+#include <float.h>
#include "__invtrigl.h"
#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-
static const long double
pS0 = 1.66666666666666666631e-01L,
pS1 = -4.16313987993683104320e-01L,
q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*(qS4+z*qS5))));
return p/q;
}
+#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
+static const long double
+pS0 = 1.66666666666666666666666666666700314e-01L,
+pS1 = -7.32816946414566252574527475428622708e-01L,
+pS2 = 1.34215708714992334609030036562143589e+00L,
+pS3 = -1.32483151677116409805070261790752040e+00L,
+pS4 = 7.61206183613632558824485341162121989e-01L,
+pS5 = -2.56165783329023486777386833928147375e-01L,
+pS6 = 4.80718586374448793411019434585413855e-02L,
+pS7 = -4.42523267167024279410230886239774718e-03L,
+pS8 = 1.44551535183911458253205638280410064e-04L,
+pS9 = -2.10558957916600254061591040482706179e-07L,
+qS1 = -4.84690167848739751544716485245697428e+00L,
+qS2 = 9.96619113536172610135016921140206980e+00L,
+qS3 = -1.13177895428973036660836798461641458e+01L,
+qS4 = 7.74004374389488266169304117714658761e+00L,
+qS5 = -3.25871986053534084709023539900339905e+00L,
+qS6 = 8.27830318881232209752469022352928864e-01L,
+qS7 = -1.18768052702942805423330715206348004e-01L,
+qS8 = 8.32600764660522313269101537926539470e-03L,
+qS9 = -1.99407384882605586705979504567947007e-04L;
+
+const long double pio2_hi = 1.57079632679489661923132169163975140L;
+const long double pio2_lo = 4.33590506506189051239852201302167613e-35L;
+
+long double __invtrigl_R(long double z)
+{
+ long double p, q;
+ p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*(pS5+z*(pS6+z*(pS7+z*(pS8+z*pS9)))))))));
+ q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*(qS4+z*(qS5+z*(pS6+z*(pS7+z*(pS8+z*pS9))))))));
+ return p/q;
+}
#endif
-#include <float.h>
-
-#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
/* shared by acosl, asinl and atan2l */
#define pio2_hi __pio2_hi
#define pio2_lo __pio2_lo
extern const long double pio2_hi, pio2_lo;
long double __invtrigl_R(long double z);
-#endif
}
#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
#include "__invtrigl.h"
+#if LDBL_MANT_DIG == 64
+#define CLEARBOTTOM(u) (u.i.m &= -1ULL << 32)
+#elif LDBL_MANT_DIG == 113
+#define CLEARBOTTOM(u) (u.i.lo = 0)
+#endif
long double acosl(long double x)
{
- union IEEEl2bits u;
- long double z, w, s, c, df;
- int16_t expsign, expt;
- u.e = x;
- expsign = u.xbits.expsign;
- expt = expsign & 0x7fff;
+ union ldshape u = {x};
+ long double z, s, c, f;
+ uint16_t e = u.i.se & 0x7fff;
+
/* |x| >= 1 or nan */
- if (expt >= 0x3fff) {
- if (expt == 0x3fff &&
- ((u.bits.manh & ~LDBL_NBIT) | u.bits.manl) == 0) {
- if (expsign > 0)
- return 0; /* acos(1) = 0 */
- return 2*pio2_hi + 0x1p-120f; /* acos(-1)= pi */
- }
- return 0/(x-x); /* acos(|x|>1) is NaN */
+ if (e >= 0x3fff) {
+ if (x == 1)
+ return 0;
+ if (x == -1)
+ return 2*pio2_hi + 0x1p-120f;
+ return 0/(x-x);
}
/* |x| < 0.5 */
- if (expt < 0x3fff - 1) {
- if (expt < 0x3fff - 65)
- return pio2_hi + 0x1p-120f; /* x < 0x1p-65: acosl(x)=pi/2 */
- return pio2_hi - (x - (pio2_lo - x * __invtrigl_R(x*x)));
+ if (e < 0x3fff - 1) {
+ if (e < 0x3fff - LDBL_MANT_DIG - 1)
+ return pio2_hi + 0x1p-120f;
+ return pio2_hi - (__invtrigl_R(x*x)*x - pio2_lo + x);
}
/* x < -0.5 */
- if (expsign < 0) {
- z = (1.0 + x) * 0.5;
+ if (u.i.se >> 15) {
+ z = (1 + x)*0.5;
s = sqrtl(z);
- w = __invtrigl_R(z) * s - pio2_lo;
- return 2*(pio2_hi - (s + w));
+ return 2*(pio2_hi - (__invtrigl_R(z)*s - pio2_lo + s));
}
/* x > 0.5 */
- z = (1.0 - x) * 0.5;
+ z = (1 - x)*0.5;
s = sqrtl(z);
- u.e = s;
- u.bits.manl = 0;
- df = u.e;
- c = (z - df * df) / (s + df);
- w = __invtrigl_R(z) * s + c;
- return 2*(df + w);
+ u.f = s;
+ CLEARBOTTOM(u);
+ f = u.f;
+ c = (z - f*f)/(s + f);
+ return 2*(__invtrigl_R(z)*s + c + f);
}
#endif
}
#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
#include "__invtrigl.h"
-/* 0.95 */
-#define THRESH ((0xe666666666666666ULL>>(64-(LDBL_MANH_SIZE-1)))|LDBL_NBIT)
+#if LDBL_MANT_DIG == 64
+#define CLOSETO1(u) (u.i.m>>56 >= 0xf7)
+#define CLEARBOTTOM(u) (u.i.m &= -1ULL << 32)
+#elif LDBL_MANT_DIG == 113
+#define CLOSETO1(u) (u.i.top >= 0xee00)
+#define CLEARBOTTOM(u) (u.i.lo = 0)
+#endif
long double asinl(long double x)
{
- union IEEEl2bits u;
- long double z,r,s;
- uint16_t expsign, expt;
+ union ldshape u = {x};
+ long double z, r, s;
+ uint16_t e = u.i.se & 0x7fff;
+ int sign = u.i.se >> 15;
- u.e = x;
- expsign = u.xbits.expsign;
- expt = expsign & 0x7fff;
- if (expt >= 0x3fff) { /* |x| >= 1 or nan */
- if (expt == 0x3fff &&
- ((u.bits.manh&~LDBL_NBIT)|u.bits.manl) == 0)
- /* asin(+-1)=+-pi/2 with inexact */
+ if (e >= 0x3fff) { /* |x| >= 1 or nan */
+ /* asin(+-1)=+-pi/2 with inexact */
+ if (x == 1 || x == -1)
return x*pio2_hi + 0x1p-120f;
return 0/(x-x);
}
- if (expt < 0x3fff - 1) { /* |x| < 0.5 */
- if (expt < 0x3fff - 32) { /* |x|<0x1p-32, asinl(x)=x */
+ if (e < 0x3fff - 1) { /* |x| < 0.5 */
+ if (e < 0x3fff - (LDBL_MANT_DIG+1)/2) {
/* return x with inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
return x;
z = (1.0 - fabsl(x))*0.5;
s = sqrtl(z);
r = __invtrigl_R(z);
- if (u.bits.manh >= THRESH) { /* if |x| is close to 1 */
+ if (CLOSETO1(u)) {
x = pio2_hi - (2*(s+s*r)-pio2_lo);
} else {
long double f, c;
- u.e = s;
- u.bits.manl = 0;
- f = u.e;
- c = (z-f*f)/(s+f);
+ u.f = s;
+ CLEARBOTTOM(u);
+ f = u.f;
+ c = (z - f*f)/(s + f);
x = 0.5*pio2_hi-(2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f));
}
- if (expsign>>15)
- return -x;
- return x;
+ return sign ? -x : x;
}
#endif
long double atan2l(long double y, long double x)
{
- union IEEEl2bits ux, uy;
+ union ldshape ux, uy;
long double z;
- int m;
- uint16_t exptx, expsignx, expty, expsigny;
+ int m, ex, ey;
if (isnan(x) || isnan(y))
return x+y;
if (x == 1)
return atanl(y);
- uy.e = y;
- expsigny = uy.xbits.expsign;
- expty = expsigny & 0x7fff;
- ux.e = x;
- expsignx = ux.xbits.expsign;
- exptx = expsignx & 0x7fff;
- m = ((expsigny>>15)&1) | ((expsignx>>14)&2); /* 2*sign(x)+sign(y) */
-
+ ux.f = x;
+ uy.f = y;
+ ex = ux.i.se & 0x7fff;
+ ey = uy.i.se & 0x7fff;
+ m = 2*(ux.i.se>>15) | uy.i.se>>15;
if (y == 0) {
switch(m) {
case 0:
case 1: return y; /* atan(+-0,+anything)=+-0 */
- case 2: return 2*pio2_hi+0x1p-120f; /* atan(+0,-anything) = pi */
- case 3: return -2*pio2_hi-0x1p-120f; /* atan(-0,-anything) =-pi */
+ case 2: return 2*pio2_hi; /* atan(+0,-anything) = pi */
+ case 3: return -2*pio2_hi; /* atan(-0,-anything) =-pi */
}
}
if (x == 0)
- return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
- /* when x is INF */
- if (exptx == 0x7fff) {
- if (expty == 0x7fff) {
+ return m&1 ? -pio2_hi : pio2_hi;
+ if (ex == 0x7fff) {
+ if (ey == 0x7fff) {
switch(m) {
- case 0: return pio2_hi*0.5+0x1p-120f; /* atan(+INF,+INF) */
- case 1: return -pio2_hi*0.5-0x1p-120f; /* atan(-INF,+INF) */
- case 2: return 1.5*pio2_hi+0x1p-120f; /* atan(+INF,-INF) */
- case 3: return -1.5*pio2_hi-0x1p-120f; /* atan(-INF,-INF) */
+ case 0: return pio2_hi/2; /* atan(+INF,+INF) */
+ case 1: return -pio2_hi/2; /* atan(-INF,+INF) */
+ case 2: return 1.5*pio2_hi; /* atan(+INF,-INF) */
+ case 3: return -1.5*pio2_hi; /* atan(-INF,-INF) */
}
} else {
switch(m) {
case 0: return 0.0; /* atan(+...,+INF) */
case 1: return -0.0; /* atan(-...,+INF) */
- case 2: return 2*pio2_hi+0x1p-120f; /* atan(+...,-INF) */
- case 3: return -2*pio2_hi-0x1p-120f; /* atan(-...,-INF) */
+ case 2: return 2*pio2_hi; /* atan(+...,-INF) */
+ case 3: return -2*pio2_hi; /* atan(-...,-INF) */
}
}
}
- /* when y is INF */
- if (exptx+120 < expty || expty == 0x7fff)
- return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
-
- if ((m&2) && expty+120 < exptx) /* |y/x| tiny, x<0 */
+ if (ex+120 < ey || ey == 0x7fff)
+ return m&1 ? -pio2_hi : pio2_hi;
+ /* z = atan(|y/x|) without spurious underflow */
+ if ((m&2) && ey+120 < ex) /* |y/x| < 0x1p-120, x<0 */
z = 0.0;
else
z = atanl(fabsl(y/x));
}
#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
+#if LDBL_MANT_DIG == 64
+#define EXPMAN(u) ((u.i.se & 0x7fff)<<8 | (u.i.m>>55 & 0xff))
+
static const long double atanhi[] = {
4.63647609000806116202e-01L,
7.85398163397448309628e-01L,
return aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] +
x * (aT[9] + x * aT[11]))));
}
+#elif LDBL_MANT_DIG == 113
+#define EXPMAN(u) ((u.i.se & 0x7fff)<<8 | u.i.top>>8)
+
+const long double atanhi[] = {
+ 4.63647609000806116214256231461214397e-01L,
+ 7.85398163397448309615660845819875699e-01L,
+ 9.82793723247329067985710611014666038e-01L,
+ 1.57079632679489661923132169163975140e+00L,
+};
+
+const long double atanlo[] = {
+ 4.89509642257333492668618435220297706e-36L,
+ 2.16795253253094525619926100651083806e-35L,
+ -2.31288434538183565909319952098066272e-35L,
+ 4.33590506506189051239852201302167613e-35L,
+};
+
+const long double aT[] = {
+ 3.33333333333333333333333333333333125e-01L,
+ -1.99999999999999999999999999999180430e-01L,
+ 1.42857142857142857142857142125269827e-01L,
+ -1.11111111111111111111110834490810169e-01L,
+ 9.09090909090909090908522355708623681e-02L,
+ -7.69230769230769230696553844935357021e-02L,
+ 6.66666666666666660390096773046256096e-02L,
+ -5.88235294117646671706582985209643694e-02L,
+ 5.26315789473666478515847092020327506e-02L,
+ -4.76190476189855517021024424991436144e-02L,
+ 4.34782608678695085948531993458097026e-02L,
+ -3.99999999632663469330634215991142368e-02L,
+ 3.70370363987423702891250829918659723e-02L,
+ -3.44827496515048090726669907612335954e-02L,
+ 3.22579620681420149871973710852268528e-02L,
+ -3.03020767654269261041647570626778067e-02L,
+ 2.85641979882534783223403715930946138e-02L,
+ -2.69824879726738568189929461383741323e-02L,
+ 2.54194698498808542954187110873675769e-02L,
+ -2.35083879708189059926183138130183215e-02L,
+ 2.04832358998165364349957325067131428e-02L,
+ -1.54489555488544397858507248612362957e-02L,
+ 8.64492360989278761493037861575248038e-03L,
+ -2.58521121597609872727919154569765469e-03L,
+};
+
+static long double T_even(long double x)
+{
+ return (aT[0] + x * (aT[2] + x * (aT[4] + x * (aT[6] + x * (aT[8] +
+ x * (aT[10] + x * (aT[12] + x * (aT[14] + x * (aT[16] +
+ x * (aT[18] + x * (aT[20] + x * aT[22])))))))))));
+}
+
+static long double T_odd(long double x)
+{
+ return (aT[1] + x * (aT[3] + x * (aT[5] + x * (aT[7] + x * (aT[9] +
+ x * (aT[11] + x * (aT[13] + x * (aT[15] + x * (aT[17] +
+ x * (aT[19] + x * (aT[21] + x * aT[23])))))))))));
+}
+#endif
long double atanl(long double x)
{
- union IEEEl2bits u;
- long double w,s1,s2,z;
+ union ldshape u = {x};
+ long double w, s1, s2, z;
int id;
- uint16_t expsign, expt;
- uint32_t expman;
+ unsigned e = u.i.se & 0x7fff;
+ unsigned sign = u.i.se >> 15;
+ unsigned expman;
- u.e = x;
- expsign = u.xbits.expsign;
- expt = expsign & 0x7fff;
- if (expt >= 0x3fff + 65) { /* if |x| is large, atan(x)~=pi/2 */
- if (expt == 0x7fff &&
- ((u.bits.manh&~LDBL_NBIT)|u.bits.manl)!=0) /* NaN */
- return x+x;
- z = atanhi[3] + 0x1p-120f;
- return expsign>>15 ? -z : z;
+ if (e >= 0x3fff + LDBL_MANT_DIG + 1) { /* if |x| is large, atan(x)~=pi/2 */
+ if (isnan(x))
+ return x;
+ return sign ? -atanhi[3] : atanhi[3];
}
/* Extract the exponent and the first few bits of the mantissa. */
- /* XXX There should be a more convenient way to do this. */
- expman = (expt << 8) | ((u.bits.manh >> (LDBL_MANH_SIZE - 9)) & 0xff);
+ expman = EXPMAN(u);
if (expman < ((0x3fff - 2) << 8) + 0xc0) { /* |x| < 0.4375 */
- if (expt < 0x3fff - 32) { /* if |x| is small, atanl(x)~=x */
+ if (e < 0x3fff - (LDBL_MANT_DIG+1)/2) { /* if |x| is small, atanl(x)~=x */
/* raise underflow if subnormal */
- if (expt == 0)
+ if (e == 0)
FORCE_EVAL((float)x);
return x;
}
if (expman < ((0x3fff + 1) << 8) + 0x38) { /* |x| < 2.4375 */
id = 2;
x = (x-1.5)/(1.0+1.5*x);
- } else { /* 2.4375 <= |x| < 2^ATAN_CONST */
+ } else { /* 2.4375 <= |x| */
id = 3;
x = -1.0/x;
}
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
- return expsign>>15 ? -z : z;
+ return sign ? -z : z;
}
#endif
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