1/*-
2 * SPDX-License-Identifier: BSD-2-Clause
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 <fenv.h>
30#include <float.h>
31
32#include "fpmath.h"
33#include "math.h"
34
35/* Return (x + ulp) for normal positive x. Assumes no overflow. */
36static inline long double
37inc(long double x)
38{
39	union IEEEl2bits u;
40
41	u.e = x;
42	if (++u.bits.manl == 0) {
43		if (++u.bits.manh == 0) {
44			u.bits.exp++;
45			u.bits.manh |= LDBL_NBIT;
46		}
47	}
48	return (u.e);
49}
50
51/* Return (x - ulp) for normal positive x. Assumes no underflow. */
52static inline long double
53dec(long double x)
54{
55	union IEEEl2bits u;
56
57	u.e = x;
58	if (u.bits.manl-- == 0) {
59		if (u.bits.manh-- == LDBL_NBIT) {
60			u.bits.exp--;
61			u.bits.manh |= LDBL_NBIT;
62		}
63	}
64	return (u.e);
65}
66
67#pragma STDC FENV_ACCESS ON
68
69/*
70 * This is slow, but simple and portable. You should use hardware sqrt
71 * if possible.
72 */
73
74long double
75sqrtl(long double x)
76{
77	union IEEEl2bits u;
78	int k, r;
79	long double lo, xn;
80	fenv_t env;
81
82	u.e = x;
83
84	/* If x = NaN, then sqrt(x) = NaN. */
85	/* If x = Inf, then sqrt(x) = Inf. */
86	/* If x = -Inf, then sqrt(x) = NaN. */
87	if (u.bits.exp == LDBL_MAX_EXP * 2 - 1)
88		return (x * x + x);
89
90	/* If x = +-0, then sqrt(x) = +-0. */
91	if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
92		return (x);
93
94	/* If x < 0, then raise invalid and return NaN */
95	if (u.bits.sign)
96		return ((x - x) / (x - x));
97
98	feholdexcept(&env);
99
100	if (u.bits.exp == 0) {
101		/* Adjust subnormal numbers. */
102		u.e *= 0x1.0p514;
103		k = -514;
104	} else {
105		k = 0;
106	}
107	/*
108	 * u.e is a normal number, so break it into u.e = e*2^n where
109	 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
110	 */
111	if ((u.bits.exp - 0x3ffe) & 1) {	/* n is odd.     */
112		k += u.bits.exp - 0x3fff;	/* 2k = n - 1.   */
113		u.bits.exp = 0x3fff;		/* u.e in [1,2). */
114	} else {
115		k += u.bits.exp - 0x4000;	/* 2k = n - 2.   */
116		u.bits.exp = 0x4000;		/* u.e in [2,4). */
117	}
118
119	/*
120	 * Newton's iteration.
121	 * Split u.e into a high and low part to achieve additional precision.
122	 */
123	xn = sqrt(u.e);			/* 53-bit estimate of sqrtl(x). */
124#if LDBL_MANT_DIG > 100
125	xn = (xn + (u.e / xn)) * 0.5;	/* 106-bit estimate. */
126#endif
127	lo = u.e;
128	u.bits.manl = 0;		/* Zero out lower bits. */
129	lo = (lo - u.e) / xn;		/* Low bits divided by xn. */
130	xn = xn + (u.e / xn);		/* High portion of estimate. */
131	u.e = xn + lo;			/* Combine everything. */
132	u.bits.exp += (k >> 1) - 1;
133
134	feclearexcept(FE_INEXACT);
135	r = fegetround();
136	fesetround(FE_TOWARDZERO);	/* Set to round-toward-zero. */
137	xn = x / u.e;			/* Chopped quotient (inexact?). */
138
139	if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
140		if (xn == u.e) {
141			fesetenv(&env);
142			return (u.e);
143		}
144		/* Round correctly for inputs like x = y**2 - ulp. */
145		xn = dec(xn);		/* xn = xn - ulp. */
146	}
147
148	if (r == FE_TONEAREST) {
149		xn = inc(xn);		/* xn = xn + ulp. */
150	} else if (r == FE_UPWARD) {
151		u.e = inc(u.e);		/* u.e = u.e + ulp. */
152		xn = inc(xn);		/* xn  = xn + ulp. */
153	}
154	u.e = u.e + xn;				/* Chopped sum. */
155	feupdateenv(&env);	/* Restore env and raise inexact */
156	u.bits.exp--;
157	return (u.e);
158}
159