first commit of the new libm!

thanks to the hard work of Szabolcs Nagy (nsz), identifying the best
(from correctness and license standpoint) implementations from freebsd
and openbsd and cleaning them up! musl should now fully support c99
float and long double math functions, and has near-complete complex
math support. tgmath should also work (fully on gcc-compatible
compilers, and mostly on any c99 compiler).

based largely on commit 0376d44a890fea261506f1fc63833e7a686dca19 from
nsz's libm git repo, with some additions (dummy versions of a few
missing long double complex functions, etc.) by me.

various cleanups still need to be made, including re-adding (if
they're correct) some asm functions that were dropped.

1 files changed, 185 insertions, 0 deletions

diff --git a/src/math/sqrt.c b/src/math/sqrt.c
new file mode 100644
index 00000000..2ebd022b
--- /dev/null
+++ b/src/math/sqrt.c
@@ -0,0 +1,185 @@
+/* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunSoft, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+/* 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
+ */
+
+#include "libm.h"
+
+static const double one = 1.0, tiny = 1.0e-300;
+
+double sqrt(double x)
+{
+	double z;
+	int32_t sign = (int)0x80000000;
+	int32_t ix0,s0,q,m,t,i;
+	uint32_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 */
+		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++;
+			ix0 -= t;
+			if (ix1 < t1)
+				ix0--;
+			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; /* raise inexact flag */
+		if (z >= one) {
+			z = one + tiny;
+			if (q1 == (uint32_t)0xffffffff) {
+				q1 = 0;
+				q++;
+			} else if (z > one) {
+				if (q1 == (uint32_t)0xfffffffe)
+					q++;
+				q1 += 2;
+			} else
+				q1 += q1 & 1;
+		}
+	}
+	ix0 = (q>>1) + 0x3fe00000;
+	ix1 = q1>>1;
+	if (q&1)
+		ix1 |= sign;
+	ix0 += m << 20;
+	INSERT_WORDS(z, ix0, ix1);
+	return z;
+}