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      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 "math_private.h"
     29 #include "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 	    /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
     78 	    fn = x*invpio2+0x1.8p112;
     79 	    fn = fn-0x1.8p112;
     80 #ifdef HAVE_EFFICIENT_I64RINT
     81 	    n  = i64rint(fn);
     82 #else
     83 	    n  = fn;
     84 #endif
     85 	    r  = x-fn*pio2_1;
     86 	    w  = fn*pio2_1t;	/* 1st round good to 180 bit */
     87 	    {
     88 		union IEEEl2bits u2;
     89 	        int ex1;
     90 	        j  = ex;
     91 	        y[0] = r-w;
     92 		u2.e = y[0];
     93 		ex1 = u2.xbits.expsign & 0x7fff;
     94 	        i = j-ex1;
     95 	        if(i>51) {  /* 2nd iteration needed, good to 248 */
     96 		    t  = r;
     97 		    w  = fn*pio2_2;
     98 		    r  = t-w;
     99 		    w  = fn*pio2_2t-((t-r)-w);
    100 		    y[0] = r-w;
    101 		    u2.e = y[0];
    102 		    ex1 = u2.xbits.expsign & 0x7fff;
    103 		    i = j-ex1;
    104 		    if(i>119) {	/* 3rd iteration need, 316 bits acc */
    105 		    	t  = r;	/* will cover all possible cases */
    106 		    	w  = fn*pio2_3;
    107 		    	r  = t-w;
    108 		    	w  = fn*pio2_3t-((t-r)-w);
    109 		    	y[0] = r-w;
    110 		    }
    111 		}
    112 	    }
    113 	    y[1] = (r-y[0])-w;
    114 	    return n;
    115 	}
    116     /*
    117      * all other (large) arguments
    118      */
    119 	if(ex==0x7fff) {		/* x is inf or NaN */
    120 	    y[0]=y[1]=x-x; return 0;
    121 	}
    122     /* set z = scalbn(|x|,ilogb(x)-23) */
    123 	u1.e = x;
    124 	e0 = ex - BIAS - 23;		/* e0 = ilogb(|x|)-23; */
    125 	u1.xbits.expsign = ex - e0;
    126 	z = u1.e;
    127 	for(i=0;i<4;i++) {
    128 		tx[i] = (double)((int32_t)(z));
    129 		z     = (z-tx[i])*two24;
    130 	}
    131 	tx[4] = z;
    132 	nx = 5;
    133 	while(tx[nx-1]==zero) nx--;	/* skip zero term */
    134 	n  =  __kernel_rem_pio2(tx,ty,e0,nx,3);
    135 	t = (long double)ty[2] + ty[1];
    136 	r = t + ty[0];
    137 	w = ty[0] - (r - t);
    138 	if(expsign<0) {y[0] = -r; y[1] = -w; return -n;}
    139 	y[0] = r; y[1] = w; return n;
    140 }
    141