1 | #include "SDL_internal.h" |
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 | /* __ieee754_sqrt(x) |
14 | * Return correctly rounded sqrt. |
15 | * ------------------------------------------ |
16 | * | Use the hardware sqrt if you have one | |
17 | * ------------------------------------------ |
18 | * Method: |
19 | * Bit by bit method using integer arithmetic. (Slow, but portable) |
20 | * 1. Normalization |
21 | * Scale x to y in [1,4) with even powers of 2: |
22 | * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then |
23 | * sqrt(x) = 2^k * sqrt(y) |
24 | * 2. Bit by bit computation |
25 | * Let q = sqrt(y) truncated to i bit after binary point (q = 1), |
26 | * i 0 |
27 | * i+1 2 |
28 | * s = 2*q , and y = 2 * ( y - q ). (1) |
29 | * i i i i |
30 | * |
31 | * To compute q from q , one checks whether |
32 | * i+1 i |
33 | * |
34 | * -(i+1) 2 |
35 | * (q + 2 ) <= y. (2) |
36 | * i |
37 | * -(i+1) |
38 | * If (2) is false, then q = q ; otherwise q = q + 2 . |
39 | * i+1 i i+1 i |
40 | * |
41 | * With some algebric manipulation, it is not difficult to see |
42 | * that (2) is equivalent to |
43 | * -(i+1) |
44 | * s + 2 <= y (3) |
45 | * i i |
46 | * |
47 | * The advantage of (3) is that s and y can be computed by |
48 | * i i |
49 | * the following recurrence formula: |
50 | * if (3) is false |
51 | * |
52 | * s = s , y = y ; (4) |
53 | * i+1 i i+1 i |
54 | * |
55 | * otherwise, |
56 | * -i -(i+1) |
57 | * s = s + 2 , y = y - s - 2 (5) |
58 | * i+1 i i+1 i i |
59 | * |
60 | * One may easily use induction to prove (4) and (5). |
61 | * Note. Since the left hand side of (3) contain only i+2 bits, |
62 | * it does not necessary to do a full (53-bit) comparison |
63 | * in (3). |
64 | * 3. Final rounding |
65 | * After generating the 53 bits result, we compute one more bit. |
66 | * Together with the remainder, we can decide whether the |
67 | * result is exact, bigger than 1/2ulp, or less than 1/2ulp |
68 | * (it will never equal to 1/2ulp). |
69 | * The rounding mode can be detected by checking whether |
70 | * huge + tiny is equal to huge, and whether huge - tiny is |
71 | * equal to huge for some floating point number "huge" and "tiny". |
72 | * |
73 | * Special cases: |
74 | * sqrt(+-0) = +-0 ... exact |
75 | * sqrt(inf) = inf |
76 | * sqrt(-ve) = NaN ... with invalid signal |
77 | * sqrt(NaN) = NaN ... with invalid signal for signaling NaN |
78 | * |
79 | * Other methods : see the appended file at the end of the program below. |
80 | *--------------- |
81 | */ |
82 | |
83 | #include "math_libm.h" |
84 | #include "math_private.h" |
85 | |
86 | static const double one = 1.0, tiny = 1.0e-300; |
87 | |
88 | double attribute_hidden __ieee754_sqrt(double x) |
89 | { |
90 | double z; |
91 | int32_t sign = (int)0x80000000; |
92 | int32_t ix0,s0,q,m,t,i; |
93 | u_int32_t r,t1,s1,ix1,q1; |
94 | |
95 | EXTRACT_WORDS(ix0,ix1,x); |
96 | |
97 | /* take care of Inf and NaN */ |
98 | if((ix0&0x7ff00000)==0x7ff00000) { |
99 | return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf |
100 | sqrt(-inf)=sNaN */ |
101 | } |
102 | /* take care of zero */ |
103 | if(ix0<=0) { |
104 | if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */ |
105 | else if(ix0<0) |
106 | return (x-x)/(x-x); /* sqrt(-ve) = sNaN */ |
107 | } |
108 | /* normalize x */ |
109 | m = (ix0>>20); |
110 | if(m==0) { /* subnormal x */ |
111 | while(ix0==0) { |
112 | m -= 21; |
113 | ix0 |= (ix1>>11); ix1 <<= 21; |
114 | } |
115 | for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1; |
116 | m -= i-1; |
117 | ix0 |= (ix1>>(32-i)); |
118 | ix1 <<= i; |
119 | } |
120 | m -= 1023; /* unbias exponent */ |
121 | ix0 = (ix0&0x000fffff)|0x00100000; |
122 | if(m&1){ /* odd m, double x to make it even */ |
123 | ix0 += ix0 + ((ix1&sign)>>31); |
124 | ix1 += ix1; |
125 | } |
126 | m >>= 1; /* m = [m/2] */ |
127 | |
128 | /* generate sqrt(x) bit by bit */ |
129 | ix0 += ix0 + ((ix1&sign)>>31); |
130 | ix1 += ix1; |
131 | q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */ |
132 | r = 0x00200000; /* r = moving bit from right to left */ |
133 | |
134 | while(r!=0) { |
135 | t = s0+r; |
136 | if(t<=ix0) { |
137 | s0 = t+r; |
138 | ix0 -= t; |
139 | q += r; |
140 | } |
141 | ix0 += ix0 + ((ix1&sign)>>31); |
142 | ix1 += ix1; |
143 | r>>=1; |
144 | } |
145 | |
146 | r = sign; |
147 | while(r!=0) { |
148 | t1 = s1+r; |
149 | t = s0; |
150 | if((t<ix0)||((t==ix0)&&(t1<=ix1))) { |
151 | s1 = t1+r; |
152 | if(((t1&sign)==(u_int32_t)sign)&&(s1&sign)==0) s0 += 1; |
153 | ix0 -= t; |
154 | if (ix1 < t1) ix0 -= 1; |
155 | ix1 -= t1; |
156 | q1 += r; |
157 | } |
158 | ix0 += ix0 + ((ix1&sign)>>31); |
159 | ix1 += ix1; |
160 | r>>=1; |
161 | } |
162 | |
163 | /* use floating add to find out rounding direction */ |
164 | if((ix0|ix1)!=0) { |
165 | z = one-tiny; /* trigger inexact flag */ |
166 | if (z>=one) { |
167 | z = one+tiny; |
168 | if (q1==(u_int32_t)0xffffffff) { q1=0; q += 1;} |
169 | else if (z>one) { |
170 | if (q1==(u_int32_t)0xfffffffe) q+=1; |
171 | q1+=2; |
172 | } else |
173 | q1 += (q1&1); |
174 | } |
175 | } |
176 | ix0 = (q>>1)+0x3fe00000; |
177 | ix1 = q1>>1; |
178 | if ((q&1)==1) ix1 |= sign; |
179 | ix0 += (m <<20); |
180 | INSERT_WORDS(z,ix0,ix1); |
181 | return z; |
182 | } |
183 | |
184 | /* |
185 | * wrapper sqrt(x) |
186 | */ |
187 | #ifndef _IEEE_LIBM |
188 | double sqrt(double x) |
189 | { |
190 | double z = __ieee754_sqrt(x); |
191 | if (_LIB_VERSION == _IEEE_ || isnan(x)) |
192 | return z; |
193 | if (x < 0.0) |
194 | return __kernel_standard(x, x, 26); /* sqrt(negative) */ |
195 | return z; |
196 | } |
197 | #else |
198 | strong_alias(__ieee754_sqrt, sqrt) |
199 | #endif |
200 | libm_hidden_def(sqrt) |
201 | |
202 | |
203 | /* |
204 | Other methods (use floating-point arithmetic) |
205 | ------------- |
206 | (This is a copy of a drafted paper by Prof W. Kahan |
207 | and K.C. Ng, written in May, 1986) |
208 | |
209 | Two algorithms are given here to implement sqrt(x) |
210 | (IEEE double precision arithmetic) in software. |
211 | Both supply sqrt(x) correctly rounded. The first algorithm (in |
212 | Section A) uses newton iterations and involves four divisions. |
213 | The second one uses reciproot iterations to avoid division, but |
214 | requires more multiplications. Both algorithms need the ability |
215 | to chop results of arithmetic operations instead of round them, |
216 | and the INEXACT flag to indicate when an arithmetic operation |
217 | is executed exactly with no roundoff error, all part of the |
218 | standard (IEEE 754-1985). The ability to perform shift, add, |
219 | subtract and logical AND operations upon 32-bit words is needed |
220 | too, though not part of the standard. |
221 | |
222 | A. sqrt(x) by Newton Iteration |
223 | |
224 | (1) Initial approximation |
225 | |
226 | Let x0 and x1 be the leading and the trailing 32-bit words of |
227 | a floating point number x (in IEEE double format) respectively |
228 | |
229 | 1 11 52 ...widths |
230 | ------------------------------------------------------ |
231 | x: |s| e | f | |
232 | ------------------------------------------------------ |
233 | msb lsb msb lsb ...order |
234 | |
235 | |
236 | ------------------------ ------------------------ |
237 | x0: |s| e | f1 | x1: | f2 | |
238 | ------------------------ ------------------------ |
239 | |
240 | By performing shifts and subtracts on x0 and x1 (both regarded |
241 | as integers), we obtain an 8-bit approximation of sqrt(x) as |
242 | follows. |
243 | |
244 | k := (x0>>1) + 0x1ff80000; |
245 | y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits |
246 | Here k is a 32-bit integer and T1[] is an integer array containing |
247 | correction terms. Now magically the floating value of y (y's |
248 | leading 32-bit word is y0, the value of its trailing word is 0) |
249 | approximates sqrt(x) to almost 8-bit. |
250 | |
251 | Value of T1: |
252 | static int T1[32]= { |
253 | 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592, |
254 | 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215, |
255 | 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581, |
256 | 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,}; |
257 | |
258 | (2) Iterative refinement |
259 | |
260 | Apply Heron's rule three times to y, we have y approximates |
261 | sqrt(x) to within 1 ulp (Unit in the Last Place): |
262 | |
263 | y := (y+x/y)/2 ... almost 17 sig. bits |
264 | y := (y+x/y)/2 ... almost 35 sig. bits |
265 | y := y-(y-x/y)/2 ... within 1 ulp |
266 | |
267 | |
268 | Remark 1. |
269 | Another way to improve y to within 1 ulp is: |
270 | |
271 | y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x) |
272 | y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x) |
273 | |
274 | 2 |
275 | (x-y )*y |
276 | y := y + 2* ---------- ...within 1 ulp |
277 | 2 |
278 | 3y + x |
279 | |
280 | |
281 | This formula has one division fewer than the one above; however, |
282 | it requires more multiplications and additions. Also x must be |
283 | scaled in advance to avoid spurious overflow in evaluating the |
284 | expression 3y*y+x. Hence it is not recommended uless division |
285 | is slow. If division is very slow, then one should use the |
286 | reciproot algorithm given in section B. |
287 | |
288 | (3) Final adjustment |
289 | |
290 | By twiddling y's last bit it is possible to force y to be |
291 | correctly rounded according to the prevailing rounding mode |
292 | as follows. Let r and i be copies of the rounding mode and |
293 | inexact flag before entering the square root program. Also we |
294 | use the expression y+-ulp for the next representable floating |
295 | numbers (up and down) of y. Note that y+-ulp = either fixed |
296 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
297 | mode. |
298 | |
299 | I := FALSE; ... reset INEXACT flag I |
300 | R := RZ; ... set rounding mode to round-toward-zero |
301 | z := x/y; ... chopped quotient, possibly inexact |
302 | If(not I) then { ... if the quotient is exact |
303 | if(z=y) { |
304 | I := i; ... restore inexact flag |
305 | R := r; ... restore rounded mode |
306 | return sqrt(x):=y. |
307 | } else { |
308 | z := z - ulp; ... special rounding |
309 | } |
310 | } |
311 | i := TRUE; ... sqrt(x) is inexact |
312 | If (r=RN) then z=z+ulp ... rounded-to-nearest |
313 | If (r=RP) then { ... round-toward-+inf |
314 | y = y+ulp; z=z+ulp; |
315 | } |
316 | y := y+z; ... chopped sum |
317 | y0:=y0-0x00100000; ... y := y/2 is correctly rounded. |
318 | I := i; ... restore inexact flag |
319 | R := r; ... restore rounded mode |
320 | return sqrt(x):=y. |
321 | |
322 | (4) Special cases |
323 | |
324 | Square root of +inf, +-0, or NaN is itself; |
325 | Square root of a negative number is NaN with invalid signal. |
326 | |
327 | |
328 | B. sqrt(x) by Reciproot Iteration |
329 | |
330 | (1) Initial approximation |
331 | |
332 | Let x0 and x1 be the leading and the trailing 32-bit words of |
333 | a floating point number x (in IEEE double format) respectively |
334 | (see section A). By performing shifs and subtracts on x0 and y0, |
335 | we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. |
336 | |
337 | k := 0x5fe80000 - (x0>>1); |
338 | y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits |
339 | |
340 | Here k is a 32-bit integer and T2[] is an integer array |
341 | containing correction terms. Now magically the floating |
342 | value of y (y's leading 32-bit word is y0, the value of |
343 | its trailing word y1 is set to zero) approximates 1/sqrt(x) |
344 | to almost 7.8-bit. |
345 | |
346 | Value of T2: |
347 | static int T2[64]= { |
348 | 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, |
349 | 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, |
350 | 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, |
351 | 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, |
352 | 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, |
353 | 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, |
354 | 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, |
355 | 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; |
356 | |
357 | (2) Iterative refinement |
358 | |
359 | Apply Reciproot iteration three times to y and multiply the |
360 | result by x to get an approximation z that matches sqrt(x) |
361 | to about 1 ulp. To be exact, we will have |
362 | -1ulp < sqrt(x)-z<1.0625ulp. |
363 | |
364 | ... set rounding mode to Round-to-nearest |
365 | y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) |
366 | y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) |
367 | ... special arrangement for better accuracy |
368 | z := x*y ... 29 bits to sqrt(x), with z*y<1 |
369 | z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) |
370 | |
371 | Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that |
372 | (a) the term z*y in the final iteration is always less than 1; |
373 | (b) the error in the final result is biased upward so that |
374 | -1 ulp < sqrt(x) - z < 1.0625 ulp |
375 | instead of |sqrt(x)-z|<1.03125ulp. |
376 | |
377 | (3) Final adjustment |
378 | |
379 | By twiddling y's last bit it is possible to force y to be |
380 | correctly rounded according to the prevailing rounding mode |
381 | as follows. Let r and i be copies of the rounding mode and |
382 | inexact flag before entering the square root program. Also we |
383 | use the expression y+-ulp for the next representable floating |
384 | numbers (up and down) of y. Note that y+-ulp = either fixed |
385 | point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
386 | mode. |
387 | |
388 | R := RZ; ... set rounding mode to round-toward-zero |
389 | switch(r) { |
390 | case RN: ... round-to-nearest |
391 | if(x<= z*(z-ulp)...chopped) z = z - ulp; else |
392 | if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; |
393 | break; |
394 | case RZ:case RM: ... round-to-zero or round-to--inf |
395 | R:=RP; ... reset rounding mod to round-to-+inf |
396 | if(x<z*z ... rounded up) z = z - ulp; else |
397 | if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; |
398 | break; |
399 | case RP: ... round-to-+inf |
400 | if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else |
401 | if(x>z*z ...chopped) z = z+ulp; |
402 | break; |
403 | } |
404 | |
405 | Remark 3. The above comparisons can be done in fixed point. For |
406 | example, to compare x and w=z*z chopped, it suffices to compare |
407 | x1 and w1 (the trailing parts of x and w), regarding them as |
408 | two's complement integers. |
409 | |
410 | ...Is z an exact square root? |
411 | To determine whether z is an exact square root of x, let z1 be the |
412 | trailing part of z, and also let x0 and x1 be the leading and |
413 | trailing parts of x. |
414 | |
415 | If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 |
416 | I := 1; ... Raise Inexact flag: z is not exact |
417 | else { |
418 | j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 |
419 | k := z1 >> 26; ... get z's 25-th and 26-th |
420 | fraction bits |
421 | I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); |
422 | } |
423 | R:= r ... restore rounded mode |
424 | return sqrt(x):=z. |
425 | |
426 | If multiplication is cheaper then the foregoing red tape, the |
427 | Inexact flag can be evaluated by |
428 | |
429 | I := i; |
430 | I := (z*z!=x) or I. |
431 | |
432 | Note that z*z can overwrite I; this value must be sensed if it is |
433 | True. |
434 | |
435 | Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be |
436 | zero. |
437 | |
438 | -------------------- |
439 | z1: | f2 | |
440 | -------------------- |
441 | bit 31 bit 0 |
442 | |
443 | Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd |
444 | or even of logb(x) have the following relations: |
445 | |
446 | ------------------------------------------------- |
447 | bit 27,26 of z1 bit 1,0 of x1 logb(x) |
448 | ------------------------------------------------- |
449 | 00 00 odd and even |
450 | 01 01 even |
451 | 10 10 odd |
452 | 10 00 even |
453 | 11 01 even |
454 | ------------------------------------------------- |
455 | |
456 | (4) Special cases (see (4) of Section A). |
457 | |
458 | */ |
459 | |