1/* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 *
12 * Optimized by Bruce D. Evans.
13 */
14/* __rem_pio2(x,y)
15 *
16 * return the remainder of x rem pi/2 in y[0]+y[1]
17 * use __rem_pio2_large() for large x
18 */
19
20#include "libm.h"
21
22#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
23#define EPS DBL_EPSILON
24#elif FLT_EVAL_METHOD==2
25#define EPS LDBL_EPSILON
26#endif
27
28/*
29 * invpio2:  53 bits of 2/pi
30 * pio2_1:   first  33 bit of pi/2
31 * pio2_1t:  pi/2 - pio2_1
32 * pio2_2:   second 33 bit of pi/2
33 * pio2_2t:  pi/2 - (pio2_1+pio2_2)
34 * pio2_3:   third  33 bit of pi/2
35 * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
36 */
37static const double
38toint   = 1.5/EPS,
39invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
40pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
41pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
42pio2_2  = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
43pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
44pio2_3  = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
45pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
46
47/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
48int __rem_pio2(double x, double *y)
49{
50	union {double f; uint64_t i;} u = {x};
51	double_t z,w,t,r,fn;
52	double tx[3],ty[2];
53	uint32_t ix;
54	int sign, n, ex, ey, i;
55
56	sign = u.i>>63;
57	ix = u.i>>32 & 0x7fffffff;
58	if (ix <= 0x400f6a7a) {  /* |x| ~<= 5pi/4 */
59		if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
60			goto medium;  /* cancellation -- use medium case */
61		if (ix <= 0x4002d97c) {  /* |x| ~<= 3pi/4 */
62			if (!sign) {
63				z = x - pio2_1;  /* one round good to 85 bits */
64				y[0] = z - pio2_1t;
65				y[1] = (z-y[0]) - pio2_1t;
66				return 1;
67			} else {
68				z = x + pio2_1;
69				y[0] = z + pio2_1t;
70				y[1] = (z-y[0]) + pio2_1t;
71				return -1;
72			}
73		} else {
74			if (!sign) {
75				z = x - 2*pio2_1;
76				y[0] = z - 2*pio2_1t;
77				y[1] = (z-y[0]) - 2*pio2_1t;
78				return 2;
79			} else {
80				z = x + 2*pio2_1;
81				y[0] = z + 2*pio2_1t;
82				y[1] = (z-y[0]) + 2*pio2_1t;
83				return -2;
84			}
85		}
86	}
87	if (ix <= 0x401c463b) {  /* |x| ~<= 9pi/4 */
88		if (ix <= 0x4015fdbc) {  /* |x| ~<= 7pi/4 */
89			if (ix == 0x4012d97c)  /* |x| ~= 3pi/2 */
90				goto medium;
91			if (!sign) {
92				z = x - 3*pio2_1;
93				y[0] = z - 3*pio2_1t;
94				y[1] = (z-y[0]) - 3*pio2_1t;
95				return 3;
96			} else {
97				z = x + 3*pio2_1;
98				y[0] = z + 3*pio2_1t;
99				y[1] = (z-y[0]) + 3*pio2_1t;
100				return -3;
101			}
102		} else {
103			if (ix == 0x401921fb)  /* |x| ~= 4pi/2 */
104				goto medium;
105			if (!sign) {
106				z = x - 4*pio2_1;
107				y[0] = z - 4*pio2_1t;
108				y[1] = (z-y[0]) - 4*pio2_1t;
109				return 4;
110			} else {
111				z = x + 4*pio2_1;
112				y[0] = z + 4*pio2_1t;
113				y[1] = (z-y[0]) + 4*pio2_1t;
114				return -4;
115			}
116		}
117	}
118	if (ix < 0x413921fb) {  /* |x| ~< 2^20*(pi/2), medium size */
119medium:
120		/* rint(x/(pi/2)), Assume round-to-nearest. */
121		fn = (double_t)x*invpio2 + toint - toint;
122		n = (int32_t)fn;
123		r = x - fn*pio2_1;
124		w = fn*pio2_1t;  /* 1st round, good to 85 bits */
125		y[0] = r - w;
126		u.f = y[0];
127		ey = u.i>>52 & 0x7ff;
128		ex = ix>>20;
129		if (ex - ey > 16) { /* 2nd round, good to 118 bits */
130			t = r;
131			w = fn*pio2_2;
132			r = t - w;
133			w = fn*pio2_2t - ((t-r)-w);
134			y[0] = r - w;
135			u.f = y[0];
136			ey = u.i>>52 & 0x7ff;
137			if (ex - ey > 49) {  /* 3rd round, good to 151 bits, covers all cases */
138				t = r;
139				w = fn*pio2_3;
140				r = t - w;
141				w = fn*pio2_3t - ((t-r)-w);
142				y[0] = r - w;
143			}
144		}
145		y[1] = (r - y[0]) - w;
146		return n;
147	}
148	/*
149	 * all other (large) arguments
150	 */
151	if (ix >= 0x7ff00000) {  /* x is inf or NaN */
152		y[0] = y[1] = x - x;
153		return 0;
154	}
155	/* set z = scalbn(|x|,-ilogb(x)+23) */
156	u.f = x;
157	u.i &= (uint64_t)-1>>12;
158	u.i |= (uint64_t)(0x3ff + 23)<<52;
159	z = u.f;
160	for (i=0; i < 2; i++) {
161		tx[i] = (double)(int32_t)z;
162		z     = (z-tx[i])*0x1p24;
163	}
164	tx[i] = z;
165	/* skip zero terms, first term is non-zero */
166	while (tx[i] == 0.0)
167		i--;
168	n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
169	if (sign) {
170		y[0] = -ty[0];
171		y[1] = -ty[1];
172		return -n;
173	}
174	y[0] = ty[0];
175	y[1] = ty[1];
176	return n;
177}
178