1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
13 #if defined(LIBM_SCCS) && !defined(lint)
14 static char rcsid
[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
18 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
19 * double x[],y[]; int e0,nx,prec; int ipio2[];
21 * __kernel_rem_pio2 return the last three digits of N with
25 * The method is to compute the integer (mod 8) and fraction parts of
26 * (2/pi)*x without doing the full multiplication. In general we
27 * skip the part of the product that are known to be a huge integer (
28 * more accurately, = 0 mod 8 ). Thus the number of operations are
29 * independent of the exponent of the input.
31 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
34 * x[] The input value (must be positive) is broken into nx
35 * pieces of 24-bit integers in double precision format.
36 * x[i] will be the i-th 24 bit of x. The scaled exponent
37 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
38 * match x's up to 24 bits.
40 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
48 * y[] output result in an array of double precision numbers.
49 * The dimension of y[] is:
54 * The actual value is the sum of them. Thus for 113-bit
55 * precision, one may have to do something like:
57 * long double t,w,r_head, r_tail;
58 * t = (long double)y[2] + (long double)y[1];
59 * w = (long double)y[0];
61 * r_tail = w - (r_head - t);
63 * e0 The exponent of x[0]
67 * prec an integer indicating the precision:
70 * 2 64 bits (extended)
74 * integer array, contains the (24*i)-th to (24*i+23)-th
75 * bit of 2/pi after binary point. The corresponding
78 * ipio2[i] * 2^(-24(i+1)).
81 * double scalbn(), floor();
84 * Here is the description of some local variables:
86 * jk jk+1 is the initial number of terms of ipio2[] needed
87 * in the computation. The recommended value is 2,3,4,
88 * 6 for single, double, extended,and quad.
90 * jz local integer variable indicating the number of
91 * terms of ipio2[] used.
95 * jv index for pointing to the suitable ipio2[] for the
96 * computation. In general, we want
97 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
99 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
100 * Hence jv = max(0,(e0-3)/24).
102 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
104 * q[] double array with integral value, representing the
105 * 24-bits chunk of the product of x and 2/pi.
107 * q0 the corresponding exponent of q[0]. Note that the
108 * exponent for q[i] would be q0-24*i.
110 * PIo2[] double precision array, obtained by cutting pi/2
111 * into 24 bits chunks.
113 * f[] ipio2[] in floating point
115 * iq[] integer array by breaking up q[] in 24-bits chunk.
117 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
119 * ih integer. If >0 it indicates q[] is >= 0.5, hence
120 * it also indicates the *sign* of the result.
127 * The hexadecimal values are the intended ones for the following
128 * constants. The decimal values may be used, provided that the
129 * compiler will convert from decimal to binary accurately enough
130 * to produce the hexadecimal values shown.
134 #include <math-narrow-eval.h>
135 #include <math_private.h>
136 #include <libc-diag.h>
138 static const int init_jk
[] = {2,3,4,6}; /* initial value for jk */
140 static const double PIo2
[] = {
141 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
142 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
143 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
144 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
145 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
146 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
147 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
148 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
154 two24
= 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
155 twon24
= 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
158 __kernel_rem_pio2 (double *x
, double *y
, int e0
, int nx
, int prec
,
159 const int32_t *ipio2
)
161 int32_t jz
, jx
, jv
, jp
, jk
, carry
, n
, iq
[20], i
, j
, k
, m
, q0
, ih
;
162 double z
, fw
, f
[20], fq
[20], q
[20];
168 /* determine jx,jv,q0, note that 3>q0 */
170 jv
= (e0
- 3) / 24; if (jv
< 0)
172 q0
= e0
- 24 * (jv
+ 1);
174 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
175 j
= jv
- jx
; m
= jx
+ jk
;
176 for (i
= 0; i
<= m
; i
++, j
++)
177 f
[i
] = (j
< 0) ? zero
: (double) ipio2
[j
];
179 /* compute q[0],q[1],...q[jk] */
180 for (i
= 0; i
<= jk
; i
++)
182 for (j
= 0, fw
= 0.0; j
<= jx
; j
++)
183 fw
+= x
[j
] * f
[jx
+ i
- j
];
189 /* distill q[] into iq[] reversingly */
190 for (i
= 0, j
= jz
, z
= q
[jz
]; j
> 0; i
++, j
--)
192 fw
= (double) ((int32_t) (twon24
* z
));
193 iq
[i
] = (int32_t) (z
- two24
* fw
);
198 z
= __scalbn (z
, q0
); /* actual value of z */
199 z
-= 8.0 * floor (z
* 0.125); /* trim off integer >= 8 */
203 if (q0
> 0) /* need iq[jz-1] to determine n */
205 i
= (iq
[jz
- 1] >> (24 - q0
)); n
+= i
;
206 iq
[jz
- 1] -= i
<< (24 - q0
);
207 ih
= iq
[jz
- 1] >> (23 - q0
);
210 ih
= iq
[jz
- 1] >> 23;
214 if (ih
> 0) /* q > 0.5 */
217 for (i
= 0; i
< jz
; i
++) /* compute 1-q */
224 carry
= 1; iq
[i
] = 0x1000000 - j
;
228 iq
[i
] = 0xffffff - j
;
230 if (q0
> 0) /* rare case: chance is 1 in 12 */
235 iq
[jz
- 1] &= 0x7fffff; break;
237 iq
[jz
- 1] &= 0x3fffff; break;
244 z
-= __scalbn (one
, q0
);
248 /* check if recomputation is needed */
252 for (i
= jz
- 1; i
>= jk
; i
--)
254 if (j
== 0) /* need recomputation */
256 /* On s390x gcc 6.1 -O3 produces the warning "array subscript is below
257 array bounds [-Werror=array-bounds]". Only __ieee754_rem_pio2l
258 calls __kernel_rem_pio2 for normal numbers and |x| > pi/4 in case
259 of ldbl-96 and |x| > 3pi/4 in case of ldbl-128[ibm].
260 Thus x can't be zero and ipio2 is not zero, too. Thus not all iq[]
261 values can't be zero. */
262 DIAG_PUSH_NEEDS_COMMENT
;
263 DIAG_IGNORE_NEEDS_COMMENT (6.1, "-Warray-bounds");
264 for (k
= 1; iq
[jk
- k
] == 0; k
++)
265 ; /* k = no. of terms needed */
266 DIAG_POP_NEEDS_COMMENT
;
268 for (i
= jz
+ 1; i
<= jz
+ k
; i
++) /* add q[jz+1] to q[jz+k] */
270 f
[jx
+ i
] = (double) ipio2
[jv
+ i
];
271 for (j
= 0, fw
= 0.0; j
<= jx
; j
++)
272 fw
+= x
[j
] * f
[jx
+ i
- j
];
280 /* chop off zero terms */
289 else /* break z into 24-bit if necessary */
291 z
= __scalbn (z
, -q0
);
294 fw
= (double) ((int32_t) (twon24
* z
));
295 iq
[jz
] = (int32_t) (z
- two24
* fw
);
297 iq
[jz
] = (int32_t) fw
;
300 iq
[jz
] = (int32_t) z
;
303 /* jz is always nonnegative here, because the result is never zero to
304 full precision (this function is not called for zero arguments).
305 Help the compiler to know it. */
306 if (jz
< 0) __builtin_unreachable ();
308 /* convert integer "bit" chunk to floating-point value */
309 fw
= __scalbn (one
, q0
);
310 for (i
= jz
; i
>= 0; i
--)
312 q
[i
] = fw
* (double) iq
[i
]; fw
*= twon24
;
315 /* compute PIo2[0,...,jp]*q[jz,...,0] */
316 for (i
= jz
; i
>= 0; i
--)
318 for (fw
= 0.0, k
= 0; k
<= jp
&& k
<= jz
- i
; k
++)
319 fw
+= PIo2
[k
] * q
[i
+ k
];
323 /* compress fq[] into y[] */
328 for (i
= jz
; i
>= 0; i
--)
330 y
[0] = (ih
== 0) ? fw
: -fw
;
335 for (i
= jz
; i
>= 0; i
--)
336 fv
= math_narrow_eval (fv
+ fq
[i
]);
337 y
[0] = (ih
== 0) ? fv
: -fv
;
338 fv
= math_narrow_eval (fq
[0] - fv
);
339 for (i
= 1; i
<= jz
; i
++)
340 fv
= math_narrow_eval (fv
+ fq
[i
]);
341 y
[1] = (ih
== 0) ? fv
: -fv
;
343 case 3: /* painful */
344 for (i
= jz
; i
> 0; i
--)
346 double fv
= math_narrow_eval (fq
[i
- 1] + fq
[i
]);
347 fq
[i
] += fq
[i
- 1] - fv
;
350 for (i
= jz
; i
> 1; i
--)
352 double fv
= math_narrow_eval (fq
[i
- 1] + fq
[i
]);
353 fq
[i
] += fq
[i
- 1] - fv
;
356 for (fw
= 0.0, i
= jz
; i
>= 2; i
--)
360 y
[0] = fq
[0]; y
[1] = fq
[1]; y
[2] = fw
;
364 y
[0] = -fq
[0]; y
[1] = -fq
[1]; y
[2] = -fw
;