kc3-lang/SDL/src/libm/k_rem_pio2.c

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• Hash : 54bb9aac
  Author :
  Date : 2020-10-08T11:51:02
  

  k_rem_pio2.c: fix typo in second memset(): iq, not q. fixes bug #5309.

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• src/libm/k_rem_pio2.c

• /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunPro, a Sun Microsystems, Inc. business.
   * Permission to use, copy, modify, and distribute this
   * software is freely granted, provided that this notice
   * is preserved.
   * ====================================================
   */
  
  /*
   * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
   * double x[],y[]; int e0,nx,prec; int ipio2[];
   *
   * __kernel_rem_pio2 return the last three digits of N with
   *		y = x - N*pi/2
   * so that |y| < pi/2.
   *
   * The method is to compute the integer (mod 8) and fraction parts of
   * (2/pi)*x without doing the full multiplication. In general we
   * skip the part of the product that are known to be a huge integer (
   * more accurately, = 0 mod 8 ). Thus the number of operations are
   * independent of the exponent of the input.
   *
   * (2/pi) is represented by an array of 24-bit integers in ipio2[].
   *
   * Input parameters:
   * 	x[]	The input value (must be positive) is broken into nx
   *		pieces of 24-bit integers in double precision format.
   *		x[i] will be the i-th 24 bit of x. The scaled exponent
   *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
   *		match x's up to 24 bits.
   *
   *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
   *			e0 = ilogb(z)-23
   *			z  = scalbn(z,-e0)
   *		for i = 0,1,2
   *			x[i] = floor(z)
   *			z    = (z-x[i])*2**24
   *
   *
   *	y[]	ouput result in an array of double precision numbers.
   *		The dimension of y[] is:
   *			24-bit  precision	1
   *			53-bit  precision	2
   *			64-bit  precision	2
   *			113-bit precision	3
   *		The actual value is the sum of them. Thus for 113-bit
   *		precison, one may have to do something like:
   *
   *		long double t,w,r_head, r_tail;
   *		t = (long double)y[2] + (long double)y[1];
   *		w = (long double)y[0];
   *		r_head = t+w;
   *		r_tail = w - (r_head - t);
   *
   *	e0	The exponent of x[0]
   *
   *	nx	dimension of x[]
   *
   *  	prec	an integer indicating the precision:
   *			0	24  bits (single)
   *			1	53  bits (double)
   *			2	64  bits (extended)
   *			3	113 bits (quad)
   *
   *	ipio2[]
   *		integer array, contains the (24*i)-th to (24*i+23)-th
   *		bit of 2/pi after binary point. The corresponding
   *		floating value is
   *
   *			ipio2[i] * 2^(-24(i+1)).
   *
   * External function:
   *	double scalbn(), floor();
   *
   *
   * Here is the description of some local variables:
   *
   * 	jk	jk+1 is the initial number of terms of ipio2[] needed
   *		in the computation. The recommended value is 2,3,4,
   *		6 for single, double, extended,and quad.
   *
   * 	jz	local integer variable indicating the number of
   *		terms of ipio2[] used.
   *
   *	jx	nx - 1
   *
   *	jv	index for pointing to the suitable ipio2[] for the
   *		computation. In general, we want
   *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
   *		is an integer. Thus
   *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
   *		Hence jv = max(0,(e0-3)/24).
   *
   *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
   *
   * 	q[]	double array with integral value, representing the
   *		24-bits chunk of the product of x and 2/pi.
   *
   *	q0	the corresponding exponent of q[0]. Note that the
   *		exponent for q[i] would be q0-24*i.
   *
   *	PIo2[]	double precision array, obtained by cutting pi/2
   *		into 24 bits chunks.
   *
   *	f[]	ipio2[] in floating point
   *
   *	iq[]	integer array by breaking up q[] in 24-bits chunk.
   *
   *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
   *
   *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
   *		it also indicates the *sign* of the result.
   *
   */
  
  
  /*
   * Constants:
   * The hexadecimal values are the intended ones for the following
   * constants. The decimal values may be used, provided that the
   * compiler will convert from decimal to binary accurately enough
   * to produce the hexadecimal values shown.
   */
  
  #include "math_libm.h"
  #include "math_private.h"
  
  #include "SDL_assert.h"
  
  static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
  
  static const double PIo2[] = {
    1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
    7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
    5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
    3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
    1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
    1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
    2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
    2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
  };
  
  static const double
  zero   = 0.0,
  one    = 1.0,
  two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
  
  int32_t attribute_hidden __kernel_rem_pio2(const double *x, double *y, int e0, int nx, const unsigned int prec, const int32_t *ipio2)
  {
  	int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
  	double z,fw,f[20],fq[20],q[20];
  
  	if (nx < 1) {
  		return 0;
  	}
  
      /* initialize jk*/
  	SDL_assert(prec < SDL_arraysize(init_jk));
  	jk = init_jk[prec];
  	SDL_assert(jk > 0);
  	jp = jk;
  
      /* determine jx,jv,q0, note that 3>q0 */
  	jx =  nx-1;
  	jv = (e0-3)/24; if(jv<0) jv=0;
  	q0 =  e0-24*(jv+1);
  
      /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
  	j = jv-jx; m = jx+jk;
  	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
  	if ((m+1) < SDL_arraysize(f)) {
  	    SDL_memset(&f[m+1], 0, sizeof (f) - ((m+1) * sizeof (f[0])));
  	}
  
      /* compute q[0],q[1],...q[jk] */
  	for (i=0;i<=jk;i++) {
  	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
  	    q[i] = fw;
  	}
  
  	jz = jk;
  recompute:
      /* distill q[] into iq[] reversingly */
  	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
  	    fw    =  (double)((int32_t)(twon24* z));
  	    iq[i] =  (int32_t)(z-two24*fw);
  	    z     =  q[j-1]+fw;
  	}
  	if (jz < SDL_arraysize(iq)) {
  	    SDL_memset(&iq[jz], 0, sizeof (iq) - (jz * sizeof (iq[0])));
  	}
  
      /* compute n */
  	z  = scalbn(z,q0);		/* actual value of z */
  	z -= 8.0*floor(z*0.125);		/* trim off integer >= 8 */
  	n  = (int32_t) z;
  	z -= (double)n;
  	ih = 0;
  	if(q0>0) {	/* need iq[jz-1] to determine n */
  	    i  = (iq[jz-1]>>(24-q0)); n += i;
  	    iq[jz-1] -= i<<(24-q0);
  	    ih = iq[jz-1]>>(23-q0);
  	}
  	else if(q0==0) ih = iq[jz-1]>>23;
  	else if(z>=0.5) ih=2;
  
  	if(ih>0) {	/* q > 0.5 */
  	    n += 1; carry = 0;
  	    for(i=0;i<jz ;i++) {	/* compute 1-q */
  		j = iq[i];
  		if(carry==0) {
  		    if(j!=0) {
  			carry = 1; iq[i] = 0x1000000- j;
  		    }
  		} else  iq[i] = 0xffffff - j;
  	    }
  	    if(q0>0) {		/* rare case: chance is 1 in 12 */
  	        switch(q0) {
  	        case 1:
  	    	   iq[jz-1] &= 0x7fffff; break;
  	    	case 2:
  	    	   iq[jz-1] &= 0x3fffff; break;
  	        }
  	    }
  	    if(ih==2) {
  		z = one - z;
  		if(carry!=0) z -= scalbn(one,q0);
  	    }
  	}
  
      /* check if recomputation is needed */
  	if(z==zero) {
  	    j = 0;
  	    for (i=jz-1;i>=jk;i--) j |= iq[i];
  	    if(j==0) { /* need recomputation */
  		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
  
  		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
  		    f[jx+i] = (double) ipio2[jv+i];
  		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
  		    q[i] = fw;
  		}
  		jz += k;
  		goto recompute;
  	    }
  	}
  
      /* chop off zero terms */
  	if(z==0.0) {
  	    jz -= 1; q0 -= 24;
  		SDL_assert(jz >= 0);
  	    while(iq[jz]==0) { jz--; SDL_assert(jz >= 0); q0-=24;}
  	} else { /* break z into 24-bit if necessary */
  	    z = scalbn(z,-q0);
  	    if(z>=two24) {
  		fw = (double)((int32_t)(twon24*z));
  		iq[jz] = (int32_t)(z-two24*fw);
  		jz += 1; q0 += 24;
  		iq[jz] = (int32_t) fw;
  	    } else iq[jz] = (int32_t) z ;
  	}
  
      /* convert integer "bit" chunk to floating-point value */
  	fw = scalbn(one,q0);
  	for(i=jz;i>=0;i--) {
  	    q[i] = fw*(double)iq[i]; fw*=twon24;
  	}
  
      /* compute PIo2[0,...,jp]*q[jz,...,0] */
  	for(i=jz;i>=0;i--) {
  	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
  	    fq[jz-i] = fw;
  	}
  	if ((jz+1) < SDL_arraysize(f)) {
  	    SDL_memset(&fq[jz+1], 0, sizeof (fq) - ((jz+1) * sizeof (fq[0])));
  	}
  
      /* compress fq[] into y[] */
  	switch(prec) {
  	    case 0:
  		fw = 0.0;
  		for (i=jz;i>=0;i--) fw += fq[i];
  		y[0] = (ih==0)? fw: -fw;
  		break;
  	    case 1:
  	    case 2:
  		fw = 0.0;
  		for (i=jz;i>=0;i--) fw += fq[i];
  		y[0] = (ih==0)? fw: -fw;
  		fw = fq[0]-fw;
  		for (i=1;i<=jz;i++) fw += fq[i];
  		y[1] = (ih==0)? fw: -fw;
  		break;
  	    case 3:	/* painful */
  		for (i=jz;i>0;i--) {
  		    fw      = fq[i-1]+fq[i];
  		    fq[i]  += fq[i-1]-fw;
  		    fq[i-1] = fw;
  		}
  		for (i=jz;i>1;i--) {
  		    fw      = fq[i-1]+fq[i];
  		    fq[i]  += fq[i-1]-fw;
  		    fq[i-1] = fw;
  		}
  		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
  		if(ih==0) {
  		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
  		} else {
  		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
  		}
  	}
  	return n&7;
  }
  

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