math: lgamma cleanup (simpler sin(pi*x) for the negative case)

* simplify sin_pi(x) (don't care about inexact here, the result is
  inexact anyway, and x is not so small to underflow)
* in lgammal add the previously removed special case for x==1 and
  x==2 (to fix the sign of zero in downward rounding mode)
* only define lgammal on supported long double platforms
* change tgamma so the generated code is a bit smaller

src/math/lgamma_r.c

@@ -82,7 +82,6 @@
 #include "libc.h"
 
 static const double
-two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
@@ -147,91 +146,62 @@ w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 
+/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
 static double sin_pi(double x)
 {
-	double y,z;
-	int n,ix;
+	int n;
 
-	GET_HIGH_WORD(ix, x);
-	ix &= 0x7fffffff;
+	/* spurious inexact if odd int */
+	x = 2.0*(x*0.5 - floor(x*0.5));  /* x mod 2.0 */
 
-	if (ix < 0x3fd00000)
-		return __sin(pi*x, 0.0, 0);
+	n = (int)(x*4.0);
+	n = (n+1)/2;
+	x -= n*0.5f;
+	x *= pi;
 
-	y = -x;  /* negative x is assumed */
-
-	/*
-	 * argument reduction, make sure inexact flag not raised if input
-	 * is an integer
-	 */
-	z = floor(y);
-	if (z != y) {    /* inexact anyway */
-		y *= 0.5;
-		y  = 2.0*(y - floor(y));   /* y = |x| mod 2.0 */
-		n  = (int)(y*4.0);
-	} else {
-		if (ix >= 0x43400000) {
-			y = 0.0;    /* y must be even */
-			n = 0;
-		} else {
-			if (ix < 0x43300000)
-				z = y + two52;  /* exact */
-			GET_LOW_WORD(n, z);
-			n &= 1;
-			y = n;
-			n <<= 2;
-		}
-	}
 	switch (n) {
-	case 0:  y =  __sin(pi*y, 0.0, 0); break;
-	case 1:
-	case 2:  y =  __cos(pi*(0.5-y), 0.0); break;
-	case 3:
-	case 4:  y =  __sin(pi*(1.0-y), 0.0, 0); break;
-	case 5:
-	case 6:  y = -__cos(pi*(y-1.5), 0.0); break;
-	default: y =  __sin(pi*(y-2.0), 0.0, 0); break;
+	default: /* case 4: */
+	case 0: return __sin(x, 0.0, 0);
+	case 1: return __cos(x, 0.0);
+	case 2: return __sin(-x, 0.0, 0);
+	case 3: return -__cos(x, 0.0);
 	}
-	return -y;
 }
 
-
 double __lgamma_r(double x, int *signgamp)
 {
-	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
-	int32_t hx;
-	int i,lx,ix;
-
-	EXTRACT_WORDS(hx, lx, x);
+	union {double f; uint64_t i;} u = {x};
+	double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
+	uint32_t ix;
+	int sign,i;
 
 	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
 	*signgamp = 1;
-	ix = hx & 0x7fffffff;
+	sign = u.i>>63;
+	ix = u.i>>32 & 0x7fffffff;
 	if (ix >= 0x7ff00000)
 		return x*x;
-	if ((ix|lx) == 0)
-		return 1.0/0.0;
-	if (ix < 0x3b900000) {  /* |x|<2**-70, return -log(|x|) */
-		if(hx < 0) {
+	if (ix < (0x3ff-70)<<20) {  /* |x|<2**-70, return -log(|x|) */
+		if(sign) {
+			x = -x;
 			*signgamp = -1;
-			return -log(-x);
 		}
 		return -log(x);
 	}
-	if (hx < 0) {
-		if (ix >= 0x43300000)  /* |x|>=2**52, must be -integer */
-			return 1.0/0.0;
+	if (sign) {
+		x = -x;
 		t = sin_pi(x);
 		if (t == 0.0) /* -integer */
-			return 1.0/0.0;
-		nadj = log(pi/fabs(t*x));
-		if (t < 0.0)
+			return 1.0/(x-x);
+		if (t > 0.0)
 			*signgamp = -1;
-		x = -x;
+		else
+			t = -t;
+		nadj = log(pi/(t*x));
 	}
 
 	/* purge off 1 and 2 */
-	if (((ix - 0x3ff00000)|lx) == 0 || ((ix - 0x40000000)|lx) == 0)
+	if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0)
 		r = 0;
 	/* for x < 2.0 */
 	else if (ix < 0x40000000) {
@@ -306,7 +276,7 @@ double __lgamma_r(double x, int *signgamp)
 		r = (x-0.5)*(t-1.0)+w;
 	} else                         /* 2**58 <= x <= inf */
 		r =  x*(log(x)-1.0);
-	if (hx < 0)
+	if (sign)
 		r = nadj - r;
 	return r;
 }

src/math/lgammaf_r.c

@@ -17,7 +17,6 @@
 #include "libc.h"
 
 static const float
-two23= 8.3886080000e+06, /* 0x4b000000 */
 pi  =  3.1415927410e+00, /* 0x40490fdb */
 a0  =  7.7215664089e-02, /* 0x3d9e233f */
 a1  =  3.2246702909e-01, /* 0x3ea51a66 */
@@ -82,87 +81,58 @@ w4  = -5.9518753551e-04, /* 0xba1c065c */
 w5  =  8.3633989561e-04, /* 0x3a5b3dd2 */
 w6  = -1.6309292987e-03; /* 0xbad5c4e8 */
 
-static float sin_pif(float x)
+/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
+static float sin_pi(float x)
 {
-	float y,z;
-	int n,ix;
+	double_t y;
+	int n;
 
-	GET_FLOAT_WORD(ix, x);
-	ix &= 0x7fffffff;
+	/* spurious inexact if odd int */
+	x = 2*(x*0.5f - floorf(x*0.5f));  /* x mod 2.0 */
 
-	if(ix < 0x3e800000)
-		return __sindf(pi*x);
-
-	y = -x;  /* negative x is assumed */
-
-	/*
-	 * argument reduction, make sure inexact flag not raised if input
-	 * is an integer
-	 */
-	z = floorf(y);
-	if (z != y) {   /* inexact anyway */
-		y *= 0.5f;
-		y  = 2.0f*(y - floorf(y));   /* y = |x| mod 2.0 */
-		n  = (int)(y*4.0f);
-	} else {
-		if (ix >= 0x4b800000) {
-			y = 0.0f;  /* y must be even */
-			n = 0;
-		} else {
-			if (ix < 0x4b000000)
-				z = y + two23;  /* exact */
-			GET_FLOAT_WORD(n, z);
-			n &= 1;
-			y = n;
-			n <<= 2;
-		}
-	}
+	n = (int)(x*4);
+	n = (n+1)/2;
+	y = x - n*0.5f;
+	y *= 3.14159265358979323846;
 	switch (n) {
-	case 0:  y =  __sindf(pi*y); break;
-	case 1:
-	case 2:  y =  __cosdf(pi*(0.5f - y)); break;
-	case 3:
-	case 4:  y =  __sindf(pi*(1.0f - y)); break;
-	case 5:
-	case 6:  y = -__cosdf(pi*(y - 1.5f)); break;
-	default: y =  __sindf(pi*(y - 2.0f)); break;
+	default: /* case 4: */
+	case 0: return __sindf(y);
+	case 1: return __cosdf(y);
+	case 2: return __sindf(-y);
+	case 3: return -__cosdf(y);
 	}
-	return -y;
 }
 
-
 float __lgammaf_r(float x, int *signgamp)
 {
+	union {float f; uint32_t i;} u = {x};
 	float t,y,z,nadj,p,p1,p2,p3,q,r,w;
-	int32_t hx;
-	int i,ix;
-
-	GET_FLOAT_WORD(hx, x);
+	uint32_t ix;
+	int i,sign;
 
 	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
 	*signgamp = 1;
-	ix = hx & 0x7fffffff;
+	sign = u.i>>31;
+	ix = u.i & 0x7fffffff;
 	if (ix >= 0x7f800000)
 		return x*x;
-	if (ix == 0)
-		return 1.0f/0.0f;
 	if (ix < 0x35000000) {  /* |x| < 2**-21, return -log(|x|) */
-		if (hx < 0) {
+		if (sign) {
 			*signgamp = -1;
-			return -logf(-x);
+			x = -x;
 		}
 		return -logf(x);
 	}
-	if (hx < 0) {
-		if (ix >= 0x4b000000)  /* |x| >= 2**23, must be -integer */
-			return 1.0f/0.0f;
-		t = sin_pif(x);
+	if (sign) {
+		x = -x;
+		t = sin_pi(x);
 		if (t == 0.0f) /* -integer */
-			return 1.0f/0.0f;
-		nadj = logf(pi/fabsf(t*x));
-		if (t < 0.0f)
+			return 1.0f/(x-x);
+		if (t > 0.0f)
 			*signgamp = -1;
-		x = -x;
+		else
+			t = -t;
+		nadj = logf(pi/(t*x));
 	}
 
 	/* purge off 1 and 2 */
@@ -241,7 +211,7 @@ float __lgammaf_r(float x, int *signgamp)
 		r = (x-0.5f)*(t-1.0f)+w;
 	} else                         /* 2**58 <= x <= inf */
 		r =  x*(logf(x)-1.0f);
-	if (hx < 0)
+	if (sign)
 		r = nadj - r;
 	return r;
 }

src/math/lgammal.c

@@ -99,7 +99,6 @@ long double __lgammal_r(long double x, int *sg)
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
 static const long double
 pi = 3.14159265358979323846264L,
-two63 = 9.223372036854775808e18L,
 
 /* lgam(1+x) = 0.5 x + x a(x)/b(x)
     -0.268402099609375 <= x <= 0
@@ -201,61 +200,27 @@ w5 =  8.412723297322498080632E-4L,
 w6 = -1.880801938119376907179E-3L,
 w7 =  4.885026142432270781165E-3L;
 
+/* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
 static long double sin_pi(long double x)
 {
-	union ldshape u = {x};
-	uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
-	long double y, z;
 	int n;
 
-	if (ix < 0x3ffd8000)  /* 0.25 */
-		return sinl(pi * x);
-	y = -x;  /* x is assume negative */
+	/* spurious inexact if odd int */
+	x *= 0.5;
+	x = 2.0*(x - floorl(x));  /* x mod 2.0 */
 
-	/*
-	 * argument reduction, make sure inexact flag not raised if input
-	 * is an integer
-	 */
-	z = floorl(y);
-	if (z != y) {  /* inexact anyway */
-		y *= 0.5;
-		y = 2.0*(y - floorl(y));/* y = |x| mod 2.0 */
-		n = (int) (y*4.0);
-	} else {
-		if (ix >= 0x403f8000) {  /* 2^64 */
-			y = 0.0;  /* y must be even */
-			n = 0;
-		} else {
-			if (ix < 0x403e8000)  /* 2^63 */
-				z = y + two63;  /* exact */
-			u.f = z;
-			n = u.i.m & 1;
-			y = n;
-			n <<= 2;
-		}
-	}
+	n = (int)(x*4.0);
+	n = (n+1)/2;
+	x -= n*0.5f;
+	x *= pi;
 
 	switch (n) {
-	case 0:
-		y = sinl(pi * y);
-		break;
-	case 1:
-	case 2:
-		y = cosl(pi * (0.5 - y));
-		break;
-	case 3:
-	case 4:
-		y = sinl(pi * (1.0 - y));
-		break;
-	case 5:
-	case 6:
-		y = -cosl(pi * (y - 1.5));
-		break;
-	default:
-		y = sinl(pi * (y - 2.0));
-		break;
+	default: /* case 4: */
+	case 0: return __sinl(x, 0.0, 0);
+	case 1: return __cosl(x, 0.0);
+	case 2: return __sinl(-x, 0.0, 0);
+	case 3: return -__cosl(x, 0.0);
 	}
-	return -y;
 }
 
 long double __lgammal_r(long double x, int *sg) {
@@ -267,31 +232,32 @@ long double __lgammal_r(long double x, int *sg) {
 
 	*sg = 1;
 
-	/* purge off +-inf, NaN, +-0, and negative arguments */
+	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
 	if (ix >= 0x7fff0000)
 		return x * x;
-	if (x == 0) {
-		*sg -= 2*sign;
-		return 1.0 / fabsl(x);
-	}
 	if (ix < 0x3fc08000) {  /* |x|<2**-63, return -log(|x|) */
 		if (sign) {
 			*sg = -1;
-			return -logl(-x);
+			x = -x;
 		}
 		return -logl(x);
 	}
 	if (sign) {
-		t = sin_pi (x);
+		x = -x;
+		t = sin_pi(x);
 		if (t == 0.0)
-			return 1.0 / fabsl(t); /* -integer */
-		nadj = logl(pi / fabsl(t * x));
-		if (t < 0.0)
+			return 1.0 / (x-x); /* -integer */
+		if (t > 0.0)
 			*sg = -1;
-		x = -x;
+		else
+			t = -t;
+		nadj = logl(pi / (t * x));
 	}
 
-	if (ix < 0x40008000) {  /* x < 2.0 */
+	/* purge off 1 and 2 (so the sign is ok with downward rounding) */
+	if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
+		r = 0;
+	} else if (ix < 0x40008000) {  /* x < 2.0 */
 		if (ix <= 0x3ffee666) {  /* 8.99993896484375e-1 */
 			/* lgamma(x) = lgamma(x+1) - log(x) */
 			r = -logl(x);
@@ -376,6 +342,7 @@ long double __lgammal_r(long double x, int *sg) {
 }
 #endif
 
+#if (LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024) || (LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384)
 extern int __signgam;
 
 long double lgammal(long double x)
@@ -384,3 +351,4 @@ long double lgammal(long double x)
 }
 
 weak_alias(__lgammal_r, lgammal_r);
+#endif

src/math/tgamma.c

@@ -26,7 +26,7 @@ most ideas and constants are from boost and python
 
 static const double pi = 3.141592653589793238462643383279502884;
 
-/* sin(pi x) with x > 0 && isnormal(x) assumption */
+/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
 static double sinpi(double x)
 {
 	int n;
@@ -49,8 +49,7 @@ static double sinpi(double x)
 	case 1:
 		return __cos(x, 0);
 	case 2:
-		/* sin(0-x) and -sin(x) have different sign at 0 */
-		return __sin(0-x, 0, 0);
+		return __sin(-x, 0, 0);
 	case 3:
 		return -__cos(x, 0);
 	}
@@ -108,35 +107,33 @@ static double S(double x)
 
 double tgamma(double x)
 {
-	double absx, y, dy, z, r;
+	union {double f; uint64_t i;} u = {x};
+	double absx, y;
+	double_t dy, z, r;
+	uint32_t ix = u.i>>32 & 0x7fffffff;
+	int sign = u.i>>63;
 
 	/* special cases */
-	if (!isfinite(x))
+	if (ix >= 0x7ff00000)
 		/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
 		return x + INFINITY;
+	if (ix < (0x3ff-54)<<20)
+		/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
+		return 1/x;
 
 	/* integer arguments */
 	/* raise inexact when non-integer */
 	if (x == floor(x)) {
-		if (x == 0)
-			/* tgamma(+-0)=+-inf with divide-by-zero */
-			return 1/x;
-		if (x < 0)
+		if (sign)
 			return 0/0.0;
 		if (x <= sizeof fact/sizeof *fact)
 			return fact[(int)x - 1];
 	}
 
-	absx = fabs(x);
-
-	/* x ~ 0: tgamma(x) ~ 1/x */
-	if (absx < 0x1p-54)
-		return 1/x;
-
 	/* x >= 172: tgamma(x)=inf with overflow */
 	/* x =< -184: tgamma(x)=+-0 with underflow */
-	if (absx >= 184) {
-		if (x < 0) {
+	if (ix >= 0x40670000) { /* |x| >= 184 */
+		if (sign) {
 			FORCE_EVAL((float)(0x1p-126/x));
 			if (floor(x) * 0.5 == floor(x * 0.5))
 				return 0;
@@ -146,6 +143,8 @@ double tgamma(double x)
 		return x;
 	}
 
+	absx = sign ? -x : x;
+
 	/* handle the error of x + g - 0.5 */
 	y = absx + gmhalf;
 	if (absx > gmhalf) {
@@ -160,20 +159,21 @@ double tgamma(double x)
 	r = S(absx) * exp(-y);
 	if (x < 0) {
 		/* reflection formula for negative x */
+		/* sinpi(absx) is not 0, integers are already handled */
 		r = -pi / (sinpi(absx) * absx * r);
 		dy = -dy;
 		z = -z;
 	}
 	r += dy * (gmhalf+0.5) * r / y;
 	z = pow(y, 0.5*z);
-	r = r * z * z;
-	return r;
+	y = r * z * z;
+	return y;
 }
 
 #if 0
 double __lgamma_r(double x, int *sign)
 {
-	double r, absx, z, zz, w;
+	double r, absx;
 
 	*sign = 1;