newlib/libm/ld80/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 /* ld80 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  * invpio2:  64 bits of 2/pi
  35  * pio2_1:   first  39 bits of pi/2
  36  * pio2_1t:  pi/2 - pio2_1
  37  * pio2_2:   second 39 bits of pi/2
  38  * pio2_2t:  pi/2 - (pio2_1+pio2_2)
  39  * pio2_3:   third  39 bits of pi/2
  40  * pio2_3t:  pi/2 - (pio2_1+pio2_2+pio2_3)
  41  */
  42
  43 static const double
  44 zero =  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  45 two24 =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  46 pio2_1  =  1.57079632679597125389e+00,  /* 0x3FF921FB, 0x54444000 */
  47 pio2_2  = -1.07463465549783099519e-12,  /* -0x12e7b967674000.0p-92 */
  48 pio2_3  =  6.36831716351370313614e-25;  /*  0x18a2e037074000.0p-133 */
  49
  50 #if defined(__amd64__) || defined(__i386__)
  51 /* Long double constants are slow on these arches, and broken on i386. */
  52 static const volatile double
  53 invpio2hi =  6.3661977236758138e-01,    /*  0x145f306dc9c883.0p-53 */
  54 invpio2lo = -3.9356538861223811e-17,    /* -0x16b00000000000.0p-107 */
  55 pio2_1thi = -1.0746346554971943e-12,    /* -0x12e7b9676733af.0p-92 */
  56 pio2_1tlo =  8.8451028997905949e-29,    /*  0x1c080000000000.0p-146 */
  57 pio2_2thi =  6.3683171635109499e-25,    /*  0x18a2e03707344a.0p-133 */
  58 pio2_2tlo =  2.3183081793789774e-41,    /*  0x10280000000000.0p-187 */
  59 pio2_3thi = -2.7529965190440717e-37,    /* -0x176b7ed8fbbacc.0p-174 */
  60 pio2_3tlo = -4.2006647512740502e-54;    /* -0x19c00000000000.0p-230 */
  61 #define invpio2 ((long double)invpio2hi + invpio2lo)
  62 #define pio2_1t ((long double)pio2_1thi + pio2_1tlo)
  63 #define pio2_2t ((long double)pio2_2thi + pio2_2tlo)
  64 #define pio2_3t ((long double)pio2_3thi + pio2_3tlo)
  65 #else
  66 static const long double
  67 invpio2 =  6.36619772367581343076e-01L, /*  0xa2f9836e4e44152a.0p-64 */
  68 pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
  69 pio2_2t =  6.36831716351095013979e-25L, /*  0xc51701b839a25205.0p-144 */
  70 pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
  71 #endif
  72
  73 static inline __always_inline int
  74 __ieee754_rem_pio2l(long double x, long double *y)
  75 {
  76         union IEEEl2bits u,u1;
  77         long double z,w,t,r,fn;
  78         double tx[3],ty[2];
  79         int e0,ex,i,j,nx,n;
  80         int16_t expsign;
  81
  82         u.e = x;
  83         expsign = u.xbits.expsign;
  84         ex = expsign & 0x7fff;
  85         if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) {
  86             /* |x| ~< 2^25*(pi/2), medium size */
  87             fn = rnintl(x*invpio2);
  88             n  = irint(fn);
  89             r  = x-fn*pio2_1;
  90             w  = fn*pio2_1t;    /* 1st round good to 102 bit */
  91             {
  92                 union IEEEl2bits u2;
  93                 int ex1;
  94                 j  = ex;
  95                 y[0] = r-w;
  96                 u2.e = y[0];
  97                 ex1 = u2.xbits.expsign & 0x7fff;
  98                 i = j-ex1;
  99                 if(i>22) {  /* 2nd iteration needed, good to 141 */
 100                     t  = r;
 101                     w  = fn*pio2_2;
 102                     r  = t-w;
 103                     w  = fn*pio2_2t-((t-r)-w);
 104                     y[0] = r-w;
 105                     u2.e = y[0];
 106                     ex1 = u2.xbits.expsign & 0x7fff;
 107                     i = j-ex1;
 108                     if(i>61) {  /* 3rd iteration need, 180 bits acc */
 109                         t  = r; /* will cover all possible cases */
 110                         w  = fn*pio2_3;
 111                         r  = t-w;
 112                         w  = fn*pio2_3t-((t-r)-w);
 113                         y[0] = r-w;
 114                     }
 115                 }
 116             }
 117             y[1] = (r-y[0])-w;
 118             return n;
 119         }
 120     /*
 121      * all other (large) arguments
 122      */
 123         if(ex==0x7fff) {                /* x is inf or NaN */
 124             y[0]=y[1]=x-x; return 0;
 125         }
 126     /* set z = scalbn(|x|,ilogb(x)-23) */
 127         u1.e = x;
 128         e0 = ex - BIAS - 23;            /* e0 = ilogb(|x|)-23; */
 129         u1.xbits.expsign = ex - e0;
 130         z = u1.e;
 131         for(i=0;i<2;i++) {
 132                 tx[i] = (double)((int32_t)(z));
 133                 z     = (z-tx[i])*two24;
 134         }
 135         tx[2] = z;
 136         nx = 3;
 137         while(tx[nx-1]==zero) nx--;     /* skip zero term */
 138         n  =  __kernel_rem_pio2(tx,ty,e0,nx,2);
 139         r = (long double)ty[0] + ty[1];
 140         w = ty[1] - (r - ty[0]);
 141         if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
 142         y[0] = r; y[1] = w; return n;
 143 }