github.com/afumu/libc@v0.0.6/musl/src/math/__rem_pio2l.c (about)

     1  /* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */
     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  #include "libm.h"
    16  #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
    17  /* ld80 and ld128 version of __rem_pio2(x,y)
    18   *
    19   * return the remainder of x rem pi/2 in y[0]+y[1]
    20   * use __rem_pio2_large() for large x
    21   */
    22  
    23  static const long double toint = 1.5/LDBL_EPSILON;
    24  
    25  #if LDBL_MANT_DIG == 64
    26  /* u ~< 0x1p25*pi/2 */
    27  #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
    28  #define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
    29  #define ROUND1 22
    30  #define ROUND2 61
    31  #define NX 3
    32  #define NY 2
    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  static const double
    43  pio2_1 =  1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
    44  pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
    45  pio2_3 =  6.36831716351370313614e-25; /*  0x18a2e037074000.0p-133 */
    46  static const long double
    47  pio4    =  0x1.921fb54442d1846ap-1L,
    48  invpio2 =  6.36619772367581343076e-01L, /*  0xa2f9836e4e44152a.0p-64 */
    49  pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
    50  pio2_2t =  6.36831716351095013979e-25L, /*  0xc51701b839a25205.0p-144 */
    51  pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
    52  #elif LDBL_MANT_DIG == 113
    53  /* u ~< 0x1p45*pi/2 */
    54  #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
    55  #define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
    56  #define ROUND1 51
    57  #define ROUND2 119
    58  #define NX 5
    59  #define NY 3
    60  static const long double
    61  pio4    =  0x1.921fb54442d18469898cc51701b8p-1L,
    62  invpio2 =  6.3661977236758134307553505349005747e-01L,	/*  0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
    63  pio2_1  =  1.5707963267948966192292994253909555e+00L,	/*  0x1921fb54442d18469800000000000.0p-112 */
    64  pio2_1t =  2.0222662487959507323996846200947577e-21L,	/*  0x13198a2e03707344a4093822299f3.0p-181 */
    65  pio2_2  =  2.0222662487959507323994779168837751e-21L,	/*  0x13198a2e03707344a400000000000.0p-181 */
    66  pio2_2t =  2.0670321098263988236496903051604844e-43L,	/*  0x127044533e63a0105df531d89cd91.0p-254 */
    67  pio2_3  =  2.0670321098263988236499468110329591e-43L,	/*  0x127044533e63a0105e00000000000.0p-254 */
    68  pio2_3t = -2.5650587247459238361625433492959285e-65L;	/* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
    69  #endif
    70  
    71  int __rem_pio2l(long double x, long double *y)
    72  {
    73  	union ldshape u,uz;
    74  	long double z,w,t,r,fn;
    75  	double tx[NX],ty[NY];
    76  	int ex,ey,n,i;
    77  
    78  	u.f = x;
    79  	ex = u.i.se & 0x7fff;
    80  	if (SMALL(u)) {
    81  		/* rint(x/(pi/2)) */
    82  		fn = x*invpio2 + toint - toint;
    83  		n = QUOBITS(fn);
    84  		r = x-fn*pio2_1;
    85  		w = fn*pio2_1t;  /* 1st round good to 102/180 bits (ld80/ld128) */
    86  		/* Matters with directed rounding. */
    87  		if (predict_false(r - w < -pio4)) {
    88  			n--;
    89  			fn--;
    90  			r = x - fn*pio2_1;
    91  			w = fn*pio2_1t;
    92  		} else if (predict_false(r - w > pio4)) {
    93  			n++;
    94  			fn++;
    95  			r = x - fn*pio2_1;
    96  			w = fn*pio2_1t;
    97  		}
    98  		y[0] = r-w;
    99  		u.f = y[0];
   100  		ey = u.i.se & 0x7fff;
   101  		if (ex - ey > ROUND1) {  /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
   102  			t = r;
   103  			w = fn*pio2_2;
   104  			r = t-w;
   105  			w = fn*pio2_2t-((t-r)-w);
   106  			y[0] = r-w;
   107  			u.f = y[0];
   108  			ey = u.i.se & 0x7fff;
   109  			if (ex - ey > ROUND2) {  /* 3rd iteration, good to 180/316 bits */
   110  				t = r; /* will cover all possible cases (not verified for ld128) */
   111  				w = fn*pio2_3;
   112  				r = t-w;
   113  				w = fn*pio2_3t-((t-r)-w);
   114  				y[0] = r-w;
   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;
   125  		return 0;
   126  	}
   127  	/* set z = scalbn(|x|,-ilogb(x)+23) */
   128  	uz.f = x;
   129  	uz.i.se = 0x3fff + 23;
   130  	z = uz.f;
   131  	for (i=0; i < NX - 1; i++) {
   132  		tx[i] = (double)(int32_t)z;
   133  		z     = (z-tx[i])*0x1p24;
   134  	}
   135  	tx[i] = z;
   136  	while (tx[i] == 0)
   137  		i--;
   138  	n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
   139  	w = ty[1];
   140  	if (NY == 3)
   141  		w += ty[2];
   142  	r = ty[0] + w;
   143  	/* TODO: for ld128 this does not follow the recommendation of the
   144  	comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
   145  	w -= r - ty[0];
   146  	if (u.i.se >> 15) {
   147  		y[0] = -r;
   148  		y[1] = -w;
   149  		return -n;
   150  	}
   151  	y[0] = r;
   152  	y[1] = w;
   153  	return n;
   154  }
   155  #endif