math: long double inverse trigonometric cleanup (acosl, asinl, atanl, atan2l)

* added ld128 support from freebsd fdlibm (untested)
* using new ldshape union instead of IEEEl2bits
* inexact status flag is not supported

src/math/__invtrigl.c

+#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,
@@ -28,4 +28,36 @@ long double __invtrigl_R(long double z)
 	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

src/math/__invtrigl.h

-#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

src/math/acosl.c

@@ -23,46 +23,45 @@ long double acosl(long double x)
 }
 #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

src/math/asinl.c

@@ -23,27 +23,29 @@ long double asinl(long double x)
 }
 #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;
@@ -54,18 +56,16 @@ long double asinl(long double 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

src/math/atan2l.c

@@ -27,56 +27,50 @@ long double atan2l(long double y, long double x)
 
 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));

src/math/atanl.c

@@ -23,6 +23,9 @@ long double atanl(long double 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,
@@ -64,32 +67,85 @@ 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]))));
 }
+#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;
 		}
@@ -108,7 +164,7 @@ long double atanl(long double 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;
 			}
@@ -123,6 +179,6 @@ long double atanl(long double 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