1 #include <stdint.h>
2 #include <math.h>
3 #include "libm.h"
4 #include "sqrt_data.h"
5
6 #define FENV_SUPPORT 1
7
8 /* returns a*b*2^-32 - e, with error 0 <= e < 1. */
mul32(uint32_t a,uint32_t b)9 static inline uint32_t mul32(uint32_t a, uint32_t b)
10 {
11 return (uint64_t)a*b >> 32;
12 }
13
14 /* returns a*b*2^-64 - e, with error 0 <= e < 3. */
mul64(uint64_t a,uint64_t b)15 static inline uint64_t mul64(uint64_t a, uint64_t b)
16 {
17 uint64_t ahi = a>>32;
18 uint64_t alo = a&0xffffffff;
19 uint64_t bhi = b>>32;
20 uint64_t blo = b&0xffffffff;
21 return ahi*bhi + (ahi*blo >> 32) + (alo*bhi >> 32);
22 }
23
sqrt(double x)24 double sqrt(double x)
25 {
26 uint64_t ix, top, m;
27
28 /* special case handling. */
29 ix = asuint64(x);
30 top = ix >> 52;
31 if (predict_false(top - 0x001 >= 0x7ff - 0x001)) {
32 /* x < 0x1p-1022 or inf or nan. */
33 if (ix * 2 == 0)
34 return x;
35 if (ix == 0x7ff0000000000000)
36 return x;
37 if (ix > 0x7ff0000000000000)
38 return __math_invalid(x);
39 /* x is subnormal, normalize it. */
40 ix = asuint64(x * 0x1p52);
41 top = ix >> 52;
42 top -= 52;
43 }
44
45 /* argument reduction:
46 x = 4^e m; with integer e, and m in [1, 4)
47 m: fixed point representation [2.62]
48 2^e is the exponent part of the result. */
49 int even = top & 1;
50 m = (ix << 11) | 0x8000000000000000;
51 if (even) m >>= 1;
52 top = (top + 0x3ff) >> 1;
53
54 /* approximate r ~ 1/sqrt(m) and s ~ sqrt(m) when m in [1,4)
55
56 initial estimate:
57 7bit table lookup (1bit exponent and 6bit significand).
58
59 iterative approximation:
60 using 2 goldschmidt iterations with 32bit int arithmetics
61 and a final iteration with 64bit int arithmetics.
62
63 details:
64
65 the relative error (e = r0 sqrt(m)-1) of a linear estimate
66 (r0 = a m + b) is |e| < 0.085955 ~ 0x1.6p-4 at best,
67 a table lookup is faster and needs one less iteration
68 6 bit lookup table (128b) gives |e| < 0x1.f9p-8
69 7 bit lookup table (256b) gives |e| < 0x1.fdp-9
70 for single and double prec 6bit is enough but for quad
71 prec 7bit is needed (or modified iterations). to avoid
72 one more iteration >=13bit table would be needed (16k).
73
74 a newton-raphson iteration for r is
75 w = r*r
76 u = 3 - m*w
77 r = r*u/2
78 can use a goldschmidt iteration for s at the end or
79 s = m*r
80
81 first goldschmidt iteration is
82 s = m*r
83 u = 3 - s*r
84 r = r*u/2
85 s = s*u/2
86 next goldschmidt iteration is
87 u = 3 - s*r
88 r = r*u/2
89 s = s*u/2
90 and at the end r is not computed only s.
91
92 they use the same amount of operations and converge at the
93 same quadratic rate, i.e. if
94 r1 sqrt(m) - 1 = e, then
95 r2 sqrt(m) - 1 = -3/2 e^2 - 1/2 e^3
96 the advantage of goldschmidt is that the mul for s and r
97 are independent (computed in parallel), however it is not
98 "self synchronizing": it only uses the input m in the
99 first iteration so rounding errors accumulate. at the end
100 or when switching to larger precision arithmetics rounding
101 errors dominate so the first iteration should be used.
102
103 the fixed point representations are
104 m: 2.30 r: 0.32, s: 2.30, d: 2.30, u: 2.30, three: 2.30
105 and after switching to 64 bit
106 m: 2.62 r: 0.64, s: 2.62, d: 2.62, u: 2.62, three: 2.62 */
107
108 static const uint64_t three = 0xc0000000;
109 uint64_t r, s, d, u, i;
110
111 i = (ix >> 46) % 128;
112 r = (uint32_t)__rsqrt_tab[i] << 16;
113 /* |r sqrt(m) - 1| < 0x1.fdp-9 */
114 s = mul32(m>>32, r);
115 /* |s/sqrt(m) - 1| < 0x1.fdp-9 */
116 d = mul32(s, r);
117 u = three - d;
118 r = mul32(r, u) << 1;
119 /* |r sqrt(m) - 1| < 0x1.7bp-16 */
120 s = mul32(s, u) << 1;
121 /* |s/sqrt(m) - 1| < 0x1.7bp-16 */
122 d = mul32(s, r);
123 u = three - d;
124 r = mul32(r, u) << 1;
125 /* |r sqrt(m) - 1| < 0x1.3704p-29 (measured worst-case) */
126 r = r << 32;
127 s = mul64(m, r);
128 d = mul64(s, r);
129 u = (three<<32) - d;
130 s = mul64(s, u); /* repr: 3.61 */
131 /* -0x1p-57 < s - sqrt(m) < 0x1.8001p-61 */
132 s = (s - 2) >> 9; /* repr: 12.52 */
133 /* -0x1.09p-52 < s - sqrt(m) < -0x1.fffcp-63 */
134
135 /* s < sqrt(m) < s + 0x1.09p-52,
136 compute nearest rounded result:
137 the nearest result to 52 bits is either s or s+0x1p-52,
138 we can decide by comparing (2^52 s + 0.5)^2 to 2^104 m. */
139 uint64_t d0, d1, d2;
140 double y, t;
141 d0 = (m << 42) - s*s;
142 d1 = s - d0;
143 d2 = d1 + s + 1;
144 s += d1 >> 63;
145 s &= 0x000fffffffffffff;
146 s |= top << 52;
147 y = asdouble(s);
148 if (FENV_SUPPORT) {
149 /* handle rounding modes and inexact exception:
150 only (s+1)^2 == 2^42 m case is exact otherwise
151 add a tiny value to cause the fenv effects. */
152 uint64_t tiny = predict_false(d2==0) ? 0 : 0x0010000000000000;
153 tiny |= (d1^d2) & 0x8000000000000000;
154 t = asdouble(tiny);
155 y = eval_as_double(y + t);
156 }
157 return y;
158 }
159