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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that the above copyright notice and this paragraph are
7: * duplicated in all such forms and that any documentation,
8: * advertising materials, and other materials related to such
9: * distribution and use acknowledge that the software was developed
10: * by the University of California, Berkeley. The name of the
11: * University may not be used to endorse or promote products derived
12: * from this software without specific prior written permission.
13: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
14: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
15: * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
16: *
17: * All recipients should regard themselves as participants in an ongoing
18: * research project and hence should feel obligated to report their
19: * experiences (good or bad) with these elementary function codes, using
20: * the sendbug(8) program, to the authors.
21: */
22:
23: #ifndef lint
24: static char sccsid[] = "@(#)support.c 5.3 (Berkeley) 6/30/88";
25: #endif /* not lint */
26:
27: /*
28: * Some IEEE standard 754 recommended functions and remainder and sqrt for
29: * supporting the C elementary functions.
30: ******************************************************************************
31: * WARNING:
32: * These codes are developed (in double) to support the C elementary
33: * functions temporarily. They are not universal, and some of them are very
34: * slow (in particular, drem and sqrt is extremely inefficient). Each
35: * computer system should have its implementation of these functions using
36: * its own assembler.
37: ******************************************************************************
38: *
39: * IEEE 754 required operations:
40: * drem(x,p)
41: * returns x REM y = x - [x/y]*y , where [x/y] is the integer
42: * nearest x/y; in half way case, choose the even one.
43: * sqrt(x)
44: * returns the square root of x correctly rounded according to
45: * the rounding mod.
46: *
47: * IEEE 754 recommended functions:
48: * (a) copysign(x,y)
49: * returns x with the sign of y.
50: * (b) scalb(x,N)
51: * returns x * (2**N), for integer values N.
52: * (c) logb(x)
53: * returns the unbiased exponent of x, a signed integer in
54: * double precision, except that logb(0) is -INF, logb(INF)
55: * is +INF, and logb(NAN) is that NAN.
56: * (d) finite(x)
57: * returns the value TRUE if -INF < x < +INF and returns
58: * FALSE otherwise.
59: *
60: *
61: * CODED IN C BY K.C. NG, 11/25/84;
62: * REVISED BY K.C. NG on 1/22/85, 2/13/85, 3/24/85.
63: */
64:
65:
66: #if defined(vax)||defined(tahoe) /* VAX D format */
67: #include <errno.h>
68: static unsigned short msign=0x7fff , mexp =0x7f80 ;
69: static short prep1=57, gap=7, bias=129 ;
70: static double novf=1.7E38, nunf=3.0E-39, zero=0.0 ;
71: #else /* defined(vax)||defined(tahoe) */
72: static unsigned short msign=0x7fff, mexp =0x7ff0 ;
73: static short prep1=54, gap=4, bias=1023 ;
74: static double novf=1.7E308, nunf=3.0E-308,zero=0.0;
75: #endif /* defined(vax)||defined(tahoe) */
76:
77: double scalb(x,N)
78: double x; int N;
79: {
80: int k;
81: double scalb();
82:
83: #ifdef national
84: unsigned short *px=(unsigned short *) &x + 3;
85: #else /* national */
86: unsigned short *px=(unsigned short *) &x;
87: #endif /* national */
88:
89: if( x == zero ) return(x);
90:
91: #if defined(vax)||defined(tahoe)
92: if( (k= *px & mexp ) != ~msign ) {
93: if (N < -260)
94: return(nunf*nunf);
95: else if (N > 260) {
96: extern double infnan(),copysign();
97: return(copysign(infnan(ERANGE),x));
98: }
99: #else /* defined(vax)||defined(tahoe) */
100: if( (k= *px & mexp ) != mexp ) {
101: if( N<-2100) return(nunf*nunf); else if(N>2100) return(novf+novf);
102: if( k == 0 ) {
103: x *= scalb(1.0,(int)prep1); N -= prep1; return(scalb(x,N));}
104: #endif /* defined(vax)||defined(tahoe) */
105:
106: if((k = (k>>gap)+ N) > 0 )
107: if( k < (mexp>>gap) ) *px = (*px&~mexp) | (k<<gap);
108: else x=novf+novf; /* overflow */
109: else
110: if( k > -prep1 )
111: /* gradual underflow */
112: {*px=(*px&~mexp)|(short)(1<<gap); x *= scalb(1.0,k-1);}
113: else
114: return(nunf*nunf);
115: }
116: return(x);
117: }
118:
119:
120: double copysign(x,y)
121: double x,y;
122: {
123: #ifdef national
124: unsigned short *px=(unsigned short *) &x+3,
125: *py=(unsigned short *) &y+3;
126: #else /* national */
127: unsigned short *px=(unsigned short *) &x,
128: *py=(unsigned short *) &y;
129: #endif /* national */
130:
131: #if defined(vax)||defined(tahoe)
132: if ( (*px & mexp) == 0 ) return(x);
133: #endif /* defined(vax)||defined(tahoe) */
134:
135: *px = ( *px & msign ) | ( *py & ~msign );
136: return(x);
137: }
138:
139: double logb(x)
140: double x;
141: {
142:
143: #ifdef national
144: short *px=(short *) &x+3, k;
145: #else /* national */
146: short *px=(short *) &x, k;
147: #endif /* national */
148:
149: #if defined(vax)||defined(tahoe)
150: return (int)(((*px&mexp)>>gap)-bias);
151: #else /* defined(vax)||defined(tahoe) */
152: if( (k= *px & mexp ) != mexp )
153: if ( k != 0 )
154: return ( (k>>gap) - bias );
155: else if( x != zero)
156: return ( -1022.0 );
157: else
158: return(-(1.0/zero));
159: else if(x != x)
160: return(x);
161: else
162: {*px &= msign; return(x);}
163: #endif /* defined(vax)||defined(tahoe) */
164: }
165:
166: finite(x)
167: double x;
168: {
169: #if defined(vax)||defined(tahoe)
170: return(1);
171: #else /* defined(vax)||defined(tahoe) */
172: #ifdef national
173: return( (*((short *) &x+3 ) & mexp ) != mexp );
174: #else /* national */
175: return( (*((short *) &x ) & mexp ) != mexp );
176: #endif /* national */
177: #endif /* defined(vax)||defined(tahoe) */
178: }
179:
180: double drem(x,p)
181: double x,p;
182: {
183: short sign;
184: double hp,dp,tmp,drem(),scalb();
185: unsigned short k;
186: #ifdef national
187: unsigned short
188: *px=(unsigned short *) &x +3,
189: *pp=(unsigned short *) &p +3,
190: *pd=(unsigned short *) &dp +3,
191: *pt=(unsigned short *) &tmp+3;
192: #else /* national */
193: unsigned short
194: *px=(unsigned short *) &x ,
195: *pp=(unsigned short *) &p ,
196: *pd=(unsigned short *) &dp ,
197: *pt=(unsigned short *) &tmp;
198: #endif /* national */
199:
200: *pp &= msign ;
201:
202: #if defined(vax)||defined(tahoe)
203: if( ( *px & mexp ) == ~msign ) /* is x a reserved operand? */
204: #else /* defined(vax)||defined(tahoe) */
205: if( ( *px & mexp ) == mexp )
206: #endif /* defined(vax)||defined(tahoe) */
207: return (x-p)-(x-p); /* create nan if x is inf */
208: if (p == zero) {
209: #if defined(vax)||defined(tahoe)
210: extern double infnan();
211: return(infnan(EDOM));
212: #else /* defined(vax)||defined(tahoe) */
213: return zero/zero;
214: #endif /* defined(vax)||defined(tahoe) */
215: }
216:
217: #if defined(vax)||defined(tahoe)
218: if( ( *pp & mexp ) == ~msign ) /* is p a reserved operand? */
219: #else /* defined(vax)||defined(tahoe) */
220: if( ( *pp & mexp ) == mexp )
221: #endif /* defined(vax)||defined(tahoe) */
222: { if (p != p) return p; else return x;}
223:
224: else if ( ((*pp & mexp)>>gap) <= 1 )
225: /* subnormal p, or almost subnormal p */
226: { double b; b=scalb(1.0,(int)prep1);
227: p *= b; x = drem(x,p); x *= b; return(drem(x,p)/b);}
228: else if ( p >= novf/2)
229: { p /= 2 ; x /= 2; return(drem(x,p)*2);}
230: else
231: {
232: dp=p+p; hp=p/2;
233: sign= *px & ~msign ;
234: *px &= msign ;
235: while ( x > dp )
236: {
237: k=(*px & mexp) - (*pd & mexp) ;
238: tmp = dp ;
239: *pt += k ;
240:
241: #if defined(vax)||defined(tahoe)
242: if( x < tmp ) *pt -= 128 ;
243: #else /* defined(vax)||defined(tahoe) */
244: if( x < tmp ) *pt -= 16 ;
245: #endif /* defined(vax)||defined(tahoe) */
246:
247: x -= tmp ;
248: }
249: if ( x > hp )
250: { x -= p ; if ( x >= hp ) x -= p ; }
251:
252: #if defined(vax)||defined(tahoe)
253: if (x)
254: #endif /* defined(vax)||defined(tahoe) */
255: *px ^= sign;
256: return( x);
257:
258: }
259: }
260: double sqrt(x)
261: double x;
262: {
263: double q,s,b,r;
264: double logb(),scalb();
265: double t,zero=0.0;
266: int m,n,i,finite();
267: #if defined(vax)||defined(tahoe)
268: int k=54;
269: #else /* defined(vax)||defined(tahoe) */
270: int k=51;
271: #endif /* defined(vax)||defined(tahoe) */
272:
273: /* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
274: if(x!=x||x==zero) return(x);
275:
276: /* sqrt(negative) is invalid */
277: if(x<zero) {
278: #if defined(vax)||defined(tahoe)
279: extern double infnan();
280: return (infnan(EDOM)); /* NaN */
281: #else /* defined(vax)||defined(tahoe) */
282: return(zero/zero);
283: #endif /* defined(vax)||defined(tahoe) */
284: }
285:
286: /* sqrt(INF) is INF */
287: if(!finite(x)) return(x);
288:
289: /* scale x to [1,4) */
290: n=logb(x);
291: x=scalb(x,-n);
292: if((m=logb(x))!=0) x=scalb(x,-m); /* subnormal number */
293: m += n;
294: n = m/2;
295: if((n+n)!=m) {x *= 2; m -=1; n=m/2;}
296:
297: /* generate sqrt(x) bit by bit (accumulating in q) */
298: q=1.0; s=4.0; x -= 1.0; r=1;
299: for(i=1;i<=k;i++) {
300: t=s+1; x *= 4; r /= 2;
301: if(t<=x) {
302: s=t+t+2, x -= t; q += r;}
303: else
304: s *= 2;
305: }
306:
307: /* generate the last bit and determine the final rounding */
308: r/=2; x *= 4;
309: if(x==zero) goto end; 100+r; /* trigger inexact flag */
310: if(s<x) {
311: q+=r; x -=s; s += 2; s *= 2; x *= 4;
312: t = (x-s)-5;
313: b=1.0+3*r/4; if(b==1.0) goto end; /* b==1 : Round-to-zero */
314: b=1.0+r/4; if(b>1.0) t=1; /* b>1 : Round-to-(+INF) */
315: if(t>=0) q+=r; } /* else: Round-to-nearest */
316: else {
317: s *= 2; x *= 4;
318: t = (x-s)-1;
319: b=1.0+3*r/4; if(b==1.0) goto end;
320: b=1.0+r/4; if(b>1.0) t=1;
321: if(t>=0) q+=r; }
322:
323: end: return(scalb(q,n));
324: }
325:
326: #if 0
327: /* DREM(X,Y)
328: * RETURN X REM Y =X-N*Y, N=[X/Y] ROUNDED (ROUNDED TO EVEN IN THE HALF WAY CASE)
329: * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
330: * INTENDED FOR ASSEMBLY LANGUAGE
331: * CODED IN C BY K.C. NG, 3/23/85, 4/8/85.
332: *
333: * Warning: this code should not get compiled in unless ALL of
334: * the following machine-dependent routines are supplied.
335: *
336: * Required machine dependent functions (not on a VAX):
337: * swapINX(i): save inexact flag and reset it to "i"
338: * swapENI(e): save inexact enable and reset it to "e"
339: */
340:
341: double drem(x,y)
342: double x,y;
343: {
344:
345: #ifdef national /* order of words in floating point number */
346: static n0=3,n1=2,n2=1,n3=0;
347: #else /* VAX, SUN, ZILOG, TAHOE */
348: static n0=0,n1=1,n2=2,n3=3;
349: #endif
350:
351: static unsigned short mexp =0x7ff0, m25 =0x0190, m57 =0x0390;
352: static double zero=0.0;
353: double hy,y1,t,t1;
354: short k;
355: long n;
356: int i,e;
357: unsigned short xexp,yexp, *px =(unsigned short *) &x ,
358: nx,nf, *py =(unsigned short *) &y ,
359: sign, *pt =(unsigned short *) &t ,
360: *pt1 =(unsigned short *) &t1 ;
361:
362: xexp = px[n0] & mexp ; /* exponent of x */
363: yexp = py[n0] & mexp ; /* exponent of y */
364: sign = px[n0] &0x8000; /* sign of x */
365:
366: /* return NaN if x is NaN, or y is NaN, or x is INF, or y is zero */
367: if(x!=x) return(x); if(y!=y) return(y); /* x or y is NaN */
368: if( xexp == mexp ) return(zero/zero); /* x is INF */
369: if(y==zero) return(y/y);
370:
371: /* save the inexact flag and inexact enable in i and e respectively
372: * and reset them to zero
373: */
374: i=swapINX(0); e=swapENI(0);
375:
376: /* subnormal number */
377: nx=0;
378: if(yexp==0) {t=1.0,pt[n0]+=m57; y*=t; nx=m57;}
379:
380: /* if y is tiny (biased exponent <= 57), scale up y to y*2**57 */
381: if( yexp <= m57 ) {py[n0]+=m57; nx+=m57; yexp+=m57;}
382:
383: nf=nx;
384: py[n0] &= 0x7fff;
385: px[n0] &= 0x7fff;
386:
387: /* mask off the least significant 27 bits of y */
388: t=y; pt[n3]=0; pt[n2]&=0xf800; y1=t;
389:
390: /* LOOP: argument reduction on x whenever x > y */
391: loop:
392: while ( x > y )
393: {
394: t=y;
395: t1=y1;
396: xexp=px[n0]&mexp; /* exponent of x */
397: k=xexp-yexp-m25;
398: if(k>0) /* if x/y >= 2**26, scale up y so that x/y < 2**26 */
399: {pt[n0]+=k;pt1[n0]+=k;}
400: n=x/t; x=(x-n*t1)-n*(t-t1);
401: }
402: /* end while (x > y) */
403:
404: if(nx!=0) {t=1.0; pt[n0]+=nx; x*=t; nx=0; goto loop;}
405:
406: /* final adjustment */
407:
408: hy=y/2.0;
409: if(x>hy||((x==hy)&&n%2==1)) x-=y;
410: px[n0] ^= sign;
411: if(nf!=0) { t=1.0; pt[n0]-=nf; x*=t;}
412:
413: /* restore inexact flag and inexact enable */
414: swapINX(i); swapENI(e);
415:
416: return(x);
417: }
418: #endif
419:
420: #if 0
421: /* SQRT
422: * RETURN CORRECTLY ROUNDED (ACCORDING TO THE ROUNDING MODE) SQRT
423: * FOR IEEE DOUBLE PRECISION ONLY, INTENDED FOR ASSEMBLY LANGUAGE
424: * CODED IN C BY K.C. NG, 3/22/85.
425: *
426: * Warning: this code should not get compiled in unless ALL of
427: * the following machine-dependent routines are supplied.
428: *
429: * Required machine dependent functions:
430: * swapINX(i) ...return the status of INEXACT flag and reset it to "i"
431: * swapRM(r) ...return the current Rounding Mode and reset it to "r"
432: * swapENI(e) ...return the status of inexact enable and reset it to "e"
433: * addc(t) ...perform t=t+1 regarding t as a 64 bit unsigned integer
434: * subc(t) ...perform t=t-1 regarding t as a 64 bit unsigned integer
435: */
436:
437: static unsigned long table[] = {
438: 0, 1204, 3062, 5746, 9193, 13348, 18162, 23592, 29598, 36145, 43202, 50740,
439: 58733, 67158, 75992, 85215, 83599, 71378, 60428, 50647, 41945, 34246, 27478,
440: 21581, 16499, 12183, 8588, 5674, 3403, 1742, 661, 130, };
441:
442: double newsqrt(x)
443: double x;
444: {
445: double y,z,t,addc(),subc(),b54=134217728.*134217728.; /* b54=2**54 */
446: long mx,scalx,mexp=0x7ff00000;
447: int i,j,r,e,swapINX(),swapRM(),swapENI();
448: unsigned long *py=(unsigned long *) &y ,
449: *pt=(unsigned long *) &t ,
450: *px=(unsigned long *) &x ;
451: #ifdef national /* ordering of word in a floating point number */
452: int n0=1, n1=0;
453: #else
454: int n0=0, n1=1;
455: #endif
456: /* Rounding Mode: RN ...round-to-nearest
457: * RZ ...round-towards 0
458: * RP ...round-towards +INF
459: * RM ...round-towards -INF
460: */
461: int RN=0,RZ=1,RP=2,RM=3;/* machine dependent: work on a Zilog Z8070
462: * and a National 32081 & 16081
463: */
464:
465: /* exceptions */
466: if(x!=x||x==0.0) return(x); /* sqrt(NaN) is NaN, sqrt(+-0) = +-0 */
467: if(x<0) return((x-x)/(x-x)); /* sqrt(negative) is invalid */
468: if((mx=px[n0]&mexp)==mexp) return(x); /* sqrt(+INF) is +INF */
469:
470: /* save, reset, initialize */
471: e=swapENI(0); /* ...save and reset the inexact enable */
472: i=swapINX(0); /* ...save INEXACT flag */
473: r=swapRM(RN); /* ...save and reset the Rounding Mode to RN */
474: scalx=0;
475:
476: /* subnormal number, scale up x to x*2**54 */
477: if(mx==0) {x *= b54 ; scalx-=0x01b00000;}
478:
479: /* scale x to avoid intermediate over/underflow:
480: * if (x > 2**512) x=x/2**512; if (x < 2**-512) x=x*2**512 */
481: if(mx>0x5ff00000) {px[n0] -= 0x20000000; scalx+= 0x10000000;}
482: if(mx<0x1ff00000) {px[n0] += 0x20000000; scalx-= 0x10000000;}
483:
484: /* magic initial approximation to almost 8 sig. bits */
485: py[n0]=(px[n0]>>1)+0x1ff80000;
486: py[n0]=py[n0]-table[(py[n0]>>15)&31];
487:
488: /* Heron's rule once with correction to improve y to almost 18 sig. bits */
489: t=x/y; y=y+t; py[n0]=py[n0]-0x00100006; py[n1]=0;
490:
491: /* triple to almost 56 sig. bits; now y approx. sqrt(x) to within 1 ulp */
492: t=y*y; z=t; pt[n0]+=0x00100000; t+=z; z=(x-z)*y;
493: t=z/(t+x) ; pt[n0]+=0x00100000; y+=t;
494:
495: /* twiddle last bit to force y correctly rounded */
496: swapRM(RZ); /* ...set Rounding Mode to round-toward-zero */
497: swapINX(0); /* ...clear INEXACT flag */
498: swapENI(e); /* ...restore inexact enable status */
499: t=x/y; /* ...chopped quotient, possibly inexact */
500: j=swapINX(i); /* ...read and restore inexact flag */
501: if(j==0) { if(t==y) goto end; else t=subc(t); } /* ...t=t-ulp */
502: b54+0.1; /* ..trigger inexact flag, sqrt(x) is inexact */
503: if(r==RN) t=addc(t); /* ...t=t+ulp */
504: else if(r==RP) { t=addc(t);y=addc(y);}/* ...t=t+ulp;y=y+ulp; */
505: y=y+t; /* ...chopped sum */
506: py[n0]=py[n0]-0x00100000; /* ...correctly rounded sqrt(x) */
507: end: py[n0]=py[n0]+scalx; /* ...scale back y */
508: swapRM(r); /* ...restore Rounding Mode */
509: return(y);
510: }
511: #endif
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