math: clean up atan2.c

* remove volatile hacks
* don't care about inexact flag for now (removed all the +-tiny)
* fix atanl to raise underflow properly
* remove signed int arithmetics
* use pi/2 instead of pi_o_2 (gcc generates the same code, which is not
correct, but it does not matter: we mainly care about nearest rounding)

src/math/atan2.c

@@ -39,75 +39,63 @@
 
 #include "libm.h"
 
-// FIXME
-static const volatile double
-tiny  = 1.0e-300;
 static const double
-pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
-pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
-pi     = 3.1415926535897931160E+00; /* 0x400921FB, 0x54442D18 */
-static const volatile double
+pi     = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */
 pi_lo  = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
 
 double atan2(double y, double x)
 {
 	double z;
-	int32_t k,m,hx,hy,ix,iy;
-	uint32_t lx,ly;
+	uint32_t m,lx,ly,ix,iy;
 
-	EXTRACT_WORDS(hx, lx, x);
-	ix = hx & 0x7fffffff;
-	EXTRACT_WORDS(hy, ly, y);
-	iy = hy & 0x7fffffff;
-	if ((ix|((lx|-lx)>>31)) > 0x7ff00000 ||
-	    (iy|((ly|-ly)>>31)) > 0x7ff00000)  /* x or y is NaN */
+	if (isnan(x) || isnan(y))
 		return x+y;
-	if ((hx-0x3ff00000 | lx) == 0)  /* x = 1.0 */
+	EXTRACT_WORDS(ix, lx, x);
+	EXTRACT_WORDS(iy, ly, y);
+	if ((ix-0x3ff00000 | lx) == 0)  /* x = 1.0 */
 		return atan(y);
-	m = ((hy>>31)&1) | ((hx>>30)&2);  /* 2*sign(x)+sign(y) */
+	m = ((iy>>31)&1) | ((ix>>30)&2);  /* 2*sign(x)+sign(y) */
+	ix = ix & 0x7fffffff;
+	iy = iy & 0x7fffffff;
 
 	/* when y = 0 */
 	if ((iy|ly) == 0) {
 		switch(m) {
 		case 0:
-		case 1: return y;        /* atan(+-0,+anything)=+-0 */
-		case 2: return  pi+tiny; /* atan(+0,-anything) = pi */
-		case 3: return -pi-tiny; /* atan(-0,-anything) =-pi */
+		case 1: return y;   /* atan(+-0,+anything)=+-0 */
+		case 2: return  pi; /* atan(+0,-anything) = pi */
+		case 3: return -pi; /* atan(-0,-anything) =-pi */
 		}
 	}
 	/* when x = 0 */
 	if ((ix|lx) == 0)
-		return hy < 0 ? -pi_o_2-tiny : pi_o_2+tiny;
+		return m&1 ? -pi/2 : pi/2;
 	/* when x is INF */
 	if (ix == 0x7ff00000) {
 		if (iy == 0x7ff00000) {
 			switch(m) {
-			case 0: return  pi_o_4+tiny; /* atan(+INF,+INF) */
-			case 1: return -pi_o_4-tiny; /* atan(-INF,+INF) */
-			case 2: return  3.0*pi_o_4+tiny; /* atan(+INF,-INF) */
-			case 3: return -3.0*pi_o_4-tiny; /* atan(-INF,-INF) */
+			case 0: return  pi/4;   /* atan(+INF,+INF) */
+			case 1: return -pi/4;   /* atan(-INF,+INF) */
+			case 2: return  3*pi/4; /* atan(+INF,-INF) */
+			case 3: return -3*pi/4; /* atan(-INF,-INF) */
 			}
 		} else {
 			switch(m) {
-			case 0: return  0.0;     /* atan(+...,+INF) */
-			case 1: return -0.0;     /* atan(-...,+INF) */
-			case 2: return  pi+tiny; /* atan(+...,-INF) */
-			case 3: return -pi-tiny; /* atan(-...,-INF) */
+			case 0: return  0.0; /* atan(+...,+INF) */
+			case 1: return -0.0; /* atan(-...,+INF) */
+			case 2: return  pi;  /* atan(+...,-INF) */
+			case 3: return -pi;  /* atan(-...,-INF) */
 			}
 		}
 	}
-	/* when y is INF */
-	if (iy == 0x7ff00000)
-		return hy < 0 ? -pi_o_2-tiny : pi_o_2+tiny;
+	/* |y/x| > 0x1p64 */
+	if (ix+(64<<20) < iy || iy == 0x7ff00000)
+		return m&1 ? -pi/2 : pi/2;
 
-	/* compute y/x */
-	k = (iy-ix)>>20;
-	if (k > 60) {                  /* |y/x| >  2**60 */
-		z = pi_o_2+0.5*pi_lo;
-		m &= 1;
-	} else if (hx < 0 && k < -60)  /* 0 > |y|/x > -2**-60 */
-		z = 0.0;
-	else                           /* safe to do y/x */
+	/* z = atan(|y/x|) without spurious underflow */
+	if ((m&2) && iy+(64<<20) < ix)  /* |y/x| < 0x1p-64, x<0 */
+		z = 0;
+	else
 		z = atan(fabs(y/x));
 	switch (m) {
 	case 0: return z;              /* atan(+,+) */

src/math/atan2f.c

@@ -15,72 +15,63 @@
 
 #include "libm.h"
 
-static const volatile float
-tiny = 1.0e-30;
 static const float
-pi_o_4 = 7.8539818525e-01, /* 0x3f490fdb */
-pi_o_2 = 1.5707963705e+00, /* 0x3fc90fdb */
-pi     = 3.1415927410e+00; /* 0x40490fdb */
-static const volatile float
+pi     = 3.1415927410e+00, /* 0x40490fdb */
 pi_lo  = -8.7422776573e-08; /* 0xb3bbbd2e */
 
 float atan2f(float y, float x)
 {
 	float z;
-	int32_t k,m,hx,hy,ix,iy;
+	uint32_t m,ix,iy;
 
-	GET_FLOAT_WORD(hx, x);
-	ix = hx & 0x7fffffff;
-	GET_FLOAT_WORD(hy, y);
-	iy = hy & 0x7fffffff;
-	if (ix > 0x7f800000 || iy > 0x7f800000)  /* x or y is NaN */
+	if (isnan(x) || isnan(y))
 		return x+y;
-	if (hx == 0x3f800000)  /* x=1.0 */
+	GET_FLOAT_WORD(ix, x);
+	GET_FLOAT_WORD(iy, y);
+	if (ix == 0x3f800000)  /* x=1.0 */
 		return atanf(y);
-	m = ((hy>>31)&1) | ((hx>>30)&2);  /* 2*sign(x)+sign(y) */
+	m = ((iy>>31)&1) | ((ix>>30)&2);  /* 2*sign(x)+sign(y) */
+	ix &= 0x7fffffff;
+	iy &= 0x7fffffff;
 
 	/* when y = 0 */
 	if (iy == 0) {
 		switch (m) {
 		case 0:
-		case 1: return y;        /* atan(+-0,+anything)=+-0 */
-		case 2: return  pi+tiny; /* atan(+0,-anything) = pi */
-		case 3: return -pi-tiny; /* atan(-0,-anything) =-pi */
+		case 1: return y;   /* atan(+-0,+anything)=+-0 */
+		case 2: return  pi; /* atan(+0,-anything) = pi */
+		case 3: return -pi; /* atan(-0,-anything) =-pi */
 		}
 	}
 	/* when x = 0 */
 	if (ix == 0)
-		return hy < 0 ? -pi_o_2-tiny : pi_o_2+tiny;
+		return m&1 ? -pi/2 : pi/2;
 	/* when x is INF */
 	if (ix == 0x7f800000) {
 		if (iy == 0x7f800000) {
 			switch (m) {
-			case 0: return  pi_o_4+tiny; /* atan(+INF,+INF) */
-			case 1: return -pi_o_4-tiny; /* atan(-INF,+INF) */
-			case 2: return 3.0f*pi_o_4+tiny;  /*atan(+INF,-INF)*/
-			case 3: return -3.0f*pi_o_4-tiny; /*atan(-INF,-INF)*/
+			case 0: return  pi/4; /* atan(+INF,+INF) */
+			case 1: return -pi/4; /* atan(-INF,+INF) */
+			case 2: return 3*pi/4;  /*atan(+INF,-INF)*/
+			case 3: return -3*pi/4; /*atan(-INF,-INF)*/
 			}
 		} else {
 			switch (m) {
 			case 0: return  0.0f;    /* atan(+...,+INF) */
 			case 1: return -0.0f;    /* atan(-...,+INF) */
-			case 2: return  pi+tiny; /* atan(+...,-INF) */
-			case 3: return -pi-tiny; /* atan(-...,-INF) */
+			case 2: return  pi; /* atan(+...,-INF) */
+			case 3: return -pi; /* atan(-...,-INF) */
 			}
 		}
 	}
-	/* when y is INF */
-	if (iy == 0x7f800000)
-		return hy < 0 ? -pi_o_2-tiny : pi_o_2+tiny;
+	/* |y/x| > 0x1p26 */
+	if (ix+(26<<23) < iy || iy == 0x7f800000)
+		return m&1 ? -pi/2 : pi/2;
 
-	/* compute y/x */
-	k = (iy-ix)>>23;
-	if (k > 26) {                  /* |y/x| >  2**26 */
-		z = pi_o_2 + 0.5f*pi_lo;
-		m &= 1;
-	} else if (k < -26 && hx < 0)  /* 0 > |y|/x > -2**-26 */
+	/* z = atan(|y/x|) with correct underflow */
+	if ((m&2) && iy+(26<<23) < ix)  /*|y/x| < 0x1p-26, x < 0 */
 		z = 0.0;
-	else                           /* safe to do y/x */
+	else
 		z = atanf(fabsf(y/x));
 	switch (m) {
 	case 0: return z;              /* atan(+,+) */

src/math/atan2l.c

@@ -29,26 +29,22 @@ long double atan2l(long double y, long double x)
 {
 	union IEEEl2bits ux, uy;
 	long double z;
-	int32_t k,m;
-	int16_t exptx, expsignx, expty, expsigny;
+	int m;
+	uint16_t exptx, expsignx, expty, expsigny;
 
+	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;
-	if ((exptx==0x7fff &&
-	     ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)!=0) || /* x is NaN */
-	    (expty==0x7fff &&
-	     ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)!=0))   /* y is NaN */
-		return x+y;
-	if (expsignx==0x3fff && ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)==0) /* x=1.0 */
-		return atanl(y);
 	m = ((expsigny>>15)&1) | ((expsignx>>14)&2);  /* 2*sign(x)+sign(y) */
 
-	/* when y = 0 */
-	if (expty==0 && ((uy.bits.manh&~LDBL_NBIT)|uy.bits.manl)==0) {
+	if (y == 0) {
 		switch(m) {
 		case 0:
 		case 1: return y;           /* atan(+-0,+anything)=+-0 */
@@ -56,9 +52,8 @@ long double atan2l(long double y, long double x)
 		case 3: return -2*pio2_hi-0x1p-120f; /* atan(-0,-anything) =-pi */
 		}
 	}
-	/* when x = 0 */
-	if (exptx==0 && ((ux.bits.manh&~LDBL_NBIT)|ux.bits.manl)==0)
-		return expsigny < 0 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
+	if (x == 0)
+		return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
 	/* when x is INF */
 	if (exptx == 0x7fff) {
 		if (expty == 0x7fff) {
@@ -78,17 +73,12 @@ long double atan2l(long double y, long double x)
 		}
 	}
 	/* when y is INF */
-	if (expty == 0x7fff)
-		return expsigny < 0 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
+	if (exptx+120 < expty || expty == 0x7fff)
+		return m&1 ? -pio2_hi-0x1p-120f : pio2_hi+0x1p-120f;
 
-	/* compute y/x */
-	k = expty-exptx;
-	if(k > LDBL_MANT_DIG+2) { /* |y/x| huge */
-		z = pio2_hi+0x1p-120f;
-		m &= 1;
-	} else if (expsignx < 0 && k < -LDBL_MANT_DIG-2) /* |y/x| tiny, x<0 */
+	if ((m&2) && expty+120 < exptx) /* |y/x| tiny, x<0 */
 		z = 0.0;
-	else                     /* safe to do y/x */
+	else
 		z = atanl(fabsl(y/x));
 	switch (m) {
 	case 0: return z;               /* atan(+,+) */

src/math/atanl.c

@@ -70,8 +70,8 @@ long double atanl(long double x)
 	union IEEEl2bits u;
 	long double w,s1,s2,z;
 	int id;
-	int16_t expsign, expt;
-	int32_t expman;
+	uint16_t expsign, expt;
+	uint32_t expman;
 
 	u.e = x;
 	expsign = u.xbits.expsign;
@@ -81,15 +81,16 @@ long double atanl(long double x)
 		    ((u.bits.manh&~LDBL_NBIT)|u.bits.manl)!=0)  /* NaN */
 			return x+x;
 		z = atanhi[3] + 0x1p-120f;
-		return expsign < 0 ? -z : z;
+		return expsign>>15 ? -z : z;
 	}
 	/* 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);
 	if (expman < ((0x3fff - 2) << 8) + 0xc0) {  /* |x| < 0.4375 */
 		if (expt < 0x3fff - 32) {   /* if |x| is small, atanl(x)~=x */
-			/* raise inexact if x!=0 */
-			FORCE_EVAL(x + 0x1p120f);
+			/* raise underflow if subnormal */
+			if (expt == 0)
+				FORCE_EVAL((float)x);
 			return x;
 		}
 		id = -1;
@@ -122,6 +123,6 @@ 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 < 0 ? -z : z;
+	return expsign>>15 ? -z : z;
 }
 #endif