newlib/libm/ld128/e_rem_pio2l.h

   1 /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  *
  13  * Optimized by Bruce D. Evans.
  14  */
  15
  16 #include <sys/cdefs.h>
  17 __FBSDID("$FreeBSD$");
  18
  19 /* ld128 version of __ieee754_rem_pio2l(x,y)
  20  *
  21  * return the remainder of x rem pi/2 in y[0]+y[1]
  22  * use __kernel_rem_pio2()
  23  */
  24
  25 #include <float.h>
  26
  27 #include "math.h"
  28 #include "../ld/math_private.h"
  29 #include "../ld/fpmath.h"
  30
  31 #define BIAS    (LDBL_MAX_EXP - 1)
  32
  33 /*
  34  * XXX need to verify that nonzero integer multiples of pi/2 within the
  35  * range get no closer to a long double than 2**-140, or that
  36  * ilogb(x) + ilogb(min_delta) < 45 - -140.
  37  */
  38 /*
  39  * invpio2:  113 bits of 2/pi
  40  * pio2_1:   first  68 bits of pi/2
  41  * pio2_1t:  pi/2 - pio2_1
  42  * pio2_2:   second 68 bits of pi/2
  43  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  44  * pio2_3:   third  68 bits of pi/2
  45  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  46  */
  47
  48 static const double
  49 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  50 two24 =  1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */
  51
  52 static const long double
  53 invpio2 =  6.3661977236758134307553505349005747e-01L,   /*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
  54 pio2_1  =  1.5707963267948966192292994253909555e+00L,   /*  0x1921fb54442d18469800000000000.0p-112 */
  55 pio2_1t =  2.0222662487959507323996846200947577e-21L,   /*  0x13198a2e03707344a4093822299f3.0p-181 */
  56 pio2_2  =  2.0222662487959507323994779168837751e-21L,   /*  0x13198a2e03707344a400000000000.0p-181 */
  57 pio2_2t =  2.0670321098263988236496903051604844e-43L,   /*  0x127044533e63a0105df531d89cd91.0p-254 */
  58 pio2_3  =  2.0670321098263988236499468110329591e-43L,   /*  0x127044533e63a0105e00000000000.0p-254 */
  59 pio2_3t = -2.5650587247459238361625433492959285e-65L;   /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
  60
  61 static inline __always_inline int
  62 __ieee754_rem_pio2l(long double x, long double *y)
  63 {
  64         union IEEEl2bits u,u1;
  65         long double z,w,t,r,fn;
  66         double tx[5],ty[3];
  67         int64_t n;
  68         int e0,ex,i,j,nx;
  69         int16_t expsign;
  70
  71         u.e = x;
  72         expsign = u.xbits.expsign;
  73         ex = expsign & 0x7fff;
  74         if (ex < BIAS + 45 || ex == BIAS + 45 &&
  75             u.bits.manh < 0x921fb54442d1LL) {
  76             /* |x| ~< 2^45*(pi/2), medium size */
  77             /* TODO: use only double precision for fn, as in expl(). */
  78             fn = rnintl(x * invpio2);
  79             n  = i64rint(fn);
  80             r  = x-fn*pio2_1;
  81             w  = fn*pio2_1t;    /* 1st round good to 180 bit */
  82             {
  83                 union IEEEl2bits u2;
  84                 int ex1;
  85                 j  = ex;
  86                 y[0] = r-w;
  87                 u2.e = y[0];
  88                 ex1 = u2.xbits.expsign & 0x7fff;
  89                 i = j-ex1;
  90                 if(i>51) {  /* 2nd iteration needed, good to 248 */
  91                     t  = r;
  92                     w  = fn*pio2_2;
  93                     r  = t-w;
  94                     w  = fn*pio2_2t-((t-r)-w);
  95                     y[0] = r-w;
  96                     u2.e = y[0];
  97                     ex1 = u2.xbits.expsign & 0x7fff;
  98                     i = j-ex1;
  99                     if(i>119) { /* 3rd iteration need, 316 bits acc */
 100                         t  = r; /* will cover all possible cases */
 101                         w  = fn*pio2_3;
 102                         r  = t-w;
 103                         w  = fn*pio2_3t-((t-r)-w);
 104                         y[0] = r-w;
 105                     }
 106                 }
 107             }
 108             y[1] = (r-y[0])-w;
 109             return n;
 110         }
 111     /*
 112      * all other (large) arguments
 113      */
 114         if(ex==0x7fff) {                /* x is inf or NaN */
 115             y[0]=y[1]=x-x; return 0;
 116         }
 117     /* set z = scalbn(|x|,ilogb(x)-23) */
 118         u1.e = x;
 119         e0 = ex - BIAS - 23;            /* e0 = ilogb(|x|)-23; */
 120         u1.xbits.expsign = ex - e0;
 121         z = u1.e;
 122         for(i=0;i<4;i++) {
 123                 tx[i] = (double)((int32_t)(z));
 124                 z     = (z-tx[i])*two24;
 125         }
 126         tx[4] = z;
 127         nx = 5;
 128         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 129         n  =  __kernel_rem_pio2(tx,ty,e0,nx,3);
 130         t = (long double)ty[2] + ty[1];
 131         r = t + ty[0];
 132         w = ty[0] - (r - t);
 133         if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
 134         y[0] = r; y[1] = w; return n;
 135 }