Expanded the libm support and put it into a separate directory.
authorSam Lantinga <slouken@libsdl.org>
Mon, 15 Sep 2008 06:33:23 +0000
changeset 2756 a98604b691c8
parent 2755 2a3ec308d995
child 2757 0581f49c9294
Expanded the libm support and put it into a separate directory.
src/libm/e_log.c
src/libm/e_pow.c
src/libm/e_rem_pio2.c
src/libm/e_sqrt.c
src/libm/k_cos.c
src/libm/k_sin.c
src/libm/math.h
src/libm/math_private.h
src/libm/s_copysign.c
src/libm/s_cos.c
src/libm/s_fabs.c
src/libm/s_scalbn.c
src/libm/s_sin.c
src/video/SDL_gamma.c
src/video/e_log.h
src/video/e_pow.h
src/video/e_sqrt.h
src/video/math_private.h
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/e_log.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,166 @@
+/* @(#)e_log.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
+#endif
+
+/* __ieee754_log(x)
+ * Return the logrithm of x
+ *
+ * Method :
+ *   1. Argument Reduction: find k and f such that
+ *			x = 2^k * (1+f),
+ *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
+ *
+ *   2. Approximation of log(1+f).
+ *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
+ *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
+ *	     	 = 2s + s*R
+ *      We use a special Reme algorithm on [0,0.1716] to generate
+ * 	a polynomial of degree 14 to approximate R The maximum error
+ *	of this polynomial approximation is bounded by 2**-58.45. In
+ *	other words,
+ *		        2      4      6      8      10      12      14
+ *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
+ *  	(the values of Lg1 to Lg7 are listed in the program)
+ *	and
+ *	    |      2          14          |     -58.45
+ *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
+ *	    |                             |
+ *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
+ *	In order to guarantee error in log below 1ulp, we compute log
+ *	by
+ *		log(1+f) = f - s*(f - R)	(if f is not too large)
+ *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
+ *
+ *	3. Finally,  log(x) = k*ln2 + log(1+f).
+ *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
+ *	   Here ln2 is split into two floating point number:
+ *			ln2_hi + ln2_lo,
+ *	   where n*ln2_hi is always exact for |n| < 2000.
+ *
+ * Special cases:
+ *	log(x) is NaN with signal if x < 0 (including -INF) ;
+ *	log(+INF) is +INF; log(0) is -INF with signal;
+ *	log(NaN) is that NaN with no signal.
+ *
+ * Accuracy:
+ *	according to an error analysis, the error is always less than
+ *	1 ulp (unit in the last place).
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
+    ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
+    two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
+    Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
+    Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
+    Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
+    Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
+    Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
+    Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
+    Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */
+
+#ifdef __STDC__
+static const double zero = 0.0;
+#else
+static double zero = 0.0;
+#endif
+
+#ifdef __STDC__
+double attribute_hidden
+__ieee754_log(double x)
+#else
+double attribute_hidden
+__ieee754_log(x)
+     double x;
+#endif
+{
+    double hfsq, f, s, z, R, w, t1, t2, dk;
+    int32_t k, hx, i, j;
+    u_int32_t lx;
+
+    EXTRACT_WORDS(hx, lx, x);
+
+    k = 0;
+    if (hx < 0x00100000) {      /* x < 2**-1022  */
+        if (((hx & 0x7fffffff) | lx) == 0)
+            return -two54 / zero;       /* log(+-0)=-inf */
+        if (hx < 0)
+            return (x - x) / zero;      /* log(-#) = NaN */
+        k -= 54;
+        x *= two54;             /* subnormal number, scale up x */
+        GET_HIGH_WORD(hx, x);
+    }
+    if (hx >= 0x7ff00000)
+        return x + x;
+    k += (hx >> 20) - 1023;
+    hx &= 0x000fffff;
+    i = (hx + 0x95f64) & 0x100000;
+    SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
+    k += (i >> 20);
+    f = x - 1.0;
+    if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
+        if (f == zero) {
+            if (k == 0)
+                return zero;
+            else {
+                dk = (double) k;
+                return dk * ln2_hi + dk * ln2_lo;
+            }
+        }
+        R = f * f * (0.5 - 0.33333333333333333 * f);
+        if (k == 0)
+            return f - R;
+        else {
+            dk = (double) k;
+            return dk * ln2_hi - ((R - dk * ln2_lo) - f);
+        }
+    }
+    s = f / (2.0 + f);
+    dk = (double) k;
+    z = s * s;
+    i = hx - 0x6147a;
+    w = z * z;
+    j = 0x6b851 - hx;
+    t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
+    t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
+    i |= j;
+    R = t2 + t1;
+    if (i > 0) {
+        hfsq = 0.5 * f * f;
+        if (k == 0)
+            return f - (hfsq - s * (hfsq + R));
+        else
+            return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
+                                  f);
+    } else {
+        if (k == 0)
+            return f - s * (f - R);
+        else
+            return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
+    }
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/e_pow.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,342 @@
+/* @(#)e_pow.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
+#endif
+
+/* __ieee754_pow(x,y) return x**y
+ *
+ *		      n
+ * Method:  Let x =  2   * (1+f)
+ *	1. Compute and return log2(x) in two pieces:
+ *		log2(x) = w1 + w2,
+ *	   where w1 has 53-24 = 29 bit trailing zeros.
+ *	2. Perform y*log2(x) = n+y' by simulating muti-precision
+ *	   arithmetic, where |y'|<=0.5.
+ *	3. Return x**y = 2**n*exp(y'*log2)
+ *
+ * Special cases:
+ *	1.  (anything) ** 0  is 1
+ *	2.  (anything) ** 1  is itself
+ *	3.  (anything) ** NAN is NAN
+ *	4.  NAN ** (anything except 0) is NAN
+ *	5.  +-(|x| > 1) **  +INF is +INF
+ *	6.  +-(|x| > 1) **  -INF is +0
+ *	7.  +-(|x| < 1) **  +INF is +0
+ *	8.  +-(|x| < 1) **  -INF is +INF
+ *	9.  +-1         ** +-INF is NAN
+ *	10. +0 ** (+anything except 0, NAN)               is +0
+ *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
+ *	12. +0 ** (-anything except 0, NAN)               is +INF
+ *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
+ *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
+ *	15. +INF ** (+anything except 0,NAN) is +INF
+ *	16. +INF ** (-anything except 0,NAN) is +0
+ *	17. -INF ** (anything)  = -0 ** (-anything)
+ *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
+ *	19. (-anything except 0 and inf) ** (non-integer) is NAN
+ *
+ * Accuracy:
+ *	pow(x,y) returns x**y nearly rounded. In particular
+ *			pow(integer,integer)
+ *	always returns the correct integer provided it is
+ *	representable.
+ *
+ * Constants :
+ * The hexadecimal values are the intended ones for the following
+ * constants. The decimal values may be used, provided that the
+ * compiler will convert from decimal to binary accurately enough
+ * to produce the hexadecimal values shown.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(scalbn)
+    libm_hidden_proto(fabs)
+#ifdef __STDC__
+     static const double
+#else
+     static double
+#endif
+       bp[] = { 1.0, 1.5, }, dp_h[] = {
+     0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
+
+         dp_l[] = {
+     0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
+
+         zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0,  /* 0x43400000, 0x00000000 */
+         huge = 1.0e300, tiny = 1.0e-300,
+         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
+         L1 = 5.99999999999994648725e-01,       /* 0x3FE33333, 0x33333303 */
+         L2 = 4.28571428578550184252e-01,       /* 0x3FDB6DB6, 0xDB6FABFF */
+         L3 = 3.33333329818377432918e-01,       /* 0x3FD55555, 0x518F264D */
+         L4 = 2.72728123808534006489e-01,       /* 0x3FD17460, 0xA91D4101 */
+         L5 = 2.30660745775561754067e-01,       /* 0x3FCD864A, 0x93C9DB65 */
+         L6 = 2.06975017800338417784e-01,       /* 0x3FCA7E28, 0x4A454EEF */
+         P1 = 1.66666666666666019037e-01,       /* 0x3FC55555, 0x5555553E */
+         P2 = -2.77777777770155933842e-03,      /* 0xBF66C16C, 0x16BEBD93 */
+         P3 = 6.61375632143793436117e-05,       /* 0x3F11566A, 0xAF25DE2C */
+         P4 = -1.65339022054652515390e-06,      /* 0xBEBBBD41, 0xC5D26BF1 */
+         P5 = 4.13813679705723846039e-08,       /* 0x3E663769, 0x72BEA4D0 */
+         lg2 = 6.93147180559945286227e-01,      /* 0x3FE62E42, 0xFEFA39EF */
+         lg2_h = 6.93147182464599609375e-01,    /* 0x3FE62E43, 0x00000000 */
+         lg2_l = -1.90465429995776804525e-09,   /* 0xBE205C61, 0x0CA86C39 */
+         ovt = 8.0085662595372944372e-0017,     /* -(1024-log2(ovfl+.5ulp)) */
+         cp = 9.61796693925975554329e-01,       /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
+         cp_h = 9.61796700954437255859e-01,     /* 0x3FEEC709, 0xE0000000 =(float)cp */
+         cp_l = -7.02846165095275826516e-09,    /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
+         ivln2 = 1.44269504088896338700e+00,    /* 0x3FF71547, 0x652B82FE =1/ln2 */
+         ivln2_h = 1.44269502162933349609e+00,  /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
+         ivln2_l = 1.92596299112661746887e-08;  /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
+
+#ifdef __STDC__
+     double attribute_hidden __ieee754_pow(double x, double y)
+#else
+     double attribute_hidden __ieee754_pow(x, y)
+     double x, y;
+#endif
+     {
+         double z, ax, z_h, z_l, p_h, p_l;
+         double y1, t1, t2, r, s, t, u, v, w;
+         int32_t i, j, k, yisint, n;
+         int32_t hx, hy, ix, iy;
+         u_int32_t lx, ly;
+
+         EXTRACT_WORDS(hx, lx, x);
+         EXTRACT_WORDS(hy, ly, y);
+         ix = hx & 0x7fffffff;
+         iy = hy & 0x7fffffff;
+
+         /* y==zero: x**0 = 1 */
+         if ((iy | ly) == 0)
+             return one;
+
+         /* +-NaN return x+y */
+         if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
+             iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
+             return x + y;
+
+         /* determine if y is an odd int when x < 0
+          * yisint = 0       ... y is not an integer
+          * yisint = 1       ... y is an odd int
+          * yisint = 2       ... y is an even int
+          */
+         yisint = 0;
+         if (hx < 0) {
+             if (iy >= 0x43400000)
+                 yisint = 2;    /* even integer y */
+             else if (iy >= 0x3ff00000) {
+                 k = (iy >> 20) - 0x3ff;        /* exponent */
+                 if (k > 20) {
+                     j = ly >> (52 - k);
+                     if ((j << (52 - k)) == ly)
+                         yisint = 2 - (j & 1);
+                 } else if (ly == 0) {
+                     j = iy >> (20 - k);
+                     if ((j << (20 - k)) == iy)
+                         yisint = 2 - (j & 1);
+                 }
+             }
+         }
+
+         /* special value of y */
+         if (ly == 0) {
+             if (iy == 0x7ff00000) {    /* y is +-inf */
+                 if (((ix - 0x3ff00000) | lx) == 0)
+                     return y - y;      /* inf**+-1 is NaN */
+                 else if (ix >= 0x3ff00000)     /* (|x|>1)**+-inf = inf,0 */
+                     return (hy >= 0) ? y : zero;
+                 else           /* (|x|<1)**-,+inf = inf,0 */
+                     return (hy < 0) ? -y : zero;
+             }
+             if (iy == 0x3ff00000) {    /* y is  +-1 */
+                 if (hy < 0)
+                     return one / x;
+                 else
+                     return x;
+             }
+             if (hy == 0x40000000)
+                 return x * x;  /* y is  2 */
+             if (hy == 0x3fe00000) {    /* y is  0.5 */
+                 if (hx >= 0)   /* x >= +0 */
+                     return __ieee754_sqrt(x);
+             }
+         }
+
+         ax = fabs(x);
+         /* special value of x */
+         if (lx == 0) {
+             if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
+                 z = ax;        /*x is +-0,+-inf,+-1 */
+                 if (hy < 0)
+                     z = one / z;       /* z = (1/|x|) */
+                 if (hx < 0) {
+                     if (((ix - 0x3ff00000) | yisint) == 0) {
+                         z = (z - z) / (z - z); /* (-1)**non-int is NaN */
+                     } else if (yisint == 1)
+                         z = -z;        /* (x<0)**odd = -(|x|**odd) */
+                 }
+                 return z;
+             }
+         }
+
+         /* (x<0)**(non-int) is NaN */
+         if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
+             return (x - x) / (x - x);
+
+         /* |y| is huge */
+         if (iy > 0x41e00000) { /* if |y| > 2**31 */
+             if (iy > 0x43f00000) {     /* if |y| > 2**64, must o/uflow */
+                 if (ix <= 0x3fefffff)
+                     return (hy < 0) ? huge * huge : tiny * tiny;
+                 if (ix >= 0x3ff00000)
+                     return (hy > 0) ? huge * huge : tiny * tiny;
+             }
+             /* over/underflow if x is not close to one */
+             if (ix < 0x3fefffff)
+                 return (hy < 0) ? huge * huge : tiny * tiny;
+             if (ix > 0x3ff00000)
+                 return (hy > 0) ? huge * huge : tiny * tiny;
+             /* now |1-x| is tiny <= 2**-20, suffice to compute
+                log(x) by x-x^2/2+x^3/3-x^4/4 */
+             t = x - 1;         /* t has 20 trailing zeros */
+             w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
+             u = ivln2_h * t;   /* ivln2_h has 21 sig. bits */
+             v = t * ivln2_l - w * ivln2;
+             t1 = u + v;
+             SET_LOW_WORD(t1, 0);
+             t2 = v - (t1 - u);
+         } else {
+             double s2, s_h, s_l, t_h, t_l;
+             n = 0;
+             /* take care subnormal number */
+             if (ix < 0x00100000) {
+                 ax *= two53;
+                 n -= 53;
+                 GET_HIGH_WORD(ix, ax);
+             }
+             n += ((ix) >> 20) - 0x3ff;
+             j = ix & 0x000fffff;
+             /* determine interval */
+             ix = j | 0x3ff00000;       /* normalize ix */
+             if (j <= 0x3988E)
+                 k = 0;         /* |x|<sqrt(3/2) */
+             else if (j < 0xBB67A)
+                 k = 1;         /* |x|<sqrt(3)   */
+             else {
+                 k = 0;
+                 n += 1;
+                 ix -= 0x00100000;
+             }
+             SET_HIGH_WORD(ax, ix);
+
+             /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
+             u = ax - bp[k];    /* bp[0]=1.0, bp[1]=1.5 */
+             v = one / (ax + bp[k]);
+             s = u * v;
+             s_h = s;
+             SET_LOW_WORD(s_h, 0);
+             /* t_h=ax+bp[k] High */
+             t_h = zero;
+             SET_HIGH_WORD(t_h,
+                           ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
+             t_l = ax - (t_h - bp[k]);
+             s_l = v * ((u - s_h * t_h) - s_h * t_l);
+             /* compute log(ax) */
+             s2 = s * s;
+             r = s2 * s2 * (L1 +
+                            s2 * (L2 +
+                                  s2 * (L3 +
+                                        s2 * (L4 + s2 * (L5 + s2 * L6)))));
+             r += s_l * (s_h + s);
+             s2 = s_h * s_h;
+             t_h = 3.0 + s2 + r;
+             SET_LOW_WORD(t_h, 0);
+             t_l = r - ((t_h - 3.0) - s2);
+             /* u+v = s*(1+...) */
+             u = s_h * t_h;
+             v = s_l * t_h + t_l * s;
+             /* 2/(3log2)*(s+...) */
+             p_h = u + v;
+             SET_LOW_WORD(p_h, 0);
+             p_l = v - (p_h - u);
+             z_h = cp_h * p_h;  /* cp_h+cp_l = 2/(3*log2) */
+             z_l = cp_l * p_h + p_l * cp + dp_l[k];
+             /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
+             t = (double) n;
+             t1 = (((z_h + z_l) + dp_h[k]) + t);
+             SET_LOW_WORD(t1, 0);
+             t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
+         }
+
+         s = one;               /* s (sign of result -ve**odd) = -1 else = 1 */
+         if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
+             s = -one;          /* (-ve)**(odd int) */
+
+         /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
+         y1 = y;
+         SET_LOW_WORD(y1, 0);
+         p_l = (y - y1) * t1 + y * t2;
+         p_h = y1 * t1;
+         z = p_l + p_h;
+         EXTRACT_WORDS(j, i, z);
+         if (j >= 0x40900000) { /* z >= 1024 */
+             if (((j - 0x40900000) | i) != 0)   /* if z > 1024 */
+                 return s * huge * huge;        /* overflow */
+             else {
+                 if (p_l + ovt > z - p_h)
+                     return s * huge * huge;    /* overflow */
+             }
+         } else if ((j & 0x7fffffff) >= 0x4090cc00) {   /* z <= -1075 */
+             if (((j - 0xc090cc00) | i) != 0)   /* z < -1075 */
+                 return s * tiny * tiny;        /* underflow */
+             else {
+                 if (p_l <= z - p_h)
+                     return s * tiny * tiny;    /* underflow */
+             }
+         }
+         /*
+          * compute 2**(p_h+p_l)
+          */
+         i = j & 0x7fffffff;
+         k = (i >> 20) - 0x3ff;
+         n = 0;
+         if (i > 0x3fe00000) {  /* if |z| > 0.5, set n = [z+0.5] */
+             n = j + (0x00100000 >> (k + 1));
+             k = ((n & 0x7fffffff) >> 20) - 0x3ff;      /* new k for n */
+             t = zero;
+             SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
+             n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
+             if (j < 0)
+                 n = -n;
+             p_h -= t;
+         }
+         t = p_l + p_h;
+         SET_LOW_WORD(t, 0);
+         u = t * lg2_h;
+         v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
+         z = u + v;
+         w = v - (z - u);
+         t = z * z;
+         t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
+         r = (z * t1) / (t1 - two) - (w + z * w);
+         z = one - (r - z);
+         GET_HIGH_WORD(j, z);
+         j += (n << 20);
+         if ((j >> 20) <= 0)
+             z = scalbn(z, n);  /* subnormal output */
+         else
+             SET_HIGH_WORD(z, j);
+         return s * z;
+     }
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/e_rem_pio2.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,201 @@
+/* @(#)e_rem_pio2.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] =
+    "$NetBSD: e_rem_pio2.c,v 1.8 1995/05/10 20:46:02 jtc Exp $";
+#endif
+
+/* __ieee754_rem_pio2(x,y)
+ *
+ * return the remainder of x rem pi/2 in y[0]+y[1]
+ * use __kernel_rem_pio2()
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(fabs)
+
+/*
+ * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
+ */
+#ifdef __STDC__
+     static const int32_t two_over_pi[] = {
+#else
+     static int32_t two_over_pi[] = {
+#endif
+         0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
+         0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
+         0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
+         0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
+         0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
+         0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
+         0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
+         0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
+         0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
+         0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
+         0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
+     };
+
+#ifdef __STDC__
+static const int32_t npio2_hw[] = {
+#else
+static int32_t npio2_hw[] = {
+#endif
+    0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
+    0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
+    0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
+    0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
+    0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
+    0x404858EB, 0x404921FB,
+};
+
+/*
+ * invpio2:  53 bits of 2/pi
+ * pio2_1:   first  33 bit of pi/2
+ * pio2_1t:  pi/2 - pio2_1
+ * pio2_2:   second 33 bit of pi/2
+ * pio2_2t:  pi/2 - (pio2_1+pio2_2)
+ * pio2_3:   third  33 bit of pi/2
+ * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
+ */
+
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+  zero = 0.00000000000000000000e+00,    /* 0x00000000, 0x00000000 */
+    half = 5.00000000000000000000e-01,  /* 0x3FE00000, 0x00000000 */
+    two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
+    invpio2 = 6.36619772367581382433e-01,       /* 0x3FE45F30, 0x6DC9C883 */
+    pio2_1 = 1.57079632673412561417e+00,        /* 0x3FF921FB, 0x54400000 */
+    pio2_1t = 6.07710050650619224932e-11,       /* 0x3DD0B461, 0x1A626331 */
+    pio2_2 = 6.07710050630396597660e-11,        /* 0x3DD0B461, 0x1A600000 */
+    pio2_2t = 2.02226624879595063154e-21,       /* 0x3BA3198A, 0x2E037073 */
+    pio2_3 = 2.02226624871116645580e-21,        /* 0x3BA3198A, 0x2E000000 */
+    pio2_3t = 8.47842766036889956997e-32;       /* 0x397B839A, 0x252049C1 */
+
+#ifdef __STDC__
+int32_t attribute_hidden
+__ieee754_rem_pio2(double x, double *y)
+#else
+int32_t attribute_hidden
+__ieee754_rem_pio2(x, y)
+     double x, y[];
+#endif
+{
+    double z = 0.0, w, t, r, fn;
+    double tx[3];
+    int32_t e0, i, j, nx, n, ix, hx;
+    u_int32_t low;
+
+    GET_HIGH_WORD(hx, x);       /* high word of x */
+    ix = hx & 0x7fffffff;
+    if (ix <= 0x3fe921fb) {     /* |x| ~<= pi/4 , no need for reduction */
+        y[0] = x;
+        y[1] = 0;
+        return 0;
+    }
+    if (ix < 0x4002d97c) {      /* |x| < 3pi/4, special case with n=+-1 */
+        if (hx > 0) {
+            z = x - pio2_1;
+            if (ix != 0x3ff921fb) {     /* 33+53 bit pi is good enough */
+                y[0] = z - pio2_1t;
+                y[1] = (z - y[0]) - pio2_1t;
+            } else {            /* near pi/2, use 33+33+53 bit pi */
+                z -= pio2_2;
+                y[0] = z - pio2_2t;
+                y[1] = (z - y[0]) - pio2_2t;
+            }
+            return 1;
+        } else {                /* negative x */
+            z = x + pio2_1;
+            if (ix != 0x3ff921fb) {     /* 33+53 bit pi is good enough */
+                y[0] = z + pio2_1t;
+                y[1] = (z - y[0]) + pio2_1t;
+            } else {            /* near pi/2, use 33+33+53 bit pi */
+                z += pio2_2;
+                y[0] = z + pio2_2t;
+                y[1] = (z - y[0]) + pio2_2t;
+            }
+            return -1;
+        }
+    }
+    if (ix <= 0x413921fb) {     /* |x| ~<= 2^19*(pi/2), medium size */
+        t = fabs(x);
+        n = (int32_t) (t * invpio2 + half);
+        fn = (double) n;
+        r = t - fn * pio2_1;
+        w = fn * pio2_1t;       /* 1st round good to 85 bit */
+        if (n < 32 && ix != npio2_hw[n - 1]) {
+            y[0] = r - w;       /* quick check no cancellation */
+        } else {
+            u_int32_t high;
+            j = ix >> 20;
+            y[0] = r - w;
+            GET_HIGH_WORD(high, y[0]);
+            i = j - ((high >> 20) & 0x7ff);
+            if (i > 16) {       /* 2nd iteration needed, good to 118 */
+                t = r;
+                w = fn * pio2_2;
+                r = t - w;
+                w = fn * pio2_2t - ((t - r) - w);
+                y[0] = r - w;
+                GET_HIGH_WORD(high, y[0]);
+                i = j - ((high >> 20) & 0x7ff);
+                if (i > 49) {   /* 3rd iteration need, 151 bits acc */
+                    t = r;      /* will cover all possible cases */
+                    w = fn * pio2_3;
+                    r = t - w;
+                    w = fn * pio2_3t - ((t - r) - w);
+                    y[0] = r - w;
+                }
+            }
+        }
+        y[1] = (r - y[0]) - w;
+        if (hx < 0) {
+            y[0] = -y[0];
+            y[1] = -y[1];
+            return -n;
+        } else
+            return n;
+    }
+    /*
+     * all other (large) arguments
+     */
+    if (ix >= 0x7ff00000) {     /* x is inf or NaN */
+        y[0] = y[1] = x - x;
+        return 0;
+    }
+    /* set z = scalbn(|x|,ilogb(x)-23) */
+    GET_LOW_WORD(low, x);
+    SET_LOW_WORD(z, low);
+    e0 = (ix >> 20) - 1046;     /* e0 = ilogb(z)-23; */
+    SET_HIGH_WORD(z, ix - ((int32_t) (e0 << 20)));
+    for (i = 0; i < 2; i++) {
+        tx[i] = (double) ((int32_t) (z));
+        z = (z - tx[i]) * two24;
+    }
+    tx[2] = z;
+    nx = 3;
+    while (tx[nx - 1] == zero)
+        nx--;                   /* skip zero term */
+    n = __kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi);
+    if (hx < 0) {
+        y[0] = -y[0];
+        y[1] = -y[1];
+        return -n;
+    }
+    return n;
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/e_sqrt.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,463 @@
+/* @(#)e_sqrt.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
+#endif
+
+/* __ieee754_sqrt(x)
+ * Return correctly rounded sqrt.
+ *           ------------------------------------------
+ *	     |  Use the hardware sqrt if you have one |
+ *           ------------------------------------------
+ * Method:
+ *   Bit by bit method using integer arithmetic. (Slow, but portable)
+ *   1. Normalization
+ *	Scale x to y in [1,4) with even powers of 2:
+ *	find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
+ *		sqrt(x) = 2^k * sqrt(y)
+ *   2. Bit by bit computation
+ *	Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
+ *	     i							 0
+ *                                     i+1         2
+ *	    s  = 2*q , and	y  =  2   * ( y - q  ).		(1)
+ *	     i      i            i                 i
+ *
+ *	To compute q    from q , one checks whether
+ *		    i+1       i
+ *
+ *			      -(i+1) 2
+ *			(q + 2      ) <= y.			(2)
+ *     			  i
+ *							      -(i+1)
+ *	If (2) is false, then q   = q ; otherwise q   = q  + 2      .
+ *		 	       i+1   i             i+1   i
+ *
+ *	With some algebric manipulation, it is not difficult to see
+ *	that (2) is equivalent to
+ *                             -(i+1)
+ *			s  +  2       <= y			(3)
+ *			 i                i
+ *
+ *	The advantage of (3) is that s  and y  can be computed by
+ *				      i      i
+ *	the following recurrence formula:
+ *	    if (3) is false
+ *
+ *	    s     =  s  ,	y    = y   ;			(4)
+ *	     i+1      i		 i+1    i
+ *
+ *	    otherwise,
+ *                         -i                     -(i+1)
+ *	    s	  =  s  + 2  ,  y    = y  -  s  - 2  		(5)
+ *           i+1      i          i+1    i     i
+ *
+ *	One may easily use induction to prove (4) and (5).
+ *	Note. Since the left hand side of (3) contain only i+2 bits,
+ *	      it does not necessary to do a full (53-bit) comparison
+ *	      in (3).
+ *   3. Final rounding
+ *	After generating the 53 bits result, we compute one more bit.
+ *	Together with the remainder, we can decide whether the
+ *	result is exact, bigger than 1/2ulp, or less than 1/2ulp
+ *	(it will never equal to 1/2ulp).
+ *	The rounding mode can be detected by checking whether
+ *	huge + tiny is equal to huge, and whether huge - tiny is
+ *	equal to huge for some floating point number "huge" and "tiny".
+ *
+ * Special cases:
+ *	sqrt(+-0) = +-0 	... exact
+ *	sqrt(inf) = inf
+ *	sqrt(-ve) = NaN		... with invalid signal
+ *	sqrt(NaN) = NaN		... with invalid signal for signaling NaN
+ *
+ * Other methods : see the appended file at the end of the program below.
+ *---------------
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const double one = 1.0, tiny = 1.0e-300;
+#else
+static double one = 1.0, tiny = 1.0e-300;
+#endif
+
+#ifdef __STDC__
+double attribute_hidden
+__ieee754_sqrt(double x)
+#else
+double attribute_hidden
+__ieee754_sqrt(x)
+     double x;
+#endif
+{
+    double z;
+    int32_t sign = (int) 0x80000000;
+    int32_t ix0, s0, q, m, t, i;
+    u_int32_t r, t1, s1, ix1, q1;
+
+    EXTRACT_WORDS(ix0, ix1, x);
+
+    /* take care of Inf and NaN */
+    if ((ix0 & 0x7ff00000) == 0x7ff00000) {
+        return x * x + x;       /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
+                                   sqrt(-inf)=sNaN */
+    }
+    /* take care of zero */
+    if (ix0 <= 0) {
+        if (((ix0 & (~sign)) | ix1) == 0)
+            return x;           /* sqrt(+-0) = +-0 */
+        else if (ix0 < 0)
+            return (x - x) / (x - x);   /* sqrt(-ve) = sNaN */
+    }
+    /* normalize x */
+    m = (ix0 >> 20);
+    if (m == 0) {               /* subnormal x */
+        while (ix0 == 0) {
+            m -= 21;
+            ix0 |= (ix1 >> 11);
+            ix1 <<= 21;
+        }
+        for (i = 0; (ix0 & 0x00100000) == 0; i++)
+            ix0 <<= 1;
+        m -= i - 1;
+        ix0 |= (ix1 >> (32 - i));
+        ix1 <<= i;
+    }
+    m -= 1023;                  /* unbias exponent */
+    ix0 = (ix0 & 0x000fffff) | 0x00100000;
+    if (m & 1) {                /* odd m, double x to make it even */
+        ix0 += ix0 + ((ix1 & sign) >> 31);
+        ix1 += ix1;
+    }
+    m >>= 1;                    /* m = [m/2] */
+
+    /* generate sqrt(x) bit by bit */
+    ix0 += ix0 + ((ix1 & sign) >> 31);
+    ix1 += ix1;
+    q = q1 = s0 = s1 = 0;       /* [q,q1] = sqrt(x) */
+    r = 0x00200000;             /* r = moving bit from right to left */
+
+    while (r != 0) {
+        t = s0 + r;
+        if (t <= ix0) {
+            s0 = t + r;
+            ix0 -= t;
+            q += r;
+        }
+        ix0 += ix0 + ((ix1 & sign) >> 31);
+        ix1 += ix1;
+        r >>= 1;
+    }
+
+    r = sign;
+    while (r != 0) {
+        t1 = s1 + r;
+        t = s0;
+        if ((t < ix0) || ((t == ix0) && (t1 <= ix1))) {
+            s1 = t1 + r;
+            if (((t1 & sign) == sign) && (s1 & sign) == 0)
+                s0 += 1;
+            ix0 -= t;
+            if (ix1 < t1)
+                ix0 -= 1;
+            ix1 -= t1;
+            q1 += r;
+        }
+        ix0 += ix0 + ((ix1 & sign) >> 31);
+        ix1 += ix1;
+        r >>= 1;
+    }
+
+    /* use floating add to find out rounding direction */
+    if ((ix0 | ix1) != 0) {
+        z = one - tiny;         /* trigger inexact flag */
+        if (z >= one) {
+            z = one + tiny;
+            if (q1 == (u_int32_t) 0xffffffff) {
+                q1 = 0;
+                q += 1;
+            } else if (z > one) {
+                if (q1 == (u_int32_t) 0xfffffffe)
+                    q += 1;
+                q1 += 2;
+            } else
+                q1 += (q1 & 1);
+        }
+    }
+    ix0 = (q >> 1) + 0x3fe00000;
+    ix1 = q1 >> 1;
+    if ((q & 1) == 1)
+        ix1 |= sign;
+    ix0 += (m << 20);
+    INSERT_WORDS(z, ix0, ix1);
+    return z;
+}
+
+/*
+Other methods  (use floating-point arithmetic)
+-------------
+(This is a copy of a drafted paper by Prof W. Kahan
+and K.C. Ng, written in May, 1986)
+
+	Two algorithms are given here to implement sqrt(x)
+	(IEEE double precision arithmetic) in software.
+	Both supply sqrt(x) correctly rounded. The first algorithm (in
+	Section A) uses newton iterations and involves four divisions.
+	The second one uses reciproot iterations to avoid division, but
+	requires more multiplications. Both algorithms need the ability
+	to chop results of arithmetic operations instead of round them,
+	and the INEXACT flag to indicate when an arithmetic operation
+	is executed exactly with no roundoff error, all part of the
+	standard (IEEE 754-1985). The ability to perform shift, add,
+	subtract and logical AND operations upon 32-bit words is needed
+	too, though not part of the standard.
+
+A.  sqrt(x) by Newton Iteration
+
+   (1)	Initial approximation
+
+	Let x0 and x1 be the leading and the trailing 32-bit words of
+	a floating point number x (in IEEE double format) respectively
+
+	    1    11		     52				  ...widths
+	   ------------------------------------------------------
+	x: |s|	  e     |	      f				|
+	   ------------------------------------------------------
+	      msb    lsb  msb				      lsb ...order
+
+
+	     ------------------------  	     ------------------------
+	x0:  |s|   e    |    f1     |	 x1: |          f2           |
+	     ------------------------  	     ------------------------
+
+	By performing shifts and subtracts on x0 and x1 (both regarded
+	as integers), we obtain an 8-bit approximation of sqrt(x) as
+	follows.
+
+		k  := (x0>>1) + 0x1ff80000;
+		y0 := k - T1[31&(k>>15)].	... y ~ sqrt(x) to 8 bits
+	Here k is a 32-bit integer and T1[] is an integer array containing
+	correction terms. Now magically the floating value of y (y's
+	leading 32-bit word is y0, the value of its trailing word is 0)
+	approximates sqrt(x) to almost 8-bit.
+
+	Value of T1:
+	static int T1[32]= {
+	0,	1024,	3062,	5746,	9193,	13348,	18162,	23592,
+	29598,	36145,	43202,	50740,	58733,	67158,	75992,	85215,
+	83599,	71378,	60428,	50647,	41945,	34246,	27478,	21581,
+	16499,	12183,	8588,	5674,	3403,	1742,	661,	130,};
+
+    (2)	Iterative refinement
+
+	Apply Heron's rule three times to y, we have y approximates
+	sqrt(x) to within 1 ulp (Unit in the Last Place):
+
+		y := (y+x/y)/2		... almost 17 sig. bits
+		y := (y+x/y)/2		... almost 35 sig. bits
+		y := y-(y-x/y)/2	... within 1 ulp
+
+
+	Remark 1.
+	    Another way to improve y to within 1 ulp is:
+
+		y := (y+x/y)		... almost 17 sig. bits to 2*sqrt(x)
+		y := y - 0x00100006	... almost 18 sig. bits to sqrt(x)
+
+				2
+			    (x-y )*y
+		y := y + 2* ----------	...within 1 ulp
+			       2
+			     3y  + x
+
+
+	This formula has one division fewer than the one above; however,
+	it requires more multiplications and additions. Also x must be
+	scaled in advance to avoid spurious overflow in evaluating the
+	expression 3y*y+x. Hence it is not recommended uless division
+	is slow. If division is very slow, then one should use the
+	reciproot algorithm given in section B.
+
+    (3) Final adjustment
+
+	By twiddling y's last bit it is possible to force y to be
+	correctly rounded according to the prevailing rounding mode
+	as follows. Let r and i be copies of the rounding mode and
+	inexact flag before entering the square root program. Also we
+	use the expression y+-ulp for the next representable floating
+	numbers (up and down) of y. Note that y+-ulp = either fixed
+	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
+	mode.
+
+		I := FALSE;	... reset INEXACT flag I
+		R := RZ;	... set rounding mode to round-toward-zero
+		z := x/y;	... chopped quotient, possibly inexact
+		If(not I) then {	... if the quotient is exact
+		    if(z=y) {
+		        I := i;	 ... restore inexact flag
+		        R := r;  ... restore rounded mode
+		        return sqrt(x):=y.
+		    } else {
+			z := z - ulp;	... special rounding
+		    }
+		}
+		i := TRUE;		... sqrt(x) is inexact
+		If (r=RN) then z=z+ulp	... rounded-to-nearest
+		If (r=RP) then {	... round-toward-+inf
+		    y = y+ulp; z=z+ulp;
+		}
+		y := y+z;		... chopped sum
+		y0:=y0-0x00100000;	... y := y/2 is correctly rounded.
+	        I := i;	 		... restore inexact flag
+	        R := r;  		... restore rounded mode
+	        return sqrt(x):=y.
+
+    (4)	Special cases
+
+	Square root of +inf, +-0, or NaN is itself;
+	Square root of a negative number is NaN with invalid signal.
+
+
+B.  sqrt(x) by Reciproot Iteration
+
+   (1)	Initial approximation
+
+	Let x0 and x1 be the leading and the trailing 32-bit words of
+	a floating point number x (in IEEE double format) respectively
+	(see section A). By performing shifs and subtracts on x0 and y0,
+	we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
+
+	    k := 0x5fe80000 - (x0>>1);
+	    y0:= k - T2[63&(k>>14)].	... y ~ 1/sqrt(x) to 7.8 bits
+
+	Here k is a 32-bit integer and T2[] is an integer array
+	containing correction terms. Now magically the floating
+	value of y (y's leading 32-bit word is y0, the value of
+	its trailing word y1 is set to zero) approximates 1/sqrt(x)
+	to almost 7.8-bit.
+
+	Value of T2:
+	static int T2[64]= {
+	0x1500,	0x2ef8,	0x4d67,	0x6b02,	0x87be,	0xa395,	0xbe7a,	0xd866,
+	0xf14a,	0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
+	0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
+	0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
+	0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
+	0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
+	0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
+	0x1527f,0x1334a,0x11051,0xe951,	0xbe01,	0x8e0d,	0x5924,	0x1edd,};
+
+    (2)	Iterative refinement
+
+	Apply Reciproot iteration three times to y and multiply the
+	result by x to get an approximation z that matches sqrt(x)
+	to about 1 ulp. To be exact, we will have
+		-1ulp < sqrt(x)-z<1.0625ulp.
+
+	... set rounding mode to Round-to-nearest
+	   y := y*(1.5-0.5*x*y*y)	... almost 15 sig. bits to 1/sqrt(x)
+	   y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
+	... special arrangement for better accuracy
+	   z := x*y			... 29 bits to sqrt(x), with z*y<1
+	   z := z + 0.5*z*(1-z*y)	... about 1 ulp to sqrt(x)
+
+	Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
+	(a) the term z*y in the final iteration is always less than 1;
+	(b) the error in the final result is biased upward so that
+		-1 ulp < sqrt(x) - z < 1.0625 ulp
+	    instead of |sqrt(x)-z|<1.03125ulp.
+
+    (3)	Final adjustment
+
+	By twiddling y's last bit it is possible to force y to be
+	correctly rounded according to the prevailing rounding mode
+	as follows. Let r and i be copies of the rounding mode and
+	inexact flag before entering the square root program. Also we
+	use the expression y+-ulp for the next representable floating
+	numbers (up and down) of y. Note that y+-ulp = either fixed
+	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
+	mode.
+
+	R := RZ;		... set rounding mode to round-toward-zero
+	switch(r) {
+	    case RN:		... round-to-nearest
+	       if(x<= z*(z-ulp)...chopped) z = z - ulp; else
+	       if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
+	       break;
+	    case RZ:case RM:	... round-to-zero or round-to--inf
+	       R:=RP;		... reset rounding mod to round-to-+inf
+	       if(x<z*z ... rounded up) z = z - ulp; else
+	       if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
+	       break;
+	    case RP:		... round-to-+inf
+	       if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
+	       if(x>z*z ...chopped) z = z+ulp;
+	       break;
+	}
+
+	Remark 3. The above comparisons can be done in fixed point. For
+	example, to compare x and w=z*z chopped, it suffices to compare
+	x1 and w1 (the trailing parts of x and w), regarding them as
+	two's complement integers.
+
+	...Is z an exact square root?
+	To determine whether z is an exact square root of x, let z1 be the
+	trailing part of z, and also let x0 and x1 be the leading and
+	trailing parts of x.
+
+	If ((z1&0x03ffffff)!=0)	... not exact if trailing 26 bits of z!=0
+	    I := 1;		... Raise Inexact flag: z is not exact
+	else {
+	    j := 1 - [(x0>>20)&1]	... j = logb(x) mod 2
+	    k := z1 >> 26;		... get z's 25-th and 26-th
+					    fraction bits
+	    I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
+	}
+	R:= r		... restore rounded mode
+	return sqrt(x):=z.
+
+	If multiplication is cheaper then the foregoing red tape, the
+	Inexact flag can be evaluated by
+
+	    I := i;
+	    I := (z*z!=x) or I.
+
+	Note that z*z can overwrite I; this value must be sensed if it is
+	True.
+
+	Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
+	zero.
+
+		    --------------------
+		z1: |        f2        |
+		    --------------------
+		bit 31		   bit 0
+
+	Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
+	or even of logb(x) have the following relations:
+
+	-------------------------------------------------
+	bit 27,26 of z1		bit 1,0 of x1	logb(x)
+	-------------------------------------------------
+	00			00		odd and even
+	01			01		even
+	10			10		odd
+	10			00		even
+	11			01		even
+	-------------------------------------------------
+
+    (4)	Special cases (see (4) of Section A).
+
+ */
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/k_cos.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,99 @@
+/* @(#)k_cos.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
+#endif
+
+/*
+ * __kernel_cos( x,  y )
+ * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
+ * Input x is assumed to be bounded by ~pi/4 in magnitude.
+ * Input y is the tail of x.
+ *
+ * Algorithm
+ *	1. Since cos(-x) = cos(x), we need only to consider positive x.
+ *	2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
+ *	3. cos(x) is approximated by a polynomial of degree 14 on
+ *	   [0,pi/4]
+ *		  	                 4            14
+ *	   	cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
+ *	   where the remez error is
+ *
+ * 	|              2     4     6     8     10    12     14 |     -58
+ * 	|cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
+ * 	|    					               |
+ *
+ * 	               4     6     8     10    12     14
+ *	4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
+ *	       cos(x) = 1 - x*x/2 + r
+ *	   since cos(x+y) ~ cos(x) - sin(x)*y
+ *			  ~ cos(x) - x*y,
+ *	   a correction term is necessary in cos(x) and hence
+ *		cos(x+y) = 1 - (x*x/2 - (r - x*y))
+ *	   For better accuracy when x > 0.3, let qx = |x|/4 with
+ *	   the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
+ *	   Then
+ *		cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
+ *	   Note that 1-qx and (x*x/2-qx) is EXACT here, and the
+ *	   magnitude of the latter is at least a quarter of x*x/2,
+ *	   thus, reducing the rounding error in the subtraction.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+  one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
+    C1 = 4.16666666666666019037e-02,    /* 0x3FA55555, 0x5555554C */
+    C2 = -1.38888888888741095749e-03,   /* 0xBF56C16C, 0x16C15177 */
+    C3 = 2.48015872894767294178e-05,    /* 0x3EFA01A0, 0x19CB1590 */
+    C4 = -2.75573143513906633035e-07,   /* 0xBE927E4F, 0x809C52AD */
+    C5 = 2.08757232129817482790e-09,    /* 0x3E21EE9E, 0xBDB4B1C4 */
+    C6 = -1.13596475577881948265e-11;   /* 0xBDA8FAE9, 0xBE8838D4 */
+
+#ifdef __STDC__
+double attribute_hidden
+__kernel_cos(double x, double y)
+#else
+double attribute_hidden
+__kernel_cos(x, y)
+     double x, y;
+#endif
+{
+    double a, hz, z, r, qx;
+    int32_t ix;
+    GET_HIGH_WORD(ix, x);
+    ix &= 0x7fffffff;           /* ix = |x|'s high word */
+    if (ix < 0x3e400000) {      /* if x < 2**27 */
+        if (((int) x) == 0)
+            return one;         /* generate inexact */
+    }
+    z = x * x;
+    r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
+    if (ix < 0x3FD33333)        /* if |x| < 0.3 */
+        return one - (0.5 * z - (z * r - x * y));
+    else {
+        if (ix > 0x3fe90000) {  /* x > 0.78125 */
+            qx = 0.28125;
+        } else {
+            INSERT_WORDS(qx, ix - 0x00200000, 0);       /* x/4 */
+        }
+        hz = 0.5 * z - qx;
+        a = one - qx;
+        return a - (hz - (z * r - x * y));
+    }
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/k_sin.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,86 @@
+/* @(#)k_sin.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: k_sin.c,v 1.8 1995/05/10 20:46:31 jtc Exp $";
+#endif
+
+/* __kernel_sin( x, y, iy)
+ * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854
+ * Input x is assumed to be bounded by ~pi/4 in magnitude.
+ * Input y is the tail of x.
+ * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
+ *
+ * Algorithm
+ *	1. Since sin(-x) = -sin(x), we need only to consider positive x.
+ *	2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0.
+ *	3. sin(x) is approximated by a polynomial of degree 13 on
+ *	   [0,pi/4]
+ *		  	         3            13
+ *	   	sin(x) ~ x + S1*x + ... + S6*x
+ *	   where
+ *
+ * 	|sin(x)         2     4     6     8     10     12  |     -58
+ * 	|----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x  +S6*x   )| <= 2
+ * 	|  x 					           |
+ *
+ *	4. sin(x+y) = sin(x) + sin'(x')*y
+ *		    ~ sin(x) + (1-x*x/2)*y
+ *	   For better accuracy, let
+ *		     3      2      2      2      2
+ *		r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
+ *	   then                   3    2
+ *		sin(x) = x + (S1*x + (x *(r-y/2)+y))
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const double
+#else
+static double
+#endif
+  half = 5.00000000000000000000e-01,    /* 0x3FE00000, 0x00000000 */
+    S1 = -1.66666666666666324348e-01,   /* 0xBFC55555, 0x55555549 */
+    S2 = 8.33333333332248946124e-03,    /* 0x3F811111, 0x1110F8A6 */
+    S3 = -1.98412698298579493134e-04,   /* 0xBF2A01A0, 0x19C161D5 */
+    S4 = 2.75573137070700676789e-06,    /* 0x3EC71DE3, 0x57B1FE7D */
+    S5 = -2.50507602534068634195e-08,   /* 0xBE5AE5E6, 0x8A2B9CEB */
+    S6 = 1.58969099521155010221e-10;    /* 0x3DE5D93A, 0x5ACFD57C */
+
+#ifdef __STDC__
+double attribute_hidden
+__kernel_sin(double x, double y, int iy)
+#else
+double attribute_hidden
+__kernel_sin(x, y, iy)
+     double x, y;
+     int iy;                    /* iy=0 if y is zero */
+#endif
+{
+    double z, r, v;
+    int32_t ix;
+    GET_HIGH_WORD(ix, x);
+    ix &= 0x7fffffff;           /* high word of x */
+    if (ix < 0x3e400000) {      /* |x| < 2**-27 */
+        if ((int) x == 0)
+            return x;
+    }                           /* generate inexact */
+    z = x * x;
+    v = z * x;
+    r = S2 + z * (S3 + z * (S4 + z * (S5 + z * S6)));
+    if (iy == 0)
+        return x + v * (S1 + z * r);
+    else
+        return x - ((z * (half * y - v * r) - y) - v * S1);
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/math.h	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,44 @@
+/*
+    SDL - Simple DirectMedia Layer
+    Copyright (C) 1997-2006 Sam Lantinga
+
+    This library is free software; you can redistribute it and/or
+    modify it under the terms of the GNU Lesser General Public
+    License as published by the Free Software Foundation; either
+    version 2.1 of the License, or (at your option) any later version.
+
+    This library is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+    Lesser General Public License for more details.
+
+    You should have received a copy of the GNU Lesser General Public
+    License along with this library; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
+
+    Sam Lantinga
+    slouken@libsdl.org
+*/
+#include "SDL_config.h"
+
+#ifdef HAVE_MATH_H
+#include <math.h>
+#else
+
+extern double __ieee754_log(double x);
+extern double __ieee754_pow(double x, double y);
+extern double __ieee754_sqrt(double x);
+
+#define log(x)      __ieee754_log(x)
+#define pow(x, y)   __ieee754_pow(x, y)
+#define sqrt(x)     __ieee754_sqrt(x)
+
+extern double copysign(double x, double y);
+extern double cos(double x);
+extern double fabs(double x);
+extern double scalbn(double x, int n);
+extern double sin(double x);
+
+#endif /* HAVE_MATH_H */
+
+/* vi: set ts=4 sw=4 expandtab: */
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/math_private.h	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,204 @@
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+/*
+ * from: @(#)fdlibm.h 5.1 93/09/24
+ * $Id: math_private.h,v 1.3 2004/02/09 07:10:38 andersen Exp $
+ */
+
+#ifndef _MATH_PRIVATE_H_
+#define _MATH_PRIVATE_H_
+
+/*#include <endian.h>*/
+#include <sys/types.h>
+
+#define attribute_hidden
+#define libm_hidden_proto(x)
+#define libm_hidden_def(x)
+
+/* The original fdlibm code used statements like:
+	n0 = ((*(int*)&one)>>29)^1;		* index of high word *
+	ix0 = *(n0+(int*)&x);			* high word of x *
+	ix1 = *((1-n0)+(int*)&x);		* low word of x *
+   to dig two 32 bit words out of the 64 bit IEEE floating point
+   value.  That is non-ANSI, and, moreover, the gcc instruction
+   scheduler gets it wrong.  We instead use the following macros.
+   Unlike the original code, we determine the endianness at compile
+   time, not at run time; I don't see much benefit to selecting
+   endianness at run time.  */
+
+/* A union which permits us to convert between a double and two 32 bit
+   ints.  */
+
+/*
+ * Math on arm is special:
+ * For FPA, float words are always big-endian.
+ * For VFP, floats words follow the memory system mode.
+ */
+
+#if (__BYTE_ORDER == __BIG_ENDIAN) || \
+    (!defined(__VFP_FP__) && (defined(__arm__) || defined(__thumb__)))
+
+typedef union
+{
+    double value;
+    struct
+    {
+        u_int32_t msw;
+        u_int32_t lsw;
+    } parts;
+} ieee_double_shape_type;
+
+#else
+
+typedef union
+{
+    double value;
+    struct
+    {
+        u_int32_t lsw;
+        u_int32_t msw;
+    } parts;
+} ieee_double_shape_type;
+
+#endif
+
+/* Get two 32 bit ints from a double.  */
+
+#define EXTRACT_WORDS(ix0,ix1,d)				\
+do {								\
+  ieee_double_shape_type ew_u;					\
+  ew_u.value = (d);						\
+  (ix0) = ew_u.parts.msw;					\
+  (ix1) = ew_u.parts.lsw;					\
+} while (0)
+
+/* Get the more significant 32 bit int from a double.  */
+
+#define GET_HIGH_WORD(i,d)					\
+do {								\
+  ieee_double_shape_type gh_u;					\
+  gh_u.value = (d);						\
+  (i) = gh_u.parts.msw;						\
+} while (0)
+
+/* Get the less significant 32 bit int from a double.  */
+
+#define GET_LOW_WORD(i,d)					\
+do {								\
+  ieee_double_shape_type gl_u;					\
+  gl_u.value = (d);						\
+  (i) = gl_u.parts.lsw;						\
+} while (0)
+
+/* Set a double from two 32 bit ints.  */
+
+#define INSERT_WORDS(d,ix0,ix1)					\
+do {								\
+  ieee_double_shape_type iw_u;					\
+  iw_u.parts.msw = (ix0);					\
+  iw_u.parts.lsw = (ix1);					\
+  (d) = iw_u.value;						\
+} while (0)
+
+/* Set the more significant 32 bits of a double from an int.  */
+
+#define SET_HIGH_WORD(d,v)					\
+do {								\
+  ieee_double_shape_type sh_u;					\
+  sh_u.value = (d);						\
+  sh_u.parts.msw = (v);						\
+  (d) = sh_u.value;						\
+} while (0)
+
+/* Set the less significant 32 bits of a double from an int.  */
+
+#define SET_LOW_WORD(d,v)					\
+do {								\
+  ieee_double_shape_type sl_u;					\
+  sl_u.value = (d);						\
+  sl_u.parts.lsw = (v);						\
+  (d) = sl_u.value;						\
+} while (0)
+
+/* A union which permits us to convert between a float and a 32 bit
+   int.  */
+
+typedef union
+{
+    float value;
+    u_int32_t word;
+} ieee_float_shape_type;
+
+/* Get a 32 bit int from a float.  */
+
+#define GET_FLOAT_WORD(i,d)					\
+do {								\
+  ieee_float_shape_type gf_u;					\
+  gf_u.value = (d);						\
+  (i) = gf_u.word;						\
+} while (0)
+
+/* Set a float from a 32 bit int.  */
+
+#define SET_FLOAT_WORD(d,i)					\
+do {								\
+  ieee_float_shape_type sf_u;					\
+  sf_u.word = (i);						\
+  (d) = sf_u.value;						\
+} while (0)
+
+/* ieee style elementary functions */
+extern double
+__ieee754_sqrt(double)
+    attribute_hidden;
+     extern double __ieee754_acos(double) attribute_hidden;
+     extern double __ieee754_acosh(double) attribute_hidden;
+     extern double __ieee754_log(double) attribute_hidden;
+     extern double __ieee754_atanh(double) attribute_hidden;
+     extern double __ieee754_asin(double) attribute_hidden;
+     extern double __ieee754_atan2(double, double) attribute_hidden;
+     extern double __ieee754_exp(double) attribute_hidden;
+     extern double __ieee754_cosh(double) attribute_hidden;
+     extern double __ieee754_fmod(double, double) attribute_hidden;
+     extern double __ieee754_pow(double, double) attribute_hidden;
+     extern double __ieee754_lgamma_r(double, int *) attribute_hidden;
+     extern double __ieee754_gamma_r(double, int *) attribute_hidden;
+     extern double __ieee754_lgamma(double) attribute_hidden;
+     extern double __ieee754_gamma(double) attribute_hidden;
+     extern double __ieee754_log10(double) attribute_hidden;
+     extern double __ieee754_sinh(double) attribute_hidden;
+     extern double __ieee754_hypot(double, double) attribute_hidden;
+     extern double __ieee754_j0(double) attribute_hidden;
+     extern double __ieee754_j1(double) attribute_hidden;
+     extern double __ieee754_y0(double) attribute_hidden;
+     extern double __ieee754_y1(double) attribute_hidden;
+     extern double __ieee754_jn(int, double) attribute_hidden;
+     extern double __ieee754_yn(int, double) attribute_hidden;
+     extern double __ieee754_remainder(double, double) attribute_hidden;
+     extern int __ieee754_rem_pio2(double, double *) attribute_hidden;
+#if defined(_SCALB_INT)
+     extern double __ieee754_scalb(double, int) attribute_hidden;
+#else
+     extern double __ieee754_scalb(double, double) attribute_hidden;
+#endif
+
+/* fdlibm kernel function */
+#ifndef _IEEE_LIBM
+     extern double __kernel_standard(double, double, int) attribute_hidden;
+#endif
+     extern double __kernel_sin(double, double, int) attribute_hidden;
+     extern double __kernel_cos(double, double) attribute_hidden;
+     extern double __kernel_tan(double, double, int) attribute_hidden;
+     extern int __kernel_rem_pio2(double *, double *, int, int, int,
+                                  const int *) attribute_hidden;
+
+#endif /* _MATH_PRIVATE_H_ */
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/s_copysign.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,42 @@
+/* @(#)s_copysign.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] =
+    "$NetBSD: s_copysign.c,v 1.8 1995/05/10 20:46:57 jtc Exp $";
+#endif
+
+/*
+ * copysign(double x, double y)
+ * copysign(x,y) returns a value with the magnitude of x and
+ * with the sign bit of y.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(copysign)
+#ifdef __STDC__
+     double copysign(double x, double y)
+#else
+     double copysign(x, y)
+     double x, y;
+#endif
+{
+    u_int32_t hx, hy;
+    GET_HIGH_WORD(hx, x);
+    GET_HIGH_WORD(hy, y);
+    SET_HIGH_WORD(x, (hx & 0x7fffffff) | (hy & 0x80000000));
+    return x;
+}
+
+libm_hidden_def(copysign)
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/s_cos.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,90 @@
+/* @(#)s_cos.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: s_cos.c,v 1.7 1995/05/10 20:47:02 jtc Exp $";
+#endif
+
+/* cos(x)
+ * Return cosine function of x.
+ *
+ * kernel function:
+ *	__kernel_sin		... sine function on [-pi/4,pi/4]
+ *	__kernel_cos		... cosine function on [-pi/4,pi/4]
+ *	__ieee754_rem_pio2	... argument reduction routine
+ *
+ * Method.
+ *      Let S,C and T denote the sin, cos and tan respectively on
+ *	[-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
+ *	in [-pi/4 , +pi/4], and let n = k mod 4.
+ *	We have
+ *
+ *          n        sin(x)      cos(x)        tan(x)
+ *     ----------------------------------------------------------
+ *	    0	       S	   C		 T
+ *	    1	       C	  -S		-1/T
+ *	    2	      -S	  -C		 T
+ *	    3	      -C	   S		-1/T
+ *     ----------------------------------------------------------
+ *
+ * Special cases:
+ *      Let trig be any of sin, cos, or tan.
+ *      trig(+-INF)  is NaN, with signals;
+ *      trig(NaN)    is that NaN;
+ *
+ * Accuracy:
+ *	TRIG(x) returns trig(x) nearly rounded
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(cos)
+#ifdef __STDC__
+     double cos(double x)
+#else
+     double cos(x)
+     double x;
+#endif
+{
+    double y[2], z = 0.0;
+    int32_t n, ix;
+
+    /* High word of x. */
+    GET_HIGH_WORD(ix, x);
+
+    /* |x| ~< pi/4 */
+    ix &= 0x7fffffff;
+    if (ix <= 0x3fe921fb)
+        return __kernel_cos(x, z);
+
+    /* cos(Inf or NaN) is NaN */
+    else if (ix >= 0x7ff00000)
+        return x - x;
+
+    /* argument reduction needed */
+    else {
+        n = __ieee754_rem_pio2(x, y);
+        switch (n & 3) {
+        case 0:
+            return __kernel_cos(y[0], y[1]);
+        case 1:
+            return -__kernel_sin(y[0], y[1], 1);
+        case 2:
+            return -__kernel_cos(y[0], y[1]);
+        default:
+            return __kernel_sin(y[0], y[1], 1);
+        }
+    }
+}
+
+libm_hidden_def(cos)
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/s_fabs.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,38 @@
+/* @(#)s_fabs.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: s_fabs.c,v 1.7 1995/05/10 20:47:13 jtc Exp $";
+#endif
+
+/*
+ * fabs(x) returns the absolute value of x.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(fabs)
+#ifdef __STDC__
+     double fabs(double x)
+#else
+     double fabs(x)
+     double x;
+#endif
+{
+    u_int32_t high;
+    GET_HIGH_WORD(high, x);
+    SET_HIGH_WORD(x, high & 0x7fffffff);
+    return x;
+}
+
+libm_hidden_def(fabs)
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/s_scalbn.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,79 @@
+/* @(#)s_scalbn.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] =
+    "$NetBSD: s_scalbn.c,v 1.8 1995/05/10 20:48:08 jtc Exp $";
+#endif
+
+/*
+ * scalbn (double x, int n)
+ * scalbn(x,n) returns x* 2**n  computed by  exponent
+ * manipulation rather than by actually performing an
+ * exponentiation or a multiplication.
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(copysign)
+#ifdef __STDC__
+     static const double
+#else
+     static double
+#endif
+       two54 = 1.80143985094819840000e+16,      /* 0x43500000, 0x00000000 */
+         twom54 = 5.55111512312578270212e-17,   /* 0x3C900000, 0x00000000 */
+         huge = 1.0e+300, tiny = 1.0e-300;
+
+libm_hidden_proto(scalbn)
+#ifdef __STDC__
+     double scalbn(double x, int n)
+#else
+     double scalbn(x, n)
+     double x;
+     int n;
+#endif
+{
+    int32_t k, hx, lx;
+    EXTRACT_WORDS(hx, lx, x);
+    k = (hx & 0x7ff00000) >> 20;        /* extract exponent */
+    if (k == 0) {               /* 0 or subnormal x */
+        if ((lx | (hx & 0x7fffffff)) == 0)
+            return x;           /* +-0 */
+        x *= two54;
+        GET_HIGH_WORD(hx, x);
+        k = ((hx & 0x7ff00000) >> 20) - 54;
+        if (n < -50000)
+            return tiny * x;    /*underflow */
+    }
+    if (k == 0x7ff)
+        return x + x;           /* NaN or Inf */
+    k = k + n;
+    if (k > 0x7fe)
+        return huge * copysign(huge, x);        /* overflow  */
+    if (k > 0) {                /* normal result */
+        SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
+        return x;
+    }
+    if (k <= -54) {
+        if (n > 50000)          /* in case integer overflow in n+k */
+            return huge * copysign(huge, x);    /*overflow */
+        else
+            return tiny * copysign(tiny, x);    /*underflow */
+    }
+    k += 54;                    /* subnormal result */
+    SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
+    return x * twom54;
+}
+
+libm_hidden_def(scalbn)
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/s_sin.c	Mon Sep 15 06:33:23 2008 +0000
@@ -0,0 +1,90 @@
+/* @(#)s_sin.c 5.1 93/09/24 */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#if defined(LIBM_SCCS) && !defined(lint)
+static char rcsid[] = "$NetBSD: s_sin.c,v 1.7 1995/05/10 20:48:15 jtc Exp $";
+#endif
+
+/* sin(x)
+ * Return sine function of x.
+ *
+ * kernel function:
+ *	__kernel_sin		... sine function on [-pi/4,pi/4]
+ *	__kernel_cos		... cose function on [-pi/4,pi/4]
+ *	__ieee754_rem_pio2	... argument reduction routine
+ *
+ * Method.
+ *      Let S,C and T denote the sin, cos and tan respectively on
+ *	[-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
+ *	in [-pi/4 , +pi/4], and let n = k mod 4.
+ *	We have
+ *
+ *          n        sin(x)      cos(x)        tan(x)
+ *     ----------------------------------------------------------
+ *	    0	       S	   C		 T
+ *	    1	       C	  -S		-1/T
+ *	    2	      -S	  -C		 T
+ *	    3	      -C	   S		-1/T
+ *     ----------------------------------------------------------
+ *
+ * Special cases:
+ *      Let trig be any of sin, cos, or tan.
+ *      trig(+-INF)  is NaN, with signals;
+ *      trig(NaN)    is that NaN;
+ *
+ * Accuracy:
+ *	TRIG(x) returns trig(x) nearly rounded
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+libm_hidden_proto(sin)
+#ifdef __STDC__
+     double sin(double x)
+#else
+     double sin(x)
+     double x;
+#endif
+{
+    double y[2], z = 0.0;
+    int32_t n, ix;
+
+    /* High word of x. */
+    GET_HIGH_WORD(ix, x);
+
+    /* |x| ~< pi/4 */
+    ix &= 0x7fffffff;
+    if (ix <= 0x3fe921fb)
+        return __kernel_sin(x, z, 0);
+
+    /* sin(Inf or NaN) is NaN */
+    else if (ix >= 0x7ff00000)
+        return x - x;
+
+    /* argument reduction needed */
+    else {
+        n = __ieee754_rem_pio2(x, y);
+        switch (n & 3) {
+        case 0:
+            return __kernel_sin(y[0], y[1], 1);
+        case 1:
+            return __kernel_cos(y[0], y[1]);
+        case 2:
+            return -__kernel_sin(y[0], y[1], 1);
+        default:
+            return -__kernel_cos(y[0], y[1]);
+        }
+    }
+}
+
+libm_hidden_def(sin)
--- a/src/video/SDL_gamma.c	Mon Sep 15 05:14:11 2008 +0000
+++ b/src/video/SDL_gamma.c	Mon Sep 15 06:33:23 2008 +0000
@@ -23,17 +23,7 @@
 
 /* Gamma correction support */
 
-#ifdef HAVE_MATH_H
-#include <math.h>               /* Used for calculating gamma ramps */
-#else
-/* Math routines from uClibc: http://www.uclibc.org */
-#include "math_private.h"
-#include "e_sqrt.h"
-#include "e_pow.h"
-#include "e_log.h"
-#define pow(x, y)	__ieee754_pow(x, y)
-#define log(x)		__ieee754_log(x)
-#endif
+#include "../libm/math.h"
 
 #include "SDL_sysvideo.h"
 
--- a/src/video/e_log.h	Mon Sep 15 05:14:11 2008 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,161 +0,0 @@
-/* @(#)e_log.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-#if defined(LIBM_SCCS) && !defined(lint)
-static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
-#endif
-
-/* __ieee754_log(x)
- * Return the logrithm of x
- *
- * Method :
- *   1. Argument Reduction: find k and f such that
- *			x = 2^k * (1+f),
- *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
- *
- *   2. Approximation of log(1+f).
- *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
- *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
- *	     	 = 2s + s*R
- *      We use a special Reme algorithm on [0,0.1716] to generate
- * 	a polynomial of degree 14 to approximate R The maximum error
- *	of this polynomial approximation is bounded by 2**-58.45. In
- *	other words,
- *		        2      4      6      8      10      12      14
- *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
- *  	(the values of Lg1 to Lg7 are listed in the program)
- *	and
- *	    |      2          14          |     -58.45
- *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
- *	    |                             |
- *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
- *	In order to guarantee error in log below 1ulp, we compute log
- *	by
- *		log(1+f) = f - s*(f - R)	(if f is not too large)
- *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
- *
- *	3. Finally,  log(x) = k*ln2 + log(1+f).
- *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
- *	   Here ln2 is split into two floating point number:
- *			ln2_hi + ln2_lo,
- *	   where n*ln2_hi is always exact for |n| < 2000.
- *
- * Special cases:
- *	log(x) is NaN with signal if x < 0 (including -INF) ;
- *	log(+INF) is +INF; log(0) is -INF with signal;
- *	log(NaN) is that NaN with no signal.
- *
- * Accuracy:
- *	according to an error analysis, the error is always less than
- *	1 ulp (unit in the last place).
- *
- * Constants:
- * The hexadecimal values are the intended ones for the following
- * constants. The decimal values may be used, provided that the
- * compiler will convert from decimal to binary accurately enough
- * to produce the hexadecimal values shown.
- */
-
-/*#include "math.h"*/
-#include "math_private.h"
-
-#ifdef __STDC__
-static const double
-#else
-static double
-#endif
-  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
-    ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
-    Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
-    Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
-    Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
-    Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
-    Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
-    Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
-    Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */
-
-#ifdef __STDC__
-double
-__ieee754_log(double x)
-#else
-double
-__ieee754_log(x)
-     double x;
-#endif
-{
-    double hfsq, f, s, z, R, w, t1, t2, dk;
-    int32_t k, hx, i, j;
-    u_int32_t lx;
-
-    EXTRACT_WORDS(hx, lx, x);
-
-    k = 0;
-    if (hx < 0x00100000) {      /* x < 2**-1022  */
-        if (((hx & 0x7fffffff) | lx) == 0)
-            return -two54 / zero;       /* log(+-0)=-inf */
-        if (hx < 0)
-            return (x - x) / zero;      /* log(-#) = NaN */
-        k -= 54;
-        x *= two54;             /* subnormal number, scale up x */
-        GET_HIGH_WORD(hx, x);
-    }
-    if (hx >= 0x7ff00000)
-        return x + x;
-    k += (hx >> 20) - 1023;
-    hx &= 0x000fffff;
-    i = (hx + 0x95f64) & 0x100000;
-    SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
-    k += (i >> 20);
-    f = x - 1.0;
-    if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
-        if (f == zero) {
-            if (k == 0)
-                return zero;
-            else {
-                dk = (double) k;
-                return dk * ln2_hi + dk * ln2_lo;
-            }
-        }
-        R = f * f * (0.5 - 0.33333333333333333 * f);
-        if (k == 0)
-            return f - R;
-        else {
-            dk = (double) k;
-            return dk * ln2_hi - ((R - dk * ln2_lo) - f);
-        }
-    }
-    s = f / (2.0 + f);
-    dk = (double) k;
-    z = s * s;
-    i = hx - 0x6147a;
-    w = z * z;
-    j = 0x6b851 - hx;
-    t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
-    t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
-    i |= j;
-    R = t2 + t1;
-    if (i > 0) {
-        hfsq = 0.5 * f * f;
-        if (k == 0)
-            return f - (hfsq - s * (hfsq + R));
-        else
-            return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
-                                  f);
-    } else {
-        if (k == 0)
-            return f - s * (f - R);
-        else
-            return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
-    }
-}
-
-/* vi: set ts=4 sw=4 expandtab: */
--- a/src/video/e_pow.h	Mon Sep 15 05:14:11 2008 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,340 +0,0 @@
-/* @(#)e_pow.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-#if defined(LIBM_SCCS) && !defined(lint)
-static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
-#endif
-
-/* __ieee754_pow(x,y) return x**y
- *
- *		      n
- * Method:  Let x =  2   * (1+f)
- *	1. Compute and return log2(x) in two pieces:
- *		log2(x) = w1 + w2,
- *	   where w1 has 53-24 = 29 bit trailing zeros.
- *	2. Perform y*log2(x) = n+y' by simulating muti-precision
- *	   arithmetic, where |y'|<=0.5.
- *	3. Return x**y = 2**n*exp(y'*log2)
- *
- * Special cases:
- *	1.  (anything) ** 0  is 1
- *	2.  (anything) ** 1  is itself
- *	3.  (anything) ** NAN is NAN
- *	4.  NAN ** (anything except 0) is NAN
- *	5.  +-(|x| > 1) **  +INF is +INF
- *	6.  +-(|x| > 1) **  -INF is +0
- *	7.  +-(|x| < 1) **  +INF is +0
- *	8.  +-(|x| < 1) **  -INF is +INF
- *	9.  +-1         ** +-INF is NAN
- *	10. +0 ** (+anything except 0, NAN)               is +0
- *	11. -0 ** (+anything except 0, NAN, odd integer)  is +0
- *	12. +0 ** (-anything except 0, NAN)               is +INF
- *	13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
- *	14. -0 ** (odd integer) = -( +0 ** (odd integer) )
- *	15. +INF ** (+anything except 0,NAN) is +INF
- *	16. +INF ** (-anything except 0,NAN) is +0
- *	17. -INF ** (anything)  = -0 ** (-anything)
- *	18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
- *	19. (-anything except 0 and inf) ** (non-integer) is NAN
- *
- * Accuracy:
- *	pow(x,y) returns x**y nearly rounded. In particular
- *			pow(integer,integer)
- *	always returns the correct integer provided it is
- *	representable.
- *
- * Constants :
- * The hexadecimal values are the intended ones for the following
- * constants. The decimal values may be used, provided that the
- * compiler will convert from decimal to binary accurately enough
- * to produce the hexadecimal values shown.
- */
-
-/*#include "math.h"*/
-#include "math_private.h"
-
-#ifdef __STDC__
-static const double
-#else
-static double
-#endif
-  bp[] = { 1.0, 1.5, }, dp_h[] = {
-0.0, 5.84962487220764160156e-01,},      /* 0x3FE2B803, 0x40000000 */
-
-    dp_l[] = {
-0.0, 1.35003920212974897128e-08,},      /* 0x3E4CFDEB, 0x43CFD006 */
-
-    /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
-    L1 = 5.99999999999994648725e-01,    /* 0x3FE33333, 0x33333303 */
-    L2 = 4.28571428578550184252e-01,    /* 0x3FDB6DB6, 0xDB6FABFF */
-    L3 = 3.33333329818377432918e-01,    /* 0x3FD55555, 0x518F264D */
-    L4 = 2.72728123808534006489e-01,    /* 0x3FD17460, 0xA91D4101 */
-    L5 = 2.30660745775561754067e-01,    /* 0x3FCD864A, 0x93C9DB65 */
-    L6 = 2.06975017800338417784e-01,    /* 0x3FCA7E28, 0x4A454EEF */
-    P1 = 1.66666666666666019037e-01,    /* 0x3FC55555, 0x5555553E */
-    P2 = -2.77777777770155933842e-03,   /* 0xBF66C16C, 0x16BEBD93 */
-    P3 = 6.61375632143793436117e-05,    /* 0x3F11566A, 0xAF25DE2C */
-    P4 = -1.65339022054652515390e-06,   /* 0xBEBBBD41, 0xC5D26BF1 */
-    P5 = 4.13813679705723846039e-08,    /* 0x3E663769, 0x72BEA4D0 */
-    lg2 = 6.93147180559945286227e-01,   /* 0x3FE62E42, 0xFEFA39EF */
-    lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
-    lg2_l = -1.90465429995776804525e-09,        /* 0xBE205C61, 0x0CA86C39 */
-    ovt = 8.0085662595372944372e-0017,  /* -(1024-log2(ovfl+.5ulp)) */
-    cp = 9.61796693925975554329e-01,    /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
-    cp_h = 9.61796700954437255859e-01,  /* 0x3FEEC709, 0xE0000000 =(float)cp */
-    cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
-    ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
-    ivln2_h = 1.44269502162933349609e+00,       /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
-    ivln2_l = 1.92596299112661746887e-08;       /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
-
-#ifdef __STDC__
-double
-__ieee754_pow(double x, double y)
-#else
-double
-__ieee754_pow(x, y)
-     double x, y;
-#endif
-{
-    double z, ax, z_h, z_l, p_h, p_l;
-    double y1, t1, t2, r, s, t, u, v, w;
-    int32_t i, j, k, yisint, n;
-    int32_t hx, hy, ix, iy;
-    u_int32_t lx, ly;
-
-    EXTRACT_WORDS(hx, lx, x);
-    EXTRACT_WORDS(hy, ly, y);
-    ix = hx & 0x7fffffff;
-    iy = hy & 0x7fffffff;
-
-    /* y==zero: x**0 = 1 */
-    if ((iy | ly) == 0)
-        return one;
-
-    /* +-NaN return x+y */
-    if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
-        iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
-        return x + y;
-
-    /* determine if y is an odd int when x < 0
-     * yisint = 0       ... y is not an integer
-     * yisint = 1       ... y is an odd int
-     * yisint = 2       ... y is an even int
-     */
-    yisint = 0;
-    if (hx < 0) {
-        if (iy >= 0x43400000)
-            yisint = 2;         /* even integer y */
-        else if (iy >= 0x3ff00000) {
-            k = (iy >> 20) - 0x3ff;     /* exponent */
-            if (k > 20) {
-                j = ly >> (52 - k);
-                if ((u_int32_t) (j << (52 - k)) == ly)
-                    yisint = 2 - (j & 1);
-            } else if (ly == 0) {
-                j = iy >> (20 - k);
-                if ((j << (20 - k)) == iy)
-                    yisint = 2 - (j & 1);
-            }
-        }
-    }
-
-    /* special value of y */
-    if (ly == 0) {
-        if (iy == 0x7ff00000) { /* y is +-inf */
-            if (((ix - 0x3ff00000) | lx) == 0)
-                return y - y;   /* inf**+-1 is NaN */
-            else if (ix >= 0x3ff00000)  /* (|x|>1)**+-inf = inf,0 */
-                return (hy >= 0) ? y : zero;
-            else                /* (|x|<1)**-,+inf = inf,0 */
-                return (hy < 0) ? -y : zero;
-        }
-        if (iy == 0x3ff00000) { /* y is  +-1 */
-            if (hy < 0)
-                return one / x;
-            else
-                return x;
-        }
-        if (hy == 0x40000000)
-            return x * x;       /* y is  2 */
-        if (hy == 0x3fe00000) { /* y is  0.5 */
-            if (hx >= 0)        /* x >= +0 */
-                return __ieee754_sqrt(x);
-        }
-    }
-
-    ax = x < 0 ? -x : x;        /*fabs(x); */
-    /* special value of x */
-    if (lx == 0) {
-        if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
-            z = ax;             /*x is +-0,+-inf,+-1 */
-            if (hy < 0)
-                z = one / z;    /* z = (1/|x|) */
-            if (hx < 0) {
-                if (((ix - 0x3ff00000) | yisint) == 0) {
-                    z = (z - z) / (z - z);      /* (-1)**non-int is NaN */
-                } else if (yisint == 1)
-                    z = -z;     /* (x<0)**odd = -(|x|**odd) */
-            }
-            return z;
-        }
-    }
-
-    /* (x<0)**(non-int) is NaN */
-    if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
-        return (x - x) / (x - x);
-
-    /* |y| is huge */
-    if (iy > 0x41e00000) {      /* if |y| > 2**31 */
-        if (iy > 0x43f00000) {  /* if |y| > 2**64, must o/uflow */
-            if (ix <= 0x3fefffff)
-                return (hy < 0) ? huge * huge : tiny * tiny;
-            if (ix >= 0x3ff00000)
-                return (hy > 0) ? huge * huge : tiny * tiny;
-        }
-        /* over/underflow if x is not close to one */
-        if (ix < 0x3fefffff)
-            return (hy < 0) ? huge * huge : tiny * tiny;
-        if (ix > 0x3ff00000)
-            return (hy > 0) ? huge * huge : tiny * tiny;
-        /* now |1-x| is tiny <= 2**-20, suffice to compute
-           log(x) by x-x^2/2+x^3/3-x^4/4 */
-        t = ax - 1;             /* t has 20 trailing zeros */
-        w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
-        u = ivln2_h * t;        /* ivln2_h has 21 sig. bits */
-        v = t * ivln2_l - w * ivln2;
-        t1 = u + v;
-        SET_LOW_WORD(t1, 0);
-        t2 = v - (t1 - u);
-    } else {
-        double s2, s_h, s_l, t_h, t_l;
-        n = 0;
-        /* take care subnormal number */
-        if (ix < 0x00100000) {
-            ax *= two53;
-            n -= 53;
-            GET_HIGH_WORD(ix, ax);
-        }
-        n += ((ix) >> 20) - 0x3ff;
-        j = ix & 0x000fffff;
-        /* determine interval */
-        ix = j | 0x3ff00000;    /* normalize ix */
-        if (j <= 0x3988E)
-            k = 0;              /* |x|<sqrt(3/2) */
-        else if (j < 0xBB67A)
-            k = 1;              /* |x|<sqrt(3)   */
-        else {
-            k = 0;
-            n += 1;
-            ix -= 0x00100000;
-        }
-        SET_HIGH_WORD(ax, ix);
-
-        /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
-        u = ax - bp[k];         /* bp[0]=1.0, bp[1]=1.5 */
-        v = one / (ax + bp[k]);
-        s = u * v;
-        s_h = s;
-        SET_LOW_WORD(s_h, 0);
-        /* t_h=ax+bp[k] High */
-        t_h = zero;
-        SET_HIGH_WORD(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
-        t_l = ax - (t_h - bp[k]);
-        s_l = v * ((u - s_h * t_h) - s_h * t_l);
-        /* compute log(ax) */
-        s2 = s * s;
-        r = s2 * s2 * (L1 +
-                       s2 * (L2 +
-                             s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
-        r += s_l * (s_h + s);
-        s2 = s_h * s_h;
-        t_h = 3.0 + s2 + r;
-        SET_LOW_WORD(t_h, 0);
-        t_l = r - ((t_h - 3.0) - s2);
-        /* u+v = s*(1+...) */
-        u = s_h * t_h;
-        v = s_l * t_h + t_l * s;
-        /* 2/(3log2)*(s+...) */
-        p_h = u + v;
-        SET_LOW_WORD(p_h, 0);
-        p_l = v - (p_h - u);
-        z_h = cp_h * p_h;       /* cp_h+cp_l = 2/(3*log2) */
-        z_l = cp_l * p_h + p_l * cp + dp_l[k];
-        /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
-        t = (double) n;
-        t1 = (((z_h + z_l) + dp_h[k]) + t);
-        SET_LOW_WORD(t1, 0);
-        t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
-    }
-
-    s = one;                    /* s (sign of result -ve**odd) = -1 else = 1 */
-    if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
-        s = -one;               /* (-ve)**(odd int) */
-
-    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
-    y1 = y;
-    SET_LOW_WORD(y1, 0);
-    p_l = (y - y1) * t1 + y * t2;
-    p_h = y1 * t1;
-    z = p_l + p_h;
-    EXTRACT_WORDS(j, i, z);
-    if (j >= 0x40900000) {      /* z >= 1024 */
-        if (((j - 0x40900000) | i) != 0)        /* if z > 1024 */
-            return s * huge * huge;     /* overflow */
-        else {
-            if (p_l + ovt > z - p_h)
-                return s * huge * huge; /* overflow */
-        }
-    } else if ((j & 0x7fffffff) >= 0x4090cc00) {        /* z <= -1075 */
-        if (((j - 0xc090cc00) | i) != 0)        /* z < -1075 */
-            return s * tiny * tiny;     /* underflow */
-        else {
-            if (p_l <= z - p_h)
-                return s * tiny * tiny; /* underflow */
-        }
-    }
-    /*
-     * compute 2**(p_h+p_l)
-     */
-    i = j & 0x7fffffff;
-    k = (i >> 20) - 0x3ff;
-    n = 0;
-    if (i > 0x3fe00000) {       /* if |z| > 0.5, set n = [z+0.5] */
-        n = j + (0x00100000 >> (k + 1));
-        k = ((n & 0x7fffffff) >> 20) - 0x3ff;   /* new k for n */
-        t = zero;
-        SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
-        n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
-        if (j < 0)
-            n = -n;
-        p_h -= t;
-    }
-    t = p_l + p_h;
-    SET_LOW_WORD(t, 0);
-    u = t * lg2_h;
-    v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
-    z = u + v;
-    w = v - (z - u);
-    t = z * z;
-    t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
-    r = (z * t1) / (t1 - two) - (w + z * w);
-    z = one - (r - z);
-    GET_HIGH_WORD(j, z);
-    j += (n << 20);
-    if ((j >> 20) <= 0)
-        z = SDL_NAME(scalbn) (z, n);    /* subnormal output */
-    else
-        SET_HIGH_WORD(z, j);
-    return s * z;
-}
-
-/* vi: set ts=4 sw=4 expandtab: */
--- a/src/video/e_sqrt.h	Mon Sep 15 05:14:11 2008 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,516 +0,0 @@
-/* @(#)e_sqrt.c 5.1 93/09/24 */
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-#if defined(LIBM_SCCS) && !defined(lint)
-static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
-#endif
-
-/* __ieee754_sqrt(x)
- * Return correctly rounded sqrt.
- *           ------------------------------------------
- *	     |  Use the hardware sqrt if you have one |
- *           ------------------------------------------
- * Method:
- *   Bit by bit method using integer arithmetic. (Slow, but portable)
- *   1. Normalization
- *	Scale x to y in [1,4) with even powers of 2:
- *	find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
- *		sqrt(x) = 2^k * sqrt(y)
- *   2. Bit by bit computation
- *	Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
- *	     i							 0
- *                                     i+1         2
- *	    s  = 2*q , and	y  =  2   * ( y - q  ).		(1)
- *	     i      i            i                 i
- *
- *	To compute q    from q , one checks whether
- *		    i+1       i
- *
- *			      -(i+1) 2
- *			(q + 2      ) <= y.			(2)
- *     			  i
- *							      -(i+1)
- *	If (2) is false, then q   = q ; otherwise q   = q  + 2      .
- *		 	       i+1   i             i+1   i
- *
- *	With some algebric manipulation, it is not difficult to see
- *	that (2) is equivalent to
- *                             -(i+1)
- *			s  +  2       <= y			(3)
- *			 i                i
- *
- *	The advantage of (3) is that s  and y  can be computed by
- *				      i      i
- *	the following recurrence formula:
- *	    if (3) is false
- *
- *	    s     =  s  ,	y    = y   ;			(4)
- *	     i+1      i		 i+1    i
- *
- *	    otherwise,
- *                         -i                     -(i+1)
- *	    s	  =  s  + 2  ,  y    = y  -  s  - 2  		(5)
- *           i+1      i          i+1    i     i
- *
- *	One may easily use induction to prove (4) and (5).
- *	Note. Since the left hand side of (3) contain only i+2 bits,
- *	      it does not necessary to do a full (53-bit) comparison
- *	      in (3).
- *   3. Final rounding
- *	After generating the 53 bits result, we compute one more bit.
- *	Together with the remainder, we can decide whether the
- *	result is exact, bigger than 1/2ulp, or less than 1/2ulp
- *	(it will never equal to 1/2ulp).
- *	The rounding mode can be detected by checking whether
- *	huge + tiny is equal to huge, and whether huge - tiny is
- *	equal to huge for some floating point number "huge" and "tiny".
- *
- * Special cases:
- *	sqrt(+-0) = +-0 	... exact
- *	sqrt(inf) = inf
- *	sqrt(-ve) = NaN		... with invalid signal
- *	sqrt(NaN) = NaN		... with invalid signal for signaling NaN
- *
- * Other methods : see the appended file at the end of the program below.
- *---------------
- */
-
-/*#include "math.h"*/
-#include "math_private.h"
-
-#ifdef __STDC__
-double SDL_NAME(copysign) (double x, double y)
-#else
-double SDL_NAME(copysign) (x, y)
-     double
-         x,
-         y;
-#endif
-{
-    u_int32_t hx, hy;
-    GET_HIGH_WORD(hx, x);
-    GET_HIGH_WORD(hy, y);
-    SET_HIGH_WORD(x, (hx & 0x7fffffff) | (hy & 0x80000000));
-    return x;
-}
-
-#ifdef __STDC__
-double SDL_NAME(scalbn) (double x, int n)
-#else
-double SDL_NAME(scalbn) (x, n)
-     double
-         x;
-     int
-         n;
-#endif
-{
-    int32_t k, hx, lx;
-    EXTRACT_WORDS(hx, lx, x);
-    k = (hx & 0x7ff00000) >> 20;        /* extract exponent */
-    if (k == 0) {               /* 0 or subnormal x */
-        if ((lx | (hx & 0x7fffffff)) == 0)
-            return x;           /* +-0 */
-        x *= two54;
-        GET_HIGH_WORD(hx, x);
-        k = ((hx & 0x7ff00000) >> 20) - 54;
-        if (n < -50000)
-            return tiny * x;    /*underflow */
-    }
-    if (k == 0x7ff)
-        return x + x;           /* NaN or Inf */
-    k = k + n;
-    if (k > 0x7fe)
-        return huge * SDL_NAME(copysign) (huge, x);     /* overflow  */
-    if (k > 0) {                /* normal result */
-        SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
-        return x;
-    }
-    if (k <= -54) {
-        if (n > 50000)          /* in case integer overflow in n+k */
-            return huge * SDL_NAME(copysign) (huge, x); /*overflow */
-        else
-            return tiny * SDL_NAME(copysign) (tiny, x); /*underflow */
-    }
-    k += 54;                    /* subnormal result */
-    SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
-    return x * twom54;
-}
-
-#ifdef __STDC__
-double
-__ieee754_sqrt(double x)
-#else
-double
-__ieee754_sqrt(x)
-     double x;
-#endif
-{
-    double z;
-    int32_t sign = (int) 0x80000000;
-    int32_t ix0, s0, q, m, t, i;
-    u_int32_t r, t1, s1, ix1, q1;
-
-    EXTRACT_WORDS(ix0, ix1, x);
-
-    /* take care of Inf and NaN */
-    if ((ix0 & 0x7ff00000) == 0x7ff00000) {
-        return x * x + x;       /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
-                                   sqrt(-inf)=sNaN */
-    }
-    /* take care of zero */
-    if (ix0 <= 0) {
-        if (((ix0 & (~sign)) | ix1) == 0)
-            return x;           /* sqrt(+-0) = +-0 */
-        else if (ix0 < 0)
-            return (x - x) / (x - x);   /* sqrt(-ve) = sNaN */
-    }
-    /* normalize x */
-    m = (ix0 >> 20);
-    if (m == 0) {               /* subnormal x */
-        while (ix0 == 0) {
-            m -= 21;
-            ix0 |= (ix1 >> 11);
-            ix1 <<= 21;
-        }
-        for (i = 0; (ix0 & 0x00100000) == 0; i++)
-            ix0 <<= 1;
-        m -= i - 1;
-        ix0 |= (ix1 >> (32 - i));
-        ix1 <<= i;
-    }
-    m -= 1023;                  /* unbias exponent */
-    ix0 = (ix0 & 0x000fffff) | 0x00100000;
-    if (m & 1) {                /* odd m, double x to make it even */
-        ix0 += ix0 + ((ix1 & sign) >> 31);
-        ix1 += ix1;
-    }
-    m >>= 1;                    /* m = [m/2] */
-
-    /* generate sqrt(x) bit by bit */
-    ix0 += ix0 + ((ix1 & sign) >> 31);
-    ix1 += ix1;
-    q = q1 = s0 = s1 = 0;       /* [q,q1] = sqrt(x) */
-    r = 0x00200000;             /* r = moving bit from right to left */
-
-    while (r != 0) {
-        t = s0 + r;
-        if (t <= ix0) {
-            s0 = t + r;
-            ix0 -= t;
-            q += r;
-        }
-        ix0 += ix0 + ((ix1 & sign) >> 31);
-        ix1 += ix1;
-        r >>= 1;
-    }
-
-    r = sign;
-    while (r != 0) {
-        t1 = s1 + r;
-        t = s0;
-        if ((t < ix0) || ((t == ix0) && (t1 <= ix1))) {
-            s1 = t1 + r;
-            if (((int32_t) (t1 & sign) == sign) && (s1 & sign) == 0)
-                s0 += 1;
-            ix0 -= t;
-            if (ix1 < t1)
-                ix0 -= 1;
-            ix1 -= t1;
-            q1 += r;
-        }
-        ix0 += ix0 + ((ix1 & sign) >> 31);
-        ix1 += ix1;
-        r >>= 1;
-    }
-
-    /* use floating add to find out rounding direction */
-    if ((ix0 | ix1) != 0) {
-        z = one - tiny;         /* trigger inexact flag */
-        if (z >= one) {
-            z = one + tiny;
-            if (q1 == (u_int32_t) 0xffffffff) {
-                q1 = 0;
-                q += 1;
-            } else if (z > one) {
-                if (q1 == (u_int32_t) 0xfffffffe)
-                    q += 1;
-                q1 += 2;
-            } else
-                q1 += (q1 & 1);
-        }
-    }
-    ix0 = (q >> 1) + 0x3fe00000;
-    ix1 = q1 >> 1;
-    if ((q & 1) == 1)
-        ix1 |= sign;
-    ix0 += (m << 20);
-    INSERT_WORDS(z, ix0, ix1);
-    return z;
-}
-
-/*
-Other methods  (use floating-point arithmetic)
--------------
-(This is a copy of a drafted paper by Prof W. Kahan
-and K.C. Ng, written in May, 1986)
-
-	Two algorithms are given here to implement sqrt(x)
-	(IEEE double precision arithmetic) in software.
-	Both supply sqrt(x) correctly rounded. The first algorithm (in
-	Section A) uses newton iterations and involves four divisions.
-	The second one uses reciproot iterations to avoid division, but
-	requires more multiplications. Both algorithms need the ability
-	to chop results of arithmetic operations instead of round them,
-	and the INEXACT flag to indicate when an arithmetic operation
-	is executed exactly with no roundoff error, all part of the
-	standard (IEEE 754-1985). The ability to perform shift, add,
-	subtract and logical AND operations upon 32-bit words is needed
-	too, though not part of the standard.
-
-A.  sqrt(x) by Newton Iteration
-
-   (1)	Initial approximation
-
-	Let x0 and x1 be the leading and the trailing 32-bit words of
-	a floating point number x (in IEEE double format) respectively
-
-	    1    11		     52				  ...widths
-	   ------------------------------------------------------
-	x: |s|	  e     |	      f				|
-	   ------------------------------------------------------
-	      msb    lsb  msb				      lsb ...order
-
-
-	     ------------------------  	     ------------------------
-	x0:  |s|   e    |    f1     |	 x1: |          f2           |
-	     ------------------------  	     ------------------------
-
-	By performing shifts and subtracts on x0 and x1 (both regarded
-	as integers), we obtain an 8-bit approximation of sqrt(x) as
-	follows.
-
-		k  := (x0>>1) + 0x1ff80000;
-		y0 := k - T1[31&(k>>15)].	... y ~ sqrt(x) to 8 bits
-	Here k is a 32-bit integer and T1[] is an integer array containing
-	correction terms. Now magically the floating value of y (y's
-	leading 32-bit word is y0, the value of its trailing word is 0)
-	approximates sqrt(x) to almost 8-bit.
-
-	Value of T1:
-	static int T1[32]= {
-	0,	1024,	3062,	5746,	9193,	13348,	18162,	23592,
-	29598,	36145,	43202,	50740,	58733,	67158,	75992,	85215,
-	83599,	71378,	60428,	50647,	41945,	34246,	27478,	21581,
-	16499,	12183,	8588,	5674,	3403,	1742,	661,	130,};
-
-    (2)	Iterative refinement
-
-	Apply Heron's rule three times to y, we have y approximates
-	sqrt(x) to within 1 ulp (Unit in the Last Place):
-
-		y := (y+x/y)/2		... almost 17 sig. bits
-		y := (y+x/y)/2		... almost 35 sig. bits
-		y := y-(y-x/y)/2	... within 1 ulp
-
-
-	Remark 1.
-	    Another way to improve y to within 1 ulp is:
-
-		y := (y+x/y)		... almost 17 sig. bits to 2*sqrt(x)
-		y := y - 0x00100006	... almost 18 sig. bits to sqrt(x)
-
-				2
-			    (x-y )*y
-		y := y + 2* ----------	...within 1 ulp
-			       2
-			     3y  + x
-
-
-	This formula has one division fewer than the one above; however,
-	it requires more multiplications and additions. Also x must be
-	scaled in advance to avoid spurious overflow in evaluating the
-	expression 3y*y+x. Hence it is not recommended uless division
-	is slow. If division is very slow, then one should use the
-	reciproot algorithm given in section B.
-
-    (3) Final adjustment
-
-	By twiddling y's last bit it is possible to force y to be
-	correctly rounded according to the prevailing rounding mode
-	as follows. Let r and i be copies of the rounding mode and
-	inexact flag before entering the square root program. Also we
-	use the expression y+-ulp for the next representable floating
-	numbers (up and down) of y. Note that y+-ulp = either fixed
-	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
-	mode.
-
-		I := FALSE;	... reset INEXACT flag I
-		R := RZ;	... set rounding mode to round-toward-zero
-		z := x/y;	... chopped quotient, possibly inexact
-		If(not I) then {	... if the quotient is exact
-		    if(z=y) {
-		        I := i;	 ... restore inexact flag
-		        R := r;  ... restore rounded mode
-		        return sqrt(x):=y.
-		    } else {
-			z := z - ulp;	... special rounding
-		    }
-		}
-		i := TRUE;		... sqrt(x) is inexact
-		If (r=RN) then z=z+ulp	... rounded-to-nearest
-		If (r=RP) then {	... round-toward-+inf
-		    y = y+ulp; z=z+ulp;
-		}
-		y := y+z;		... chopped sum
-		y0:=y0-0x00100000;	... y := y/2 is correctly rounded.
-	        I := i;	 		... restore inexact flag
-	        R := r;  		... restore rounded mode
-	        return sqrt(x):=y.
-
-    (4)	Special cases
-
-	Square root of +inf, +-0, or NaN is itself;
-	Square root of a negative number is NaN with invalid signal.
-
-
-B.  sqrt(x) by Reciproot Iteration
-
-   (1)	Initial approximation
-
-	Let x0 and x1 be the leading and the trailing 32-bit words of
-	a floating point number x (in IEEE double format) respectively
-	(see section A). By performing shifs and subtracts on x0 and y0,
-	we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
-
-	    k := 0x5fe80000 - (x0>>1);
-	    y0:= k - T2[63&(k>>14)].	... y ~ 1/sqrt(x) to 7.8 bits
-
-	Here k is a 32-bit integer and T2[] is an integer array
-	containing correction terms. Now magically the floating
-	value of y (y's leading 32-bit word is y0, the value of
-	its trailing word y1 is set to zero) approximates 1/sqrt(x)
-	to almost 7.8-bit.
-
-	Value of T2:
-	static int T2[64]= {
-	0x1500,	0x2ef8,	0x4d67,	0x6b02,	0x87be,	0xa395,	0xbe7a,	0xd866,
-	0xf14a,	0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
-	0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
-	0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
-	0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
-	0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
-	0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
-	0x1527f,0x1334a,0x11051,0xe951,	0xbe01,	0x8e0d,	0x5924,	0x1edd,};
-
-    (2)	Iterative refinement
-
-	Apply Reciproot iteration three times to y and multiply the
-	result by x to get an approximation z that matches sqrt(x)
-	to about 1 ulp. To be exact, we will have
-		-1ulp < sqrt(x)-z<1.0625ulp.
-
-	... set rounding mode to Round-to-nearest
-	   y := y*(1.5-0.5*x*y*y)	... almost 15 sig. bits to 1/sqrt(x)
-	   y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
-	... special arrangement for better accuracy
-	   z := x*y			... 29 bits to sqrt(x), with z*y<1
-	   z := z + 0.5*z*(1-z*y)	... about 1 ulp to sqrt(x)
-
-	Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
-	(a) the term z*y in the final iteration is always less than 1;
-	(b) the error in the final result is biased upward so that
-		-1 ulp < sqrt(x) - z < 1.0625 ulp
-	    instead of |sqrt(x)-z|<1.03125ulp.
-
-    (3)	Final adjustment
-
-	By twiddling y's last bit it is possible to force y to be
-	correctly rounded according to the prevailing rounding mode
-	as follows. Let r and i be copies of the rounding mode and
-	inexact flag before entering the square root program. Also we
-	use the expression y+-ulp for the next representable floating
-	numbers (up and down) of y. Note that y+-ulp = either fixed
-	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
-	mode.
-
-	R := RZ;		... set rounding mode to round-toward-zero
-	switch(r) {
-	    case RN:		... round-to-nearest
-	       if(x<= z*(z-ulp)...chopped) z = z - ulp; else
-	       if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
-	       break;
-	    case RZ:case RM:	... round-to-zero or round-to--inf
-	       R:=RP;		... reset rounding mod to round-to-+inf
-	       if(x<z*z ... rounded up) z = z - ulp; else
-	       if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
-	       break;
-	    case RP:		... round-to-+inf
-	       if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
-	       if(x>z*z ...chopped) z = z+ulp;
-	       break;
-	}
-
-	Remark 3. The above comparisons can be done in fixed point. For
-	example, to compare x and w=z*z chopped, it suffices to compare
-	x1 and w1 (the trailing parts of x and w), regarding them as
-	two's complement integers.
-
-	...Is z an exact square root?
-	To determine whether z is an exact square root of x, let z1 be the
-	trailing part of z, and also let x0 and x1 be the leading and
-	trailing parts of x.
-
-	If ((z1&0x03ffffff)!=0)	... not exact if trailing 26 bits of z!=0
-	    I := 1;		... Raise Inexact flag: z is not exact
-	else {
-	    j := 1 - [(x0>>20)&1]	... j = logb(x) mod 2
-	    k := z1 >> 26;		... get z's 25-th and 26-th
-					    fraction bits
-	    I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
-	}
-	R:= r		... restore rounded mode
-	return sqrt(x):=z.
-
-	If multiplication is cheaper then the foregoing red tape, the
-	Inexact flag can be evaluated by
-
-	    I := i;
-	    I := (z*z!=x) or I.
-
-	Note that z*z can overwrite I; this value must be sensed if it is
-	True.
-
-	Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
-	zero.
-
-		    --------------------
-		z1: |        f2        |
-		    --------------------
-		bit 31		   bit 0
-
-	Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
-	or even of logb(x) have the following relations:
-
-	-------------------------------------------------
-	bit 27,26 of z1		bit 1,0 of x1	logb(x)
-	-------------------------------------------------
-	00			00		odd and even
-	01			01		even
-	10			10		odd
-	10			00		even
-	11			01		even
-	-------------------------------------------------
-
-    (4)	Special cases (see (4) of Section A).
-
- */
-/* vi: set ts=4 sw=4 expandtab: */
--- a/src/video/math_private.h	Mon Sep 15 05:14:11 2008 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,171 +0,0 @@
-/*
- * ====================================================
- * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
- *
- * Developed at SunPro, a Sun Microsystems, Inc. business.
- * Permission to use, copy, modify, and distribute this
- * software is freely granted, provided that this notice
- * is preserved.
- * ====================================================
- */
-
-/*
- * from: @(#)fdlibm.h 5.1 93/09/24
- * $Id$
- */
-
-#ifndef _MATH_PRIVATE_H_
-#define _MATH_PRIVATE_H_
-
-#include "SDL_name.h"
-#include "SDL_endian.h"
-
-#define huge		really_big      /* huge is a reserved keyword in VC++ 6.0 */
-#define u_int32_t	uint32_t
-
-/* The original fdlibm code used statements like:
-	n0 = ((*(int*)&one)>>29)^1;		* index of high word *
-	ix0 = *(n0+(int*)&x);			* high word of x *
-	ix1 = *((1-n0)+(int*)&x);		* low word of x *
-   to dig two 32 bit words out of the 64 bit IEEE floating point
-   value.  That is non-ANSI, and, moreover, the gcc instruction
-   scheduler gets it wrong.  We instead use the following macros.
-   Unlike the original code, we determine the endianness at compile
-   time, not at run time; I don't see much benefit to selecting
-   endianness at run time.  */
-
-/* A union which permits us to convert between a double and two 32 bit
-   ints.  */
-
-/*
- * Math on arm is special:
- * For FPA, float words are always big-endian.
- * For VFP, floats words follow the memory system mode.
- */
-
-#if (SDL_BYTEORDER == SDL_BIG_ENDIAN) || \
-    (!defined(__VFP_FP__) && (defined(__arm__) || defined(__thumb__)))
-
-typedef union
-{
-    double value;
-    struct
-    {
-        u_int32_t msw;
-        u_int32_t lsw;
-    } parts;
-} ieee_double_shape_type;
-
-#else
-
-typedef union
-{
-    double value;
-    struct
-    {
-        u_int32_t lsw;
-        u_int32_t msw;
-    } parts;
-} ieee_double_shape_type;
-
-#endif
-
-/* Get two 32 bit ints from a double.  */
-
-#define EXTRACT_WORDS(ix0,ix1,d)				\
-do {								\
-  ieee_double_shape_type ew_u;					\
-  ew_u.value = (d);						\
-  (ix0) = ew_u.parts.msw;					\
-  (ix1) = ew_u.parts.lsw;					\
-} while (0)
-
-/* Get the more significant 32 bit int from a double.  */
-
-#define GET_HIGH_WORD(i,d)					\
-do {								\
-  ieee_double_shape_type gh_u;					\
-  gh_u.value = (d);						\
-  (i) = gh_u.parts.msw;						\
-} while (0)
-
-/* Get the less significant 32 bit int from a double.  */
-
-#define GET_LOW_WORD(i,d)					\
-do {								\
-  ieee_double_shape_type gl_u;					\
-  gl_u.value = (d);						\
-  (i) = gl_u.parts.lsw;						\
-} while (0)
-
-/* Set a double from two 32 bit ints.  */
-
-#define INSERT_WORDS(d,ix0,ix1)					\
-do {								\
-  ieee_double_shape_type iw_u;					\
-  iw_u.parts.msw = (ix0);					\
-  iw_u.parts.lsw = (ix1);					\
-  (d) = iw_u.value;						\
-} while (0)
-
-/* Set the more significant 32 bits of a double from an int.  */
-
-#define SET_HIGH_WORD(d,v)					\
-do {								\
-  ieee_double_shape_type sh_u;					\
-  sh_u.value = (d);						\
-  sh_u.parts.msw = (v);						\
-  (d) = sh_u.value;						\
-} while (0)
-
-/* Set the less significant 32 bits of a double from an int.  */
-
-#define SET_LOW_WORD(d,v)					\
-do {								\
-  ieee_double_shape_type sl_u;					\
-  sl_u.value = (d);						\
-  sl_u.parts.lsw = (v);						\
-  (d) = sl_u.value;						\
-} while (0)
-
-/* A union which permits us to convert between a float and a 32 bit
-   int.  */
-
-typedef union
-{
-    float value;
-    u_int32_t word;
-} ieee_float_shape_type;
-
-/* Get a 32 bit int from a float.  */
-
-#define GET_FLOAT_WORD(i,d)					\
-do {								\
-  ieee_float_shape_type gf_u;					\
-  gf_u.value = (d);						\
-  (i) = gf_u.word;						\
-} while (0)
-
-/* Set a float from a 32 bit int.  */
-
-#define SET_FLOAT_WORD(d,i)					\
-do {								\
-  ieee_float_shape_type sf_u;					\
-  sf_u.word = (i);						\
-  (d) = sf_u.value;						\
-} while (0)
-
-
-#ifdef __STDC__
-static const double
-#else
-static double
-#endif
-  zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
-    two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
-    twom54 = 5.55111512312578270212e-17,        /* 0x3C900000, 0x00000000 */
-    huge = 1.0e+300, tiny = 1.0e-300;
-
-#endif /* _MATH_PRIVATE_H_ */
-
-/* vi: set ts=4 sw=4 expandtab: */