sqrt (t1 * y1 - (w * (-w) - (t1 * y2 + t2 * b)));
}
if (k != 0)
{
u_int32_t high;
t1 = 1.0;
GET_HIGH_WORD (high, t1);
SET_HIGH_WORD (t1, high + (k << 20));
w *= t1;
math_check_force_underflow_nonneg (w);
return w;
}
else
return w;
}
double expm1(double x)
{
double y, hi, lo, c=0, t, e, hxs, hfx, r1;
s32_t k, xsb;
u32_t hx;
GET_HIGH_WORD(hx,x);
xsb = hx & 0x80000000;
if (xsb == 0)
y = x;
else
y = -x;
hx &= 0x7fffffff;
if (hx >= 0x4043687A)
{
if (hx >= 0x40862E42)
{
if (hx >= 0x7ff00000)
{
u32_t low;
GET_LOW_WORD(low,x);
if (((hx & 0xfffff) | low) != 0)
return x + x;
else
return (xsb == 0) ? x : -1.0;
}
if (x > o_threshold)
return huge * huge;
}
if (xsb != 0)
{
if (x + tiny < 0.0)
return tiny - one;
}
}
if (hx > 0x3fd62e42)
{
if (hx < 0x3FF0A2B2)
{
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;
lo = t * ln2_lo;
}
x = hi - lo;
c = (hi - x) - lo;
}
else if (hx < 0x3c900000)
{
t = huge + x;
return x - (t - (huge + x));
}
else
k = 0;
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);
else
{
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)
{
u32_t high;
y = one - (e - x);
GET_HIGH_WORD(high,y);
SET_HIGH_WORD(y,high+(k<<20));
return y - one;
}
t = one;
if (k < 20)
{
u32_t high;
SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));
y = t - (e - x);
GET_HIGH_WORD(high,y);
SET_HIGH_WORD(y,high+(k<<20));
}
else
{
double attribute_hidden __ieee754_hypot(double x, double y)
{
double a=x,b=y,t1,t2,y1,y2,w;
int32_t j,k,ha,hb;
GET_HIGH_WORD(ha,x);
ha &= 0x7fffffff;
GET_HIGH_WORD(hb,y);
hb &= 0x7fffffff;
if(hb > ha) {
a=y;
b=x;
j=ha;
ha=hb;
hb=j;
}
else {
a=x;
b=y;
}
SET_HIGH_WORD(a,ha); /* a <- |a| */
SET_HIGH_WORD(b,hb); /* b <- |b| */
if((ha-hb)>0x3c00000) {
return a+b; /* x/y > 2**60 */
}
k=0;
if(ha > 0x5f300000) { /* a>2**500 */
if(ha >= 0x7ff00000) { /* Inf or NaN */
u_int32_t low;
w = a+b; /* for sNaN */
GET_LOW_WORD(low,a);
if(((ha&0xfffff)|low)==0) w = a;
GET_LOW_WORD(low,b);
if(((hb^0x7ff00000)|low)==0) w = b;
return w;
}
/* scale a and b by 2**-600 */
ha -= 0x25800000;
hb -= 0x25800000;
k += 600;
SET_HIGH_WORD(a,ha);
SET_HIGH_WORD(b,hb);
}
if(hb < 0x20b00000) { /* b < 2**-500 */
if(hb <= 0x000fffff) { /* subnormal b or 0 */
u_int32_t low;
GET_LOW_WORD(low,b);
if((hb|low)==0) return a;
t1=0;
SET_HIGH_WORD(t1,0x7fd00000); /* t1=2^1022 */
b *= t1;
a *= t1;
k -= 1022;
} else { /* scale a and b by 2^600 */
ha += 0x25800000; /* a *= 2^600 */
hb += 0x25800000; /* b *= 2^600 */
k -= 600;
SET_HIGH_WORD(a,ha);
SET_HIGH_WORD(b,hb);
}
}
/* medium size a and b */
w = a-b;
if (w>b) {
t1 = 0;
SET_HIGH_WORD(t1,ha);
t2 = a-t1;
w = __ieee754_sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
} else {
a = a+a;
y1 = 0;
SET_HIGH_WORD(y1,hb);
y2 = b - y1;
t1 = 0;
SET_HIGH_WORD(t1,ha+0x00100000);
t2 = a - t1;
w = __ieee754_sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
}
if(k!=0) {
u_int32_t high;
t1 = 1.0;
GET_HIGH_WORD(high,t1);
SET_HIGH_WORD(t1,high+(k<<20));
return t1*w;
} else return w;
}
