Initial import

libc/math/sqrt.c

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+/* 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 "libc/math/libm.h"
+
+static const double 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 = 1.0 - tiny; /* raise inexact flag */
+		if (z >= 1.0) {
+			z = 1.0 + tiny;
+			if (q1 == (uint32_t)0xffffffff) {
+				q1 = 0;
+				q++;
+			} else if (z > 1.0) {
+				if (q1 == (uint32_t)0xfffffffe)
+					q++;
+				q1 += 2;
+			} else
+				q1 += q1 & 1;
+		}
+	}
+	ix0 = (q>>1) + 0x3fe00000;
+	ix1 = q1>>1;
+	if (q&1)
+		ix1 |= sign;
+	INSERT_WORDS(z, ix0 + ((uint32_t)m << 20), ix1);
+	return z;
+}