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