diff options
author | Franklin Wei <git@fwei.tk> | 2017-01-21 15:18:31 -0500 |
---|---|---|
committer | Franklin Wei <git@fwei.tk> | 2017-12-23 21:01:26 -0500 |
commit | a855d6202536ff28e5aae4f22a0f31d8f5b325d0 (patch) | |
tree | 8c75f224dd64ed360505afa8843d016b0d75000b /apps/plugins/sdl/src/video/e_sqrt.h | |
parent | 01c6dcf6c7b9bb1ad2fa0450f99bacc5f3d3e04b (diff) | |
download | rockbox-a855d6202536ff28e5aae4f22a0f31d8f5b325d0.tar.gz rockbox-a855d6202536ff28e5aae4f22a0f31d8f5b325d0.zip |
Port of Duke Nukem 3D
This ports Fabien Sanglard's Chocolate Duke to run on a version of SDL
for Rockbox.
Change-Id: I8f2c4c78af19de10c1633ed7bb7a997b43256dd9
Diffstat (limited to 'apps/plugins/sdl/src/video/e_sqrt.h')
-rw-r--r-- | apps/plugins/sdl/src/video/e_sqrt.h | 493 |
1 files changed, 493 insertions, 0 deletions
diff --git a/apps/plugins/sdl/src/video/e_sqrt.h b/apps/plugins/sdl/src/video/e_sqrt.h new file mode 100644 index 0000000000..657380ea43 --- /dev/null +++ b/apps/plugins/sdl/src/video/e_sqrt.h | |||
@@ -0,0 +1,493 @@ | |||
1 | /* @(#)e_sqrt.c 5.1 93/09/24 */ | ||
2 | /* | ||
3 | * ==================================================== | ||
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | ||
5 | * | ||
6 | * Developed at SunPro, a Sun Microsystems, Inc. business. | ||
7 | * Permission to use, copy, modify, and distribute this | ||
8 | * software is freely granted, provided that this notice | ||
9 | * is preserved. | ||
10 | * ==================================================== | ||
11 | */ | ||
12 | |||
13 | #if defined(LIBM_SCCS) && !defined(lint) | ||
14 | static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $"; | ||
15 | #endif | ||
16 | |||
17 | /* __ieee754_sqrt(x) | ||
18 | * Return correctly rounded sqrt. | ||
19 | * ------------------------------------------ | ||
20 | * | Use the hardware sqrt if you have one | | ||
21 | * ------------------------------------------ | ||
22 | * Method: | ||
23 | * Bit by bit method using integer arithmetic. (Slow, but portable) | ||
24 | * 1. Normalization | ||
25 | * Scale x to y in [1,4) with even powers of 2: | ||
26 | * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then | ||
27 | * sqrt(x) = 2^k * sqrt(y) | ||
28 | * 2. Bit by bit computation | ||
29 | * Let q = sqrt(y) truncated to i bit after binary point (q = 1), | ||
30 | * i 0 | ||
31 | * i+1 2 | ||
32 | * s = 2*q , and y = 2 * ( y - q ). (1) | ||
33 | * i i i i | ||
34 | * | ||
35 | * To compute q from q , one checks whether | ||
36 | * i+1 i | ||
37 | * | ||
38 | * -(i+1) 2 | ||
39 | * (q + 2 ) <= y. (2) | ||
40 | * i | ||
41 | * -(i+1) | ||
42 | * If (2) is false, then q = q ; otherwise q = q + 2 . | ||
43 | * i+1 i i+1 i | ||
44 | * | ||
45 | * With some algebric manipulation, it is not difficult to see | ||
46 | * that (2) is equivalent to | ||
47 | * -(i+1) | ||
48 | * s + 2 <= y (3) | ||
49 | * i i | ||
50 | * | ||
51 | * The advantage of (3) is that s and y can be computed by | ||
52 | * i i | ||
53 | * the following recurrence formula: | ||
54 | * if (3) is false | ||
55 | * | ||
56 | * s = s , y = y ; (4) | ||
57 | * i+1 i i+1 i | ||
58 | * | ||
59 | * otherwise, | ||
60 | * -i -(i+1) | ||
61 | * s = s + 2 , y = y - s - 2 (5) | ||
62 | * i+1 i i+1 i i | ||
63 | * | ||
64 | * One may easily use induction to prove (4) and (5). | ||
65 | * Note. Since the left hand side of (3) contain only i+2 bits, | ||
66 | * it does not necessary to do a full (53-bit) comparison | ||
67 | * in (3). | ||
68 | * 3. Final rounding | ||
69 | * After generating the 53 bits result, we compute one more bit. | ||
70 | * Together with the remainder, we can decide whether the | ||
71 | * result is exact, bigger than 1/2ulp, or less than 1/2ulp | ||
72 | * (it will never equal to 1/2ulp). | ||
73 | * The rounding mode can be detected by checking whether | ||
74 | * huge + tiny is equal to huge, and whether huge - tiny is | ||
75 | * equal to huge for some floating point number "huge" and "tiny". | ||
76 | * | ||
77 | * Special cases: | ||
78 | * sqrt(+-0) = +-0 ... exact | ||
79 | * sqrt(inf) = inf | ||
80 | * sqrt(-ve) = NaN ... with invalid signal | ||
81 | * sqrt(NaN) = NaN ... with invalid signal for signaling NaN | ||
82 | * | ||
83 | * Other methods : see the appended file at the end of the program below. | ||
84 | *--------------- | ||
85 | */ | ||
86 | |||
87 | /*#include "math.h"*/ | ||
88 | #include "math_private.h" | ||
89 | |||
90 | #ifdef __STDC__ | ||
91 | double SDL_NAME(copysign)(double x, double y) | ||
92 | #else | ||
93 | double SDL_NAME(copysign)(x,y) | ||
94 | double x,y; | ||
95 | #endif | ||
96 | { | ||
97 | u_int32_t hx,hy; | ||
98 | GET_HIGH_WORD(hx,x); | ||
99 | GET_HIGH_WORD(hy,y); | ||
100 | SET_HIGH_WORD(x,(hx&0x7fffffff)|(hy&0x80000000)); | ||
101 | return x; | ||
102 | } | ||
103 | |||
104 | #ifdef __STDC__ | ||
105 | double SDL_NAME(scalbn) (double x, int n) | ||
106 | #else | ||
107 | double SDL_NAME(scalbn) (x,n) | ||
108 | double x; int n; | ||
109 | #endif | ||
110 | { | ||
111 | int32_t k,hx,lx; | ||
112 | EXTRACT_WORDS(hx,lx,x); | ||
113 | k = (hx&0x7ff00000)>>20; /* extract exponent */ | ||
114 | if (k==0) { /* 0 or subnormal x */ | ||
115 | if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ | ||
116 | x *= two54; | ||
117 | GET_HIGH_WORD(hx,x); | ||
118 | k = ((hx&0x7ff00000)>>20) - 54; | ||
119 | if (n< -50000) return tiny*x; /*underflow*/ | ||
120 | } | ||
121 | if (k==0x7ff) return x+x; /* NaN or Inf */ | ||
122 | k = k+n; | ||
123 | if (k > 0x7fe) return huge*SDL_NAME(copysign)(huge,x); /* overflow */ | ||
124 | if (k > 0) /* normal result */ | ||
125 | {SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); return x;} | ||
126 | if (k <= -54) { | ||
127 | if (n > 50000) /* in case integer overflow in n+k */ | ||
128 | return huge*SDL_NAME(copysign)(huge,x); /*overflow*/ | ||
129 | else return tiny*SDL_NAME(copysign)(tiny,x); /*underflow*/ | ||
130 | } | ||
131 | k += 54; /* subnormal result */ | ||
132 | SET_HIGH_WORD(x,(hx&0x800fffff)|(k<<20)); | ||
133 | return x*twom54; | ||
134 | } | ||
135 | |||
136 | #ifdef __STDC__ | ||
137 | double __ieee754_sqrt(double x) | ||
138 | #else | ||
139 | double __ieee754_sqrt(x) | ||
140 | double x; | ||
141 | #endif | ||
142 | { | ||
143 | double z; | ||
144 | int32_t sign = (int)0x80000000; | ||
145 | int32_t ix0,s0,q,m,t,i; | ||
146 | u_int32_t r,t1,s1,ix1,q1; | ||
147 | |||
148 | EXTRACT_WORDS(ix0,ix1,x); | ||
149 | |||
150 | /* take care of Inf and NaN */ | ||
151 | if((ix0&0x7ff00000)==0x7ff00000) { | ||
152 | return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf | ||
153 | sqrt(-inf)=sNaN */ | ||
154 | } | ||
155 | /* take care of zero */ | ||
156 | if(ix0<=0) { | ||
157 | if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ | ||
158 | else if(ix0<0) | ||
159 | return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ | ||
160 | } | ||
161 | /* normalize x */ | ||
162 | m = (ix0>>20); | ||
163 | if(m==0) { /* subnormal x */ | ||
164 | while(ix0==0) { | ||
165 | m -= 21; | ||
166 | ix0 |= (ix1>>11); ix1 <<= 21; | ||
167 | } | ||
168 | for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; | ||
169 | m -= i-1; | ||
170 | ix0 |= (ix1>>(32-i)); | ||
171 | ix1 <<= i; | ||
172 | } | ||
173 | m -= 1023; /* unbias exponent */ | ||
174 | ix0 = (ix0&0x000fffff)|0x00100000; | ||
175 | if(m&1){ /* odd m, double x to make it even */ | ||
176 | ix0 += ix0 + ((ix1&sign)>>31); | ||
177 | ix1 += ix1; | ||
178 | } | ||
179 | m >>= 1; /* m = [m/2] */ | ||
180 | |||
181 | /* generate sqrt(x) bit by bit */ | ||
182 | ix0 += ix0 + ((ix1&sign)>>31); | ||
183 | ix1 += ix1; | ||
184 | q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ | ||
185 | r = 0x00200000; /* r = moving bit from right to left */ | ||
186 | |||
187 | while(r!=0) { | ||
188 | t = s0+r; | ||
189 | if(t<=ix0) { | ||
190 | s0 = t+r; | ||
191 | ix0 -= t; | ||
192 | q += r; | ||
193 | } | ||
194 | ix0 += ix0 + ((ix1&sign)>>31); | ||
195 | ix1 += ix1; | ||
196 | r>>=1; | ||
197 | } | ||
198 | |||
199 | r = sign; | ||
200 | while(r!=0) { | ||
201 | t1 = s1+r; | ||
202 | t = s0; | ||
203 | if((t<ix0)||((t==ix0)&&(t1<=ix1))) { | ||
204 | s1 = t1+r; | ||
205 | if(((int32_t)(t1&sign)==sign)&&(s1&sign)==0) s0 += 1; | ||
206 | ix0 -= t; | ||
207 | if (ix1 < t1) ix0 -= 1; | ||
208 | ix1 -= t1; | ||
209 | q1 += r; | ||
210 | } | ||
211 | ix0 += ix0 + ((ix1&sign)>>31); | ||
212 | ix1 += ix1; | ||
213 | r>>=1; | ||
214 | } | ||
215 | |||
216 | /* use floating add to find out rounding direction */ | ||
217 | if((ix0|ix1)!=0) { | ||
218 | z = one-tiny; /* trigger inexact flag */ | ||
219 | if (z>=one) { | ||
220 | z = one+tiny; | ||
221 | if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} | ||
222 | else if (z>one) { | ||
223 | if (q1==(u_int32_t)0xfffffffe) q+=1; | ||
224 | q1+=2; | ||
225 | } else | ||
226 | q1 += (q1&1); | ||
227 | } | ||
228 | } | ||
229 | ix0 = (q>>1)+0x3fe00000; | ||
230 | ix1 = q1>>1; | ||
231 | if ((q&1)==1) ix1 |= sign; | ||
232 | ix0 += (m <<20); | ||
233 | INSERT_WORDS(z,ix0,ix1); | ||
234 | return z; | ||
235 | } | ||
236 | |||
237 | /* | ||
238 | Other methods (use floating-point arithmetic) | ||
239 | ------------- | ||
240 | (This is a copy of a drafted paper by Prof W. Kahan | ||
241 | and K.C. Ng, written in May, 1986) | ||
242 | |||
243 | Two algorithms are given here to implement sqrt(x) | ||
244 | (IEEE double precision arithmetic) in software. | ||
245 | Both supply sqrt(x) correctly rounded. The first algorithm (in | ||
246 | Section A) uses newton iterations and involves four divisions. | ||
247 | The second one uses reciproot iterations to avoid division, but | ||
248 | requires more multiplications. Both algorithms need the ability | ||
249 | to chop results of arithmetic operations instead of round them, | ||
250 | and the INEXACT flag to indicate when an arithmetic operation | ||
251 | is executed exactly with no roundoff error, all part of the | ||
252 | standard (IEEE 754-1985). The ability to perform shift, add, | ||
253 | subtract and logical AND operations upon 32-bit words is needed | ||
254 | too, though not part of the standard. | ||
255 | |||
256 | A. sqrt(x) by Newton Iteration | ||
257 | |||
258 | (1) Initial approximation | ||
259 | |||
260 | Let x0 and x1 be the leading and the trailing 32-bit words of | ||
261 | a floating point number x (in IEEE double format) respectively | ||
262 | |||
263 | 1 11 52 ...widths | ||
264 | ------------------------------------------------------ | ||
265 | x: |s| e | f | | ||
266 | ------------------------------------------------------ | ||
267 | msb lsb msb lsb ...order | ||
268 | |||
269 | |||
270 | ------------------------ ------------------------ | ||
271 | x0: |s| e | f1 | x1: | f2 | | ||
272 | ------------------------ ------------------------ | ||
273 | |||
274 | By performing shifts and subtracts on x0 and x1 (both regarded | ||
275 | as integers), we obtain an 8-bit approximation of sqrt(x) as | ||
276 | follows. | ||
277 | |||
278 | k := (x0>>1) + 0x1ff80000; | ||
279 | y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits | ||
280 | Here k is a 32-bit integer and T1[] is an integer array containing | ||
281 | correction terms. Now magically the floating value of y (y's | ||
282 | leading 32-bit word is y0, the value of its trailing word is 0) | ||
283 | approximates sqrt(x) to almost 8-bit. | ||
284 | |||
285 | Value of T1: | ||
286 | static int T1[32]= { | ||
287 | 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, | ||
288 | 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, | ||
289 | 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, | ||
290 | 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; | ||
291 | |||
292 | (2) Iterative refinement | ||
293 | |||
294 | Apply Heron's rule three times to y, we have y approximates | ||
295 | sqrt(x) to within 1 ulp (Unit in the Last Place): | ||
296 | |||
297 | y := (y+x/y)/2 ... almost 17 sig. bits | ||
298 | y := (y+x/y)/2 ... almost 35 sig. bits | ||
299 | y := y-(y-x/y)/2 ... within 1 ulp | ||
300 | |||
301 | |||
302 | Remark 1. | ||
303 | Another way to improve y to within 1 ulp is: | ||
304 | |||
305 | y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) | ||
306 | y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) | ||
307 | |||
308 | 2 | ||
309 | (x-y )*y | ||
310 | y := y + 2* ---------- ...within 1 ulp | ||
311 | 2 | ||
312 | 3y + x | ||
313 | |||
314 | |||
315 | This formula has one division fewer than the one above; however, | ||
316 | it requires more multiplications and additions. Also x must be | ||
317 | scaled in advance to avoid spurious overflow in evaluating the | ||
318 | expression 3y*y+x. Hence it is not recommended uless division | ||
319 | is slow. If division is very slow, then one should use the | ||
320 | reciproot algorithm given in section B. | ||
321 | |||
322 | (3) Final adjustment | ||
323 | |||
324 | By twiddling y's last bit it is possible to force y to be | ||
325 | correctly rounded according to the prevailing rounding mode | ||
326 | as follows. Let r and i be copies of the rounding mode and | ||
327 | inexact flag before entering the square root program. Also we | ||
328 | use the expression y+-ulp for the next representable floating | ||
329 | numbers (up and down) of y. Note that y+-ulp = either fixed | ||
330 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped | ||
331 | mode. | ||
332 | |||
333 | I := FALSE; ... reset INEXACT flag I | ||
334 | R := RZ; ... set rounding mode to round-toward-zero | ||
335 | z := x/y; ... chopped quotient, possibly inexact | ||
336 | If(not I) then { ... if the quotient is exact | ||
337 | if(z=y) { | ||
338 | I := i; ... restore inexact flag | ||
339 | R := r; ... restore rounded mode | ||
340 | return sqrt(x):=y. | ||
341 | } else { | ||
342 | z := z - ulp; ... special rounding | ||
343 | } | ||
344 | } | ||
345 | i := TRUE; ... sqrt(x) is inexact | ||
346 | If (r=RN) then z=z+ulp ... rounded-to-nearest | ||
347 | If (r=RP) then { ... round-toward-+inf | ||
348 | y = y+ulp; z=z+ulp; | ||
349 | } | ||
350 | y := y+z; ... chopped sum | ||
351 | y0:=y0-0x00100000; ... y := y/2 is correctly rounded. | ||
352 | I := i; ... restore inexact flag | ||
353 | R := r; ... restore rounded mode | ||
354 | return sqrt(x):=y. | ||
355 | |||
356 | (4) Special cases | ||
357 | |||
358 | Square root of +inf, +-0, or NaN is itself; | ||
359 | Square root of a negative number is NaN with invalid signal. | ||
360 | |||
361 | |||
362 | B. sqrt(x) by Reciproot Iteration | ||
363 | |||
364 | (1) Initial approximation | ||
365 | |||
366 | Let x0 and x1 be the leading and the trailing 32-bit words of | ||
367 | a floating point number x (in IEEE double format) respectively | ||
368 | (see section A). By performing shifs and subtracts on x0 and y0, | ||
369 | we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. | ||
370 | |||
371 | k := 0x5fe80000 - (x0>>1); | ||
372 | y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits | ||
373 | |||
374 | Here k is a 32-bit integer and T2[] is an integer array | ||
375 | containing correction terms. Now magically the floating | ||
376 | value of y (y's leading 32-bit word is y0, the value of | ||
377 | its trailing word y1 is set to zero) approximates 1/sqrt(x) | ||
378 | to almost 7.8-bit. | ||
379 | |||
380 | Value of T2: | ||
381 | static int T2[64]= { | ||
382 | 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, | ||
383 | 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, | ||
384 | 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, | ||
385 | 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, | ||
386 | 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, | ||
387 | 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, | ||
388 | 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, | ||
389 | 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; | ||
390 | |||
391 | (2) Iterative refinement | ||
392 | |||
393 | Apply Reciproot iteration three times to y and multiply the | ||
394 | result by x to get an approximation z that matches sqrt(x) | ||
395 | to about 1 ulp. To be exact, we will have | ||
396 | -1ulp < sqrt(x)-z<1.0625ulp. | ||
397 | |||
398 | ... set rounding mode to Round-to-nearest | ||
399 | y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) | ||
400 | y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) | ||
401 | ... special arrangement for better accuracy | ||
402 | z := x*y ... 29 bits to sqrt(x), with z*y<1 | ||
403 | z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) | ||
404 | |||
405 | Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that | ||
406 | (a) the term z*y in the final iteration is always less than 1; | ||
407 | (b) the error in the final result is biased upward so that | ||
408 | -1 ulp < sqrt(x) - z < 1.0625 ulp | ||
409 | instead of |sqrt(x)-z|<1.03125ulp. | ||
410 | |||
411 | (3) Final adjustment | ||
412 | |||
413 | By twiddling y's last bit it is possible to force y to be | ||
414 | correctly rounded according to the prevailing rounding mode | ||
415 | as follows. Let r and i be copies of the rounding mode and | ||
416 | inexact flag before entering the square root program. Also we | ||
417 | use the expression y+-ulp for the next representable floating | ||
418 | numbers (up and down) of y. Note that y+-ulp = either fixed | ||
419 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped | ||
420 | mode. | ||
421 | |||
422 | R := RZ; ... set rounding mode to round-toward-zero | ||
423 | switch(r) { | ||
424 | case RN: ... round-to-nearest | ||
425 | if(x<= z*(z-ulp)...chopped) z = z - ulp; else | ||
426 | if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; | ||
427 | break; | ||
428 | case RZ:case RM: ... round-to-zero or round-to--inf | ||
429 | R:=RP; ... reset rounding mod to round-to-+inf | ||
430 | if(x<z*z ... rounded up) z = z - ulp; else | ||
431 | if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; | ||
432 | break; | ||
433 | case RP: ... round-to-+inf | ||
434 | if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else | ||
435 | if(x>z*z ...chopped) z = z+ulp; | ||
436 | break; | ||
437 | } | ||
438 | |||
439 | Remark 3. The above comparisons can be done in fixed point. For | ||
440 | example, to compare x and w=z*z chopped, it suffices to compare | ||
441 | x1 and w1 (the trailing parts of x and w), regarding them as | ||
442 | two's complement integers. | ||
443 | |||
444 | ...Is z an exact square root? | ||
445 | To determine whether z is an exact square root of x, let z1 be the | ||
446 | trailing part of z, and also let x0 and x1 be the leading and | ||
447 | trailing parts of x. | ||
448 | |||
449 | If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 | ||
450 | I := 1; ... Raise Inexact flag: z is not exact | ||
451 | else { | ||
452 | j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 | ||
453 | k := z1 >> 26; ... get z's 25-th and 26-th | ||
454 | fraction bits | ||
455 | I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); | ||
456 | } | ||
457 | R:= r ... restore rounded mode | ||
458 | return sqrt(x):=z. | ||
459 | |||
460 | If multiplication is cheaper then the foregoing red tape, the | ||
461 | Inexact flag can be evaluated by | ||
462 | |||
463 | I := i; | ||
464 | I := (z*z!=x) or I. | ||
465 | |||
466 | Note that z*z can overwrite I; this value must be sensed if it is | ||
467 | True. | ||
468 | |||
469 | Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be | ||
470 | zero. | ||
471 | |||
472 | -------------------- | ||
473 | z1: | f2 | | ||
474 | -------------------- | ||
475 | bit 31 bit 0 | ||
476 | |||
477 | Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd | ||
478 | or even of logb(x) have the following relations: | ||
479 | |||
480 | ------------------------------------------------- | ||
481 | bit 27,26 of z1 bit 1,0 of x1 logb(x) | ||
482 | ------------------------------------------------- | ||
483 | 00 00 odd and even | ||
484 | 01 01 even | ||
485 | 10 10 odd | ||
486 | 10 00 even | ||
487 | 11 01 even | ||
488 | ------------------------------------------------- | ||
489 | |||
490 | (4) Special cases (see (4) of Section A). | ||
491 | |||
492 | */ | ||
493 | |||