math: tan cleanups

* use unsigned arithmetics on the representation
* store arg reduction quotient in unsigned (so n%2 would work like n&1)
* use different convention to pass the arg reduction bit to __tan
  (this argument used to be 1 for even and -1 for odd reduction
  which meant obscure bithacks, the new n&1 is cleaner)
* raise inexact and underflow flags correctly for small x
  (tanl(x) may still raise spurious underflow for small but normal x)
  (this exception raising code increases codesize a bit, similar fixes
  are needed in many other places, it may worth investigating at some
  point if the inexact and underflow flags are worth raising correctly
  as this is not strictly required by the standard)
* tanf manual reduction optimization is kept for now
* tanl code path is cleaned up to follow similar logic to tan and tanf

1 files changed, 24 insertions, 33 deletions

diff --git a/src/math/__tan.c b/src/math/__tan.c
index fc739f95..8019844d 100644
--- a/src/math/__tan.c
+++ b/src/math/__tan.c
@@ -12,7 +12,7 @@
  * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  * Input y is the tail of x.
- * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned.
+ * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
  *
  * Algorithm
  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
@@ -63,21 +63,22 @@ static const double T[] = {
 pio4 =       7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
 pio4lo =     3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
 
-double __tan(double x, double y, int iy)
+double __tan(double x, double y, int odd)
 {
-	double_t z, r, v, w, s, sign;
-	int32_t ix, hx;
+	double_t z, r, v, w, s, a;
+	double w0, a0;
+	uint32_t hx;
+	int big, sign;
 
 	GET_HIGH_WORD(hx,x);
-	ix = hx & 0x7fffffff;    /* high word of |x| */
-	if (ix >= 0x3FE59428) {  /* |x| >= 0.6744 */
-		if (hx < 0) {
+	big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
+	if (big) {
+		sign = hx>>31;
+		if (sign) {
 			x = -x;
 			y = -y;
 		}
-		z = pio4 - x;
-		w = pio4lo - y;
-		x = z + w;
+		x = (pio4 - x) + (pio4lo - y);
 		y = 0.0;
 	}
 	z = x * x;
@@ -90,30 +91,20 @@ double __tan(double x, double y, int iy)
 	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;
+	r = y + z*(s*(r + v) + y) + s*T[0];
 	w = x + r;
-	if (ix >= 0x3FE59428) {
-		v = iy;
-		sign = 1 - ((hx >> 30) & 2);
-		return sign * (v - 2.0 * (x - (w * w / (w + v) - r)));
+	if (big) {
+		s = 1 - 2*odd;
+		v = s - 2.0 * (x + (r - w*w/(w + s)));
+		return sign ? -v : v;
 	}
-	if (iy == 1)
+	if (!odd)
 		return w;
-	else {
-		/*
-		 * if allow error up to 2 ulp, simply return
-		 * -1.0 / (x+r) here
-		 */
-		/* compute -1.0 / (x+r) accurately */
-		double_t a;
-		double z, t;
-		z = w;
-		SET_LOW_WORD(z,0);
-		v = r - (z - x);        /* z+v = r+x */
-		t = a = -1.0 / w;       /* a = -1.0/w */
-		SET_LOW_WORD(t,0);
-		s = 1.0 + t * z;
-		return t + a * (s + t * v);
-	}
+	/* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
+	w0 = w;
+	SET_LOW_WORD(w0, 0);
+	v = r - (w0 - x);       /* w0+v = r+x */
+	a0 = a = -1.0 / w;
+	SET_LOW_WORD(a0, 0);
+	return a0 + a*(1.0 + a0*w0 + a0*v);
 }