1/*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2007 Steven G. Kargl
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice unmodified, this list of conditions, and the following
12 *    disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 *    notice, this list of conditions and the following disclaimer in the
15 *    documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29#include <sys/cdefs.h>
30__FBSDID("$FreeBSD$");
31
32#include <fenv.h>
33#include <float.h>
34
35#include "fpmath.h"
36#include "math.h"
37
38/* Return (x + ulp) for normal positive x. Assumes no overflow. */
39static inline long double
40inc(long double x)
41{
42	union IEEEl2bits u;
43
44	u.e = x;
45	if (++u.bits.manl == 0) {
46		if (++u.bits.manh == 0) {
47			u.bits.exp++;
48			u.bits.manh |= LDBL_NBIT;
49		}
50	}
51	return (u.e);
52}
53
54/* Return (x - ulp) for normal positive x. Assumes no underflow. */
55static inline long double
56dec(long double x)
57{
58	union IEEEl2bits u;
59
60	u.e = x;
61	if (u.bits.manl-- == 0) {
62		if (u.bits.manh-- == LDBL_NBIT) {
63			u.bits.exp--;
64			u.bits.manh |= LDBL_NBIT;
65		}
66	}
67	return (u.e);
68}
69
70#pragma STDC FENV_ACCESS ON
71
72/*
73 * This is slow, but simple and portable. You should use hardware sqrt
74 * if possible.
75 */
76
77long double
78sqrtl(long double x)
79{
80	union IEEEl2bits u;
81	int k, r;
82	long double lo, xn;
83	fenv_t env;
84
85	u.e = x;
86
87	/* If x = NaN, then sqrt(x) = NaN. */
88	/* If x = Inf, then sqrt(x) = Inf. */
89	/* If x = -Inf, then sqrt(x) = NaN. */
90	if (u.bits.exp == LDBL_MAX_EXP * 2 - 1)
91		return (x * x + x);
92
93	/* If x = +-0, then sqrt(x) = +-0. */
94	if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
95		return (x);
96
97	/* If x < 0, then raise invalid and return NaN */
98	if (u.bits.sign)
99		return ((x - x) / (x - x));
100
101	feholdexcept(&env);
102
103	if (u.bits.exp == 0) {
104		/* Adjust subnormal numbers. */
105		u.e *= 0x1.0p514;
106		k = -514;
107	} else {
108		k = 0;
109	}
110	/*
111	 * u.e is a normal number, so break it into u.e = e*2^n where
112	 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
113	 */
114	if ((u.bits.exp - 0x3ffe) & 1) {	/* n is odd.     */
115		k += u.bits.exp - 0x3fff;	/* 2k = n - 1.   */
116		u.bits.exp = 0x3fff;		/* u.e in [1,2). */
117	} else {
118		k += u.bits.exp - 0x4000;	/* 2k = n - 2.   */
119		u.bits.exp = 0x4000;		/* u.e in [2,4). */
120	}
121
122	/*
123	 * Newton's iteration.
124	 * Split u.e into a high and low part to achieve additional precision.
125	 */
126	xn = sqrt(u.e);			/* 53-bit estimate of sqrtl(x). */
127#if LDBL_MANT_DIG > 100
128	xn = (xn + (u.e / xn)) * 0.5;	/* 106-bit estimate. */
129#endif
130	lo = u.e;
131	u.bits.manl = 0;		/* Zero out lower bits. */
132	lo = (lo - u.e) / xn;		/* Low bits divided by xn. */
133	xn = xn + (u.e / xn);		/* High portion of estimate. */
134	u.e = xn + lo;			/* Combine everything. */
135	u.bits.exp += (k >> 1) - 1;
136
137	feclearexcept(FE_INEXACT);
138	r = fegetround();
139	fesetround(FE_TOWARDZERO);	/* Set to round-toward-zero. */
140	xn = x / u.e;			/* Chopped quotient (inexact?). */
141
142	if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
143		if (xn == u.e) {
144			fesetenv(&env);
145			return (u.e);
146		}
147		/* Round correctly for inputs like x = y**2 - ulp. */
148		xn = dec(xn);		/* xn = xn - ulp. */
149	}
150
151	if (r == FE_TONEAREST) {
152		xn = inc(xn);		/* xn = xn + ulp. */
153	} else if (r == FE_UPWARD) {
154		u.e = inc(u.e);		/* u.e = u.e + ulp. */
155		xn = inc(xn);		/* xn  = xn + ulp. */
156	}
157	u.e = u.e + xn;				/* Chopped sum. */
158	feupdateenv(&env);	/* Restore env and raise inexact */
159	u.bits.exp--;
160	return (u.e);
161}
162