add license and other comments to existing files

library/compiler-builtins/libm/src/math/sqrt.rs

@@ -1,3 +1,81 @@
+/* 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
+ */
+
 use core::f64;
 
 const TINY: f64 = 1.0e-300;