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      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: head/lib/msun/src/e_sqrtl.c 326219 2017-11-26 02:00:33Z pfg $");
     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. */
     39 static inline long double
     40 inc(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. */
     55 static inline long double
     56 dec(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 
     77 long double
     78 sqrtl(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