The value small=2**-(p+3), where p is the precision, can be determine from

lgamma(x) = -log(x) - log(1+x) + x*(1-g) + x**2*P(x) with g = 0.57...
being the Euler constant and P(x) a polynomial.  Substitution of small
into the RHS shows that the last 3 terms are negligible in comparison to
the leading term.  The choice of 3 may be conservative.

The value large=2**(p+3) is detemined from Stirling's approximation
lgamma(x) = x*(log(x)-1) - log(x)/2 + log(2*pi)/2 + P(1/x)/x
Again, substitution of large into the RHS reveals the last 3 terms
are negligible in comparison to the leading term.

Move the x=+-0 special case into the |x|<small block.

In the ld80 and ld128 implementaion, use fdlibm compatible comparisons
involving ix, lx, and llx.  This replaces several floating point
comparisons (some involving fabsl()) and also fixes the special cases
x=1 and x=2.

While here
  . Remove unnecessary parentheses.
  . Fix/improve comments due to the above changes.
  . Fix nearby whitespace.

* src/e_lgamma_r.c:
  . Sort declaration.
  . Remove unneeded explicit cast for type conversion.
  . Replace a double literal constant by an integer literal constant.

* src/e_lgammaf_r.c:
  . Sort declaration.

* ld128/e_lgammal_r.c:
  . Replace a long double literal constant by a double literal constant.

* ld80/e_lgammal_r.c:
  . Remove unused '#include float.h'
  . Replace a long double literal constant by a double literal constant.

Requested by:	bde

1 files changed, 47 insertions, 38 deletions

diff --git a/lib/msun/ld80/e_lgammal_r.c b/lib/msun/ld80/e_lgammal_r.c
index 621b04e..1e65769 100644
--- a/lib/msun/ld80/e_lgammal_r.c
+++ b/lib/msun/ld80/e_lgammal_r.c
@@ -14,12 +14,11 @@
 __FBSDID("$FreeBSD$");
 
 /*
- * See s_lgamma_r.c for complete comments.
+ * See e_lgamma_r.c for complete comments.
  *
  * Converted to long double by Steven G. Kargl.
  */
 
-#include <float.h>
 #ifdef __i386__
 #include <ieeefp.h>
 #endif
@@ -219,8 +218,8 @@ sin_pil(long double x)
 
 	y = -x;
 
-	vz = y+0x1p63L;
-	z = vz-0x1p63L;
+	vz = y+0x1p63;
+	z = vz-0x1p63;
 	if (z == y)
 	    return zero;
 
@@ -243,7 +242,7 @@ sin_pil(long double x)
 	    case 5:
 	    case 6:   y = -__kernel_cosl(pi*(y-1.5),zero); break;
 	    default:  y =  __kernel_sinl(pi*(y-2.0),zero,0); break;
-	}
+	    }
 	return -y;
 }
 
@@ -253,31 +252,29 @@ lgammal_r(long double x, int *signgamp)
 	long double nadj,p,p1,p2,p3,q,r,t,w,y,z;
 	uint64_t lx;
 	int i;
-	uint16_t hx;
+	uint16_t hx,ix;
 
 	EXTRACT_LDBL80_WORDS(hx,lx,x);
 
-    /* purge off +-inf, NaN, +-0 */
+    /* purge +-Inf and NaNs */
 	*signgamp = 1;
-	if((hx & 0x7fff) == 0x7fff)	/* x is +-Inf or NaN */
-		return x*x;
-	if((hx==0||hx==0x8000)&&lx==0) {
-	    if (hx&0x8000)
-		*signgamp = -1;
-	    return one/vzero;
-	}
+	ix = hx&0x7fff;
+	if(ix==0x7fff) return x*x;
 
 	ENTERI();
 
-    /* purge off tiny and negative arguments */
-	if(fabsl(x)<0x1p-70L) {		/* |x|<2**-70, return -log(|x|) */
-	    if(hx&0x8000) {
-	        *signgamp = -1;
-	        RETURNI(-logl(-x));
-	    } else RETURNI(-logl(x));
+    /* purge +-0 and tiny arguments */
+	*signgamp = 1-2*(hx>>15);
+	if(ix<0x3fff-67) {		/* |x|<2**-(p+3), return -log(|x|) */
+	    if((ix|lx)==0)
+		RETURNI(one/vzero);
+	    RETURNI(-logl(fabsl(x)));
 	}
+
+    /* purge negative integers and start evaluation for other x < 0 */
 	if(hx&0x8000) {
-	    if(fabsl(x)>=0x1p63) 	/* |x|>=2**(p-1), must be -integer */
+	    *signgamp = 1;
+	    if(ix>=0x3fff+63) 		/* |x|>=2**(p-1), must be -integer */
 		RETURNI(one/vzero);
 	    t = sin_pil(x);
 	    if(t==zero) RETURNI(one/vzero); /* -integer */
@@ -286,19 +283,30 @@ lgammal_r(long double x, int *signgamp)
 	    x = -x;
 	}
 
-    /* purge off 1 and 2 */
-	if(x == 1 || x == 2) r = 0;
+    /* purge 1 and 2 */
+	if((ix==0x3fff || ix==0x4000) && lx==0x8000000000000000ULL) r = 0;
     /* for x < 2.0 */
-	else if(x<2) {
-	    if(x<=0.8999996185302734) {	/* lgamma(x) = lgamma(x+1)-log(x) */
-		r = - logl(x);
-		if(x>=0.7315998077392578) {y = 1-x; i= 0;}
-		else if(x>=0.2316399812698364) {y= x-(tc-1); i=1;}
-	  	else {y = x; i=2;}
+	else if(ix<0x4000) {
+    /*
+     * XXX Supposedly, one can use the following information to replace the
+     * XXX FP rational expressions.  A similar approach is appropriate
+     * XXX for ld128, but one (may need?) needs to consider llx, too.
+     *
+     * 8.9999961853027344e-01 3ffe e666600000000000
+     * 7.3159980773925781e-01 3ffe bb4a200000000000
+     * 2.3163998126983643e-01 3ffc ed33080000000000
+     * 1.7316312789916992e+00 3fff dda6180000000000
+     * 1.2316322326660156e+00 3fff 9da6200000000000
+     */
+	    if(x<8.9999961853027344e-01) {
+		r = -logl(x);
+		if(x>=7.3159980773925781e-01) {y = 1-x; i= 0;}
+		else if(x>=2.3163998126983643e-01) {y= x-(tc-1); i=1;}
+		else {y = x; i=2;}
 	    } else {
 		r = 0;
-	        if(x>=1.7316312789916992) {y=2-x;i=0;}
-	        else if(x>=1.2316322326660156) {y=x-tc;i=1;}
+		if(x>=1.7316312789916992e+00) {y=2-x;i=0;}
+		else if(x>=1.2316322326660156e+00) {y=x-tc;i=1;}
 		else {y=x-1;i=2;}
 	    }
 	    switch(i) {
@@ -307,19 +315,20 @@ lgammal_r(long double x, int *signgamp)
 		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*(a10+z*a12)))));
 		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*(a11+z*a13))))));
 		p  = y*p1+p2;
-		r  += (p-y/2); break;
+		r  += p-y/2; break;
 	      case 1:
 		p = t0+y*t1+tt+y*y*(t2+y*(t3+y*(t4+y*(t5+y*(t6+y*(t7+y*(t8+
 		    y*(t9+y*(t10+y*(t11+y*(t12+y*(t13+y*(t14+y*(t15+y*(t16+
 		    y*(t17+y*t18))))))))))))))));
-		r += (tf + p); break;
+		r += tf + p; break;
 	      case 2:
 		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*u6))))));
 		p2 = 1+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*v6)))));
-		r += (-y/2 + p1/p2);
+		r += p1/p2-y/2;
 	    }
 	}
-	else if(x<8) {
+    /* x < 8.0 */
+	else if(ix<0x4002) {
 	    i = x;
 	    y = x-i;
 	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
@@ -334,15 +343,15 @@ lgammal_r(long double x, int *signgamp)
 	    case 3: z *= (y+2);		/* FALLTHRU */
 		    r += logl(z); break;
 	    }
-    /* 8.0 <= x < 2**70 */
-	} else if (x < 0x1p70L) {
+    /* 8.0 <= x < 2**(p+3) */
+	} else if (ix<0x3fff+67) {
 	    t = logl(x);
 	    z = one/x;
 	    y = z*z;
 	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*(w6+y*(w7+y*w8)))))));
 	    r = (x-half)*(t-one)+w;
+    /* 2**(p+3) <= x <= inf */
 	} else 
-    /* 2**70 <= x <= inf */
 	    r =  x*(logl(x)-1);
 	if(hx&0x8000) r = nadj - r;
 	RETURNI(r);