1*4882a593Smuzhiyun // SPDX-License-Identifier: GPL-2.0-only
2*4882a593Smuzhiyun /* IEEE754 floating point arithmetic
3*4882a593Smuzhiyun * double precision square root
4*4882a593Smuzhiyun */
5*4882a593Smuzhiyun /*
6*4882a593Smuzhiyun * MIPS floating point support
7*4882a593Smuzhiyun * Copyright (C) 1994-2000 Algorithmics Ltd.
8*4882a593Smuzhiyun */
9*4882a593Smuzhiyun
10*4882a593Smuzhiyun #include "ieee754dp.h"
11*4882a593Smuzhiyun
12*4882a593Smuzhiyun static const unsigned int table[] = {
13*4882a593Smuzhiyun 0, 1204, 3062, 5746, 9193, 13348, 18162, 23592,
14*4882a593Smuzhiyun 29598, 36145, 43202, 50740, 58733, 67158, 75992,
15*4882a593Smuzhiyun 85215, 83599, 71378, 60428, 50647, 41945, 34246,
16*4882a593Smuzhiyun 27478, 21581, 16499, 12183, 8588, 5674, 3403,
17*4882a593Smuzhiyun 1742, 661, 130
18*4882a593Smuzhiyun };
19*4882a593Smuzhiyun
ieee754dp_sqrt(union ieee754dp x)20*4882a593Smuzhiyun union ieee754dp ieee754dp_sqrt(union ieee754dp x)
21*4882a593Smuzhiyun {
22*4882a593Smuzhiyun struct _ieee754_csr oldcsr;
23*4882a593Smuzhiyun union ieee754dp y, z, t;
24*4882a593Smuzhiyun unsigned int scalx, yh;
25*4882a593Smuzhiyun COMPXDP;
26*4882a593Smuzhiyun
27*4882a593Smuzhiyun EXPLODEXDP;
28*4882a593Smuzhiyun ieee754_clearcx();
29*4882a593Smuzhiyun FLUSHXDP;
30*4882a593Smuzhiyun
31*4882a593Smuzhiyun /* x == INF or NAN? */
32*4882a593Smuzhiyun switch (xc) {
33*4882a593Smuzhiyun case IEEE754_CLASS_SNAN:
34*4882a593Smuzhiyun return ieee754dp_nanxcpt(x);
35*4882a593Smuzhiyun
36*4882a593Smuzhiyun case IEEE754_CLASS_QNAN:
37*4882a593Smuzhiyun /* sqrt(Nan) = Nan */
38*4882a593Smuzhiyun return x;
39*4882a593Smuzhiyun
40*4882a593Smuzhiyun case IEEE754_CLASS_ZERO:
41*4882a593Smuzhiyun /* sqrt(0) = 0 */
42*4882a593Smuzhiyun return x;
43*4882a593Smuzhiyun
44*4882a593Smuzhiyun case IEEE754_CLASS_INF:
45*4882a593Smuzhiyun if (xs) {
46*4882a593Smuzhiyun /* sqrt(-Inf) = Nan */
47*4882a593Smuzhiyun ieee754_setcx(IEEE754_INVALID_OPERATION);
48*4882a593Smuzhiyun return ieee754dp_indef();
49*4882a593Smuzhiyun }
50*4882a593Smuzhiyun /* sqrt(+Inf) = Inf */
51*4882a593Smuzhiyun return x;
52*4882a593Smuzhiyun
53*4882a593Smuzhiyun case IEEE754_CLASS_DNORM:
54*4882a593Smuzhiyun DPDNORMX;
55*4882a593Smuzhiyun fallthrough;
56*4882a593Smuzhiyun case IEEE754_CLASS_NORM:
57*4882a593Smuzhiyun if (xs) {
58*4882a593Smuzhiyun /* sqrt(-x) = Nan */
59*4882a593Smuzhiyun ieee754_setcx(IEEE754_INVALID_OPERATION);
60*4882a593Smuzhiyun return ieee754dp_indef();
61*4882a593Smuzhiyun }
62*4882a593Smuzhiyun break;
63*4882a593Smuzhiyun }
64*4882a593Smuzhiyun
65*4882a593Smuzhiyun /* save old csr; switch off INX enable & flag; set RN rounding */
66*4882a593Smuzhiyun oldcsr = ieee754_csr;
67*4882a593Smuzhiyun ieee754_csr.mx &= ~IEEE754_INEXACT;
68*4882a593Smuzhiyun ieee754_csr.sx &= ~IEEE754_INEXACT;
69*4882a593Smuzhiyun ieee754_csr.rm = FPU_CSR_RN;
70*4882a593Smuzhiyun
71*4882a593Smuzhiyun /* adjust exponent to prevent overflow */
72*4882a593Smuzhiyun scalx = 0;
73*4882a593Smuzhiyun if (xe > 512) { /* x > 2**-512? */
74*4882a593Smuzhiyun xe -= 512; /* x = x / 2**512 */
75*4882a593Smuzhiyun scalx += 256;
76*4882a593Smuzhiyun } else if (xe < -512) { /* x < 2**-512? */
77*4882a593Smuzhiyun xe += 512; /* x = x * 2**512 */
78*4882a593Smuzhiyun scalx -= 256;
79*4882a593Smuzhiyun }
80*4882a593Smuzhiyun
81*4882a593Smuzhiyun x = builddp(0, xe + DP_EBIAS, xm & ~DP_HIDDEN_BIT);
82*4882a593Smuzhiyun y = x;
83*4882a593Smuzhiyun
84*4882a593Smuzhiyun /* magic initial approximation to almost 8 sig. bits */
85*4882a593Smuzhiyun yh = y.bits >> 32;
86*4882a593Smuzhiyun yh = (yh >> 1) + 0x1ff80000;
87*4882a593Smuzhiyun yh = yh - table[(yh >> 15) & 31];
88*4882a593Smuzhiyun y.bits = ((u64) yh << 32) | (y.bits & 0xffffffff);
89*4882a593Smuzhiyun
90*4882a593Smuzhiyun /* Heron's rule once with correction to improve to ~18 sig. bits */
91*4882a593Smuzhiyun /* t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0; */
92*4882a593Smuzhiyun t = ieee754dp_div(x, y);
93*4882a593Smuzhiyun y = ieee754dp_add(y, t);
94*4882a593Smuzhiyun y.bits -= 0x0010000600000000LL;
95*4882a593Smuzhiyun y.bits &= 0xffffffff00000000LL;
96*4882a593Smuzhiyun
97*4882a593Smuzhiyun /* triple to almost 56 sig. bits: y ~= sqrt(x) to within 1 ulp */
98*4882a593Smuzhiyun /* t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y; */
99*4882a593Smuzhiyun t = ieee754dp_mul(y, y);
100*4882a593Smuzhiyun z = t;
101*4882a593Smuzhiyun t.bexp += 0x001;
102*4882a593Smuzhiyun t = ieee754dp_add(t, z);
103*4882a593Smuzhiyun z = ieee754dp_mul(ieee754dp_sub(x, z), y);
104*4882a593Smuzhiyun
105*4882a593Smuzhiyun /* t=z/(t+x) ; pt[n0]+=0x00100000; y+=t; */
106*4882a593Smuzhiyun t = ieee754dp_div(z, ieee754dp_add(t, x));
107*4882a593Smuzhiyun t.bexp += 0x001;
108*4882a593Smuzhiyun y = ieee754dp_add(y, t);
109*4882a593Smuzhiyun
110*4882a593Smuzhiyun /* twiddle last bit to force y correctly rounded */
111*4882a593Smuzhiyun
112*4882a593Smuzhiyun /* set RZ, clear INEX flag */
113*4882a593Smuzhiyun ieee754_csr.rm = FPU_CSR_RZ;
114*4882a593Smuzhiyun ieee754_csr.sx &= ~IEEE754_INEXACT;
115*4882a593Smuzhiyun
116*4882a593Smuzhiyun /* t=x/y; ...chopped quotient, possibly inexact */
117*4882a593Smuzhiyun t = ieee754dp_div(x, y);
118*4882a593Smuzhiyun
119*4882a593Smuzhiyun if (ieee754_csr.sx & IEEE754_INEXACT || t.bits != y.bits) {
120*4882a593Smuzhiyun
121*4882a593Smuzhiyun if (!(ieee754_csr.sx & IEEE754_INEXACT))
122*4882a593Smuzhiyun /* t = t-ulp */
123*4882a593Smuzhiyun t.bits -= 1;
124*4882a593Smuzhiyun
125*4882a593Smuzhiyun /* add inexact to result status */
126*4882a593Smuzhiyun oldcsr.cx |= IEEE754_INEXACT;
127*4882a593Smuzhiyun oldcsr.sx |= IEEE754_INEXACT;
128*4882a593Smuzhiyun
129*4882a593Smuzhiyun switch (oldcsr.rm) {
130*4882a593Smuzhiyun case FPU_CSR_RU:
131*4882a593Smuzhiyun y.bits += 1;
132*4882a593Smuzhiyun fallthrough;
133*4882a593Smuzhiyun case FPU_CSR_RN:
134*4882a593Smuzhiyun t.bits += 1;
135*4882a593Smuzhiyun break;
136*4882a593Smuzhiyun }
137*4882a593Smuzhiyun
138*4882a593Smuzhiyun /* y=y+t; ...chopped sum */
139*4882a593Smuzhiyun y = ieee754dp_add(y, t);
140*4882a593Smuzhiyun
141*4882a593Smuzhiyun /* adjust scalx for correctly rounded sqrt(x) */
142*4882a593Smuzhiyun scalx -= 1;
143*4882a593Smuzhiyun }
144*4882a593Smuzhiyun
145*4882a593Smuzhiyun /* py[n0]=py[n0]+scalx; ...scale back y */
146*4882a593Smuzhiyun y.bexp += scalx;
147*4882a593Smuzhiyun
148*4882a593Smuzhiyun /* restore rounding mode, possibly set inexact */
149*4882a593Smuzhiyun ieee754_csr = oldcsr;
150*4882a593Smuzhiyun
151*4882a593Smuzhiyun return y;
152*4882a593Smuzhiyun }
153