math: long double trigonometric cleanup (cosl, sinl, sincosl, tanl)

ld128 support was added to internal kernel functions (__cosl, __sinl,
__tanl, __rem_pio2l) from freebsd (not tested, but should be a good
start for when ld128 arch arrives)

__rem_pio2l had some code cleanup, the freebsd ld128 code seems to
gather the results of a large reduction with precision loss (fixed
the bug but a todo comment was added for later investigation)

the old copyright was removed from the non-kernel wrapper functions
(cosl, sinl, sincosl, tanl) since these are trivial and the interesting
parts and comments had been already rewritten.

1 files changed, 64 insertions, 53 deletions

diff --git a/src/math/__rem_pio2l.c b/src/math/__rem_pio2l.c
index f0cb99ac..8b15b7b2 100644
--- a/src/math/__rem_pio2l.c
+++ b/src/math/__rem_pio2l.c
@@ -13,15 +13,22 @@
  * Optimized by Bruce D. Evans.
  */
 #include "libm.h"
-#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-/* ld80 version of __rem_pio2(x,y)
+#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
+/* ld80 and ld128 version of __rem_pio2(x,y)
  *
  * return the remainder of x rem pi/2 in y[0]+y[1]
  * use __rem_pio2_large() for large x
  */
 
-#define BIAS    (LDBL_MAX_EXP - 1)
-
+#if LDBL_MANT_DIG == 64
+/* u ~< 0x1p25*pi/2 */
+#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
+#define TOINT 0x1.8p63
+#define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
+#define ROUND1 22
+#define ROUND2 61
+#define NX 3
+#define NY 2
 /*
  * invpio2:  64 bits of 2/pi
  * pio2_1:   first  39 bits of pi/2
@@ -32,60 +39,61 @@
  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  */
 static const double
-two24  =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
 pio2_1 =  1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
 pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
 pio2_3 =  6.36831716351370313614e-25; /*  0x18a2e037074000.0p-133 */
-
 static const long double
 invpio2 =  6.36619772367581343076e-01L, /*  0xa2f9836e4e44152a.0p-64 */
 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
 pio2_2t =  6.36831716351095013979e-25L, /*  0xc51701b839a25205.0p-144 */
 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
+#elif LDBL_MANT_DIG == 113
+/* u ~< 0x1p45*pi/2 */
+#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
+#define TOINT 0x1.8p112
+#define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
+#define ROUND1 51
+#define ROUND2 119
+#define NX 5
+#define NY 3
+static const long double
+invpio2 =  6.3661977236758134307553505349005747e-01L,	/*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
+pio2_1  =  1.5707963267948966192292994253909555e+00L,	/*  0x1921fb54442d18469800000000000.0p-112 */
+pio2_1t =  2.0222662487959507323996846200947577e-21L,	/*  0x13198a2e03707344a4093822299f3.0p-181 */
+pio2_2  =  2.0222662487959507323994779168837751e-21L,	/*  0x13198a2e03707344a400000000000.0p-181 */
+pio2_2t =  2.0670321098263988236496903051604844e-43L,	/*  0x127044533e63a0105df531d89cd91.0p-254 */
+pio2_3  =  2.0670321098263988236499468110329591e-43L,	/*  0x127044533e63a0105e00000000000.0p-254 */
+pio2_3t = -2.5650587247459238361625433492959285e-65L;	/* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
+#endif
 
 int __rem_pio2l(long double x, long double *y)
 {
-	union IEEEl2bits u,u1;
+	union ldshape u,uz;
 	long double z,w,t,r,fn;
-	double tx[3],ty[2];
-	int e0,ex,i,j,nx,n;
-	int16_t expsign;
-
-	u.e = x;
-	expsign = u.xbits.expsign;
-	ex = expsign & 0x7fff;
-	if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) {
-		union IEEEl2bits u2;
-		int ex1;
+	double tx[NX],ty[NY];
+	int ex,ey,n,i;
 
-		/* |x| ~< 2^25*(pi/2), medium size */
-		/* Use a specialized rint() to get fn.  Assume round-to-nearest. */
-		fn = x*invpio2 + 0x1.8p63;
-		fn = fn - 0x1.8p63;
-// FIXME
-//#ifdef HAVE_EFFICIENT_IRINT
-//		n = irint(fn);
-//#else
-		n = fn;
-//#endif
+	u.f = x;
+	ex = u.i.se & 0x7fff;
+	if (SMALL(u)) {
+		/* rint(x/(pi/2)), Assume round-to-nearest. */
+		fn = x*invpio2 + TOINT - TOINT;
+		n = QUOBITS(fn);
 		r = x-fn*pio2_1;
-		w = fn*pio2_1t;    /* 1st round good to 102 bit */
-		j = ex;
+		w = fn*pio2_1t;  /* 1st round good to 102/180 bits (ld80/ld128) */
 		y[0] = r-w;
-		u2.e = y[0];
-		ex1 = u2.xbits.expsign & 0x7fff;
-		i = j-ex1;
-		if (i > 22) {  /* 2nd iteration needed, good to 141 */
+		u.f = y[0];
+		ey = u.i.se & 0x7fff;
+		if (ex - ey > ROUND1) {  /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
 			t = r;
 			w = fn*pio2_2;
 			r = t-w;
 			w = fn*pio2_2t-((t-r)-w);
 			y[0] = r-w;
-			u2.e = y[0];
-			ex1 = u2.xbits.expsign & 0x7fff;
-			i = j-ex1;
-			if (i > 61) {  /* 3rd iteration need, 180 bits acc */
-				t = r; /* will cover all possible cases */
+			u.f = y[0];
+			ey = u.i.se & 0x7fff;
+			if (ex - ey > ROUND2) {  /* 3rd iteration, good to 180/316 bits */
+				t = r; /* will cover all possible cases (not verified for ld128) */
 				w = fn*pio2_3;
 				r = t-w;
 				w = fn*pio2_3t-((t-r)-w);
@@ -102,23 +110,26 @@ int __rem_pio2l(long double x, long double *y)
 		y[0] = y[1] = x - x;
 		return 0;
 	}
-	/* set z = scalbn(|x|,ilogb(x)-23) */
-	u1.e = x;
-	e0 = ex - BIAS - 23;            /* e0 = ilogb(|x|)-23; */
-	u1.xbits.expsign = ex - e0;
-	z = u1.e;
-	for (i=0; i<2; i++) {
+	/* set z = scalbn(|x|,-ilogb(x)+23) */
+	uz.f = x;
+	uz.i.se = 0x3fff + 23;
+	z = uz.f;
+	for (i=0; i < NX - 1; i++) {
 		tx[i] = (double)(int32_t)z;
-		z     = (z-tx[i])*two24;
+		z     = (z-tx[i])*0x1p24;
 	}
-	tx[2] = z;
-	nx = 3;
-	while (tx[nx-1] == 0.0)
-		nx--;     /* skip zero term */
-	n = __rem_pio2_large(tx,ty,e0,nx,2);
-	r = (long double)ty[0] + ty[1];
-	w = ty[1] - (r - ty[0]);
-	if (expsign < 0) {
+	tx[i] = z;
+	while (tx[i] == 0)
+		i--;
+	n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
+	w = ty[1];
+	if (NY == 3)
+		w += ty[2];
+	r = ty[0] + w;
+	/* TODO: for ld128 this does not follow the recommendation of the
+	comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
+	w -= r - ty[0];
+	if (u.i.se >> 15) {
 		y[0] = -r;
 		y[1] = -w;
 		return -n;