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 pio4
= 0x1.921fb54442d18p
-1,
40 invpio2
= 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
41 pio2_1
= 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
42 pio2_1t
= 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
43 pio2_2
= 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
44 pio2_2t
= 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
45 pio2_3
= 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
46 pio2_3t
= 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
48 /* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
49 int __rem_pio2(double x
, double *y
)
51 union {double f
; uint64_t i
;} u
= {x
};
55 int sign
, n
, ex
, ey
, i
;
58 ix
= u
.i
>>32 & 0x7fffffff;
59 if (ix
<= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
60 if ((ix
& 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
61 goto medium
; /* cancellation -- use medium case */
62 if (ix
<= 0x4002d97c) { /* |x| ~<= 3pi/4 */
64 z
= x
- pio2_1
; /* one round good to 85 bits */
66 y
[1] = (z
-y
[0]) - pio2_1t
;
71 y
[1] = (z
-y
[0]) + pio2_1t
;
78 y
[1] = (z
-y
[0]) - 2*pio2_1t
;
83 y
[1] = (z
-y
[0]) + 2*pio2_1t
;
88 if (ix
<= 0x401c463b) { /* |x| ~<= 9pi/4 */
89 if (ix
<= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
90 if (ix
== 0x4012d97c) /* |x| ~= 3pi/2 */
95 y
[1] = (z
-y
[0]) - 3*pio2_1t
;
100 y
[1] = (z
-y
[0]) + 3*pio2_1t
;
104 if (ix
== 0x401921fb) /* |x| ~= 4pi/2 */
108 y
[0] = z
- 4*pio2_1t
;
109 y
[1] = (z
-y
[0]) - 4*pio2_1t
;
113 y
[0] = z
+ 4*pio2_1t
;
114 y
[1] = (z
-y
[0]) + 4*pio2_1t
;
119 if (ix
< 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
122 fn
= (double_t
)x
*invpio2
+ toint
- toint
;
125 w
= fn
*pio2_1t
; /* 1st round, good to 85 bits */
126 /* Matters with directed rounding. */
127 if (predict_false(r
- w
< -pio4
)) {
132 } else if (predict_false(r
- w
> pio4
)) {
140 ey
= u
.i
>>52 & 0x7ff;
142 if (ex
- ey
> 16) { /* 2nd round, good to 118 bits */
146 w
= fn
*pio2_2t
- ((t
-r
)-w
);
149 ey
= u
.i
>>52 & 0x7ff;
150 if (ex
- ey
> 49) { /* 3rd round, good to 151 bits, covers all cases */
154 w
= fn
*pio2_3t
- ((t
-r
)-w
);
158 y
[1] = (r
- y
[0]) - w
;
162 * all other (large) arguments
164 if (ix
>= 0x7ff00000) { /* x is inf or NaN */
168 /* set z = scalbn(|x|,-ilogb(x)+23) */
170 u
.i
&= (uint64_t)-1>>12;
171 u
.i
|= (uint64_t)(0x3ff + 23)<<52;
173 for (i
=0; i
< 2; i
++) {
174 tx
[i
] = (double)(int32_t)z
;
175 z
= (z
-tx
[i
])*0x1p
24;
178 /* skip zero terms, first term is non-zero */
181 n
= __rem_pio2_large(tx
,ty
,(int)(ix
>>20)-(0x3ff+23),i
+1,1);