1 /* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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
10 * ====================================================
12 * Optimized by Bruce D. Evans.
16 * return the remainder of x rem pi/2 in y[0]+y[1]
17 * use __rem_pio2_large() for large x
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
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)
39 invpio2
= 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
40 pio2_1
= 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
41 pio2_1t
= 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
42 pio2_2
= 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
43 pio2_2t
= 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
44 pio2_3
= 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
45 pio2_3t
= 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
47 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
48 int __rem_pio2(double x
, double *y
)
50 union {double f
; uint64_t i
;} u
= {x
};
54 int sign
, n
, ex
, ey
, i
;
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 */
63 z
= x
- pio2_1
; /* one round good to 85 bits */
65 y
[1] = (z
-y
[0]) - pio2_1t
;
70 y
[1] = (z
-y
[0]) + pio2_1t
;
77 y
[1] = (z
-y
[0]) - 2*pio2_1t
;
82 y
[1] = (z
-y
[0]) + 2*pio2_1t
;
87 if (ix
<= 0x401c463b) { /* |x| ~<= 9pi/4 */
88 if (ix
<= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
89 if (ix
== 0x4012d97c) /* |x| ~= 3pi/2 */
94 y
[1] = (z
-y
[0]) - 3*pio2_1t
;
99 y
[1] = (z
-y
[0]) + 3*pio2_1t
;
103 if (ix
== 0x401921fb) /* |x| ~= 4pi/2 */
107 y
[0] = z
- 4*pio2_1t
;
108 y
[1] = (z
-y
[0]) - 4*pio2_1t
;
112 y
[0] = z
+ 4*pio2_1t
;
113 y
[1] = (z
-y
[0]) + 4*pio2_1t
;
118 if (ix
< 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
120 /* rint(x/(pi/2)), Assume round-to-nearest. */
121 fn
= (double_t
)x
*invpio2
+ toint
- toint
;
124 w
= fn
*pio2_1t
; /* 1st round, good to 85 bits */
127 ey
= u
.i
>>52 & 0x7ff;
129 if (ex
- ey
> 16) { /* 2nd round, good to 118 bits */
133 w
= fn
*pio2_2t
- ((t
-r
)-w
);
136 ey
= u
.i
>>52 & 0x7ff;
137 if (ex
- ey
> 49) { /* 3rd round, good to 151 bits, covers all cases */
141 w
= fn
*pio2_3t
- ((t
-r
)-w
);
145 y
[1] = (r
- y
[0]) - w
;
149 * all other (large) arguments
151 if (ix
>= 0x7ff00000) { /* x is inf or NaN */
155 /* set z = scalbn(|x|,-ilogb(x)+23) */
157 u
.i
&= (uint64_t)-1>>12;
158 u
.i
|= (uint64_t)(0x3ff + 23)<<52;
160 for (i
=0; i
< 2; i
++) {
161 tx
[i
] = (double)(int32_t)z
;
162 z
= (z
-tx
[i
])*0x1p
24;
165 /* skip zero terms, first term is non-zero */
168 n
= __rem_pio2_large(tx
,ty
,(int)(ix
>>20)-(0x3ff+23),i
+1,1);