File : s-rannum.adb
1 ------------------------------------------------------------------------------
2 -- --
3 -- GNAT RUN-TIME COMPONENTS --
4 -- --
5 -- S Y S T E M . R A N D O M _ N U M B E R S --
6 -- --
7 -- B o d y --
8 -- --
9 -- Copyright (C) 2007-2015, Free Software Foundation, Inc. --
10 -- --
11 -- GNAT is free software; you can redistribute it and/or modify it under --
12 -- terms of the GNU General Public License as published by the Free Soft- --
13 -- ware Foundation; either version 3, or (at your option) any later ver- --
14 -- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
15 -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
16 -- or FITNESS FOR A PARTICULAR PURPOSE. --
17 -- --
18 -- --
19 -- --
20 -- --
21 -- --
22 -- You should have received a copy of the GNU General Public License and --
23 -- a copy of the GCC Runtime Library Exception along with this program; --
24 -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
25 -- <http://www.gnu.org/licenses/>. --
26 -- --
27 -- GNAT was originally developed by the GNAT team at New York University. --
28 -- Extensive contributions were provided by Ada Core Technologies Inc. --
29 -- --
30 ------------------------------------------------------------------------------
31
32 ------------------------------------------------------------------------------
33 -- --
34 -- The implementation here is derived from a C-program for MT19937, with --
35 -- initialization improved 2002/1/26. As required, the following notice is --
36 -- copied from the original program. --
37 -- --
38 -- Copyright (C) 1997 - 2002, Makoto Matsumoto and Takuji Nishimura, --
39 -- All rights reserved. --
40 -- --
41 -- Redistribution and use in source and binary forms, with or without --
42 -- modification, are permitted provided that the following conditions --
43 -- are met: --
44 -- --
45 -- 1. Redistributions of source code must retain the above copyright --
46 -- notice, this list of conditions and the following disclaimer. --
47 -- --
48 -- 2. Redistributions in binary form must reproduce the above copyright --
49 -- notice, this list of conditions and the following disclaimer in the --
50 -- documentation and/or other materials provided with the distribution.--
51 -- --
52 -- 3. The names of its contributors may not be used to endorse or promote --
53 -- products derived from this software without specific prior written --
54 -- permission. --
55 -- --
56 -- THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS --
57 -- "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT --
58 -- LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR --
59 -- A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT --
60 -- OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, --
61 -- SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED --
62 -- TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --
63 -- PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --
64 -- LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --
65 -- NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --
66 -- SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. --
67 -- --
68 ------------------------------------------------------------------------------
69
70 ------------------------------------------------------------------------------
71 -- --
72 -- This is an implementation of the Mersenne Twister, twisted generalized --
73 -- feedback shift register of rational normal form, with state-bit --
74 -- reflection and tempering. This version generates 32-bit integers with a --
75 -- period of 2**19937 - 1 (a Mersenne prime, hence the name). For --
76 -- applications requiring more than 32 bits (up to 64), we concatenate two --
77 -- 32-bit numbers. --
78 -- --
79 -- See http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html for --
80 -- details. --
81 -- --
82 -- In contrast to the original code, we do not generate random numbers in --
83 -- batches of N. Measurement seems to show this has very little if any --
84 -- effect on performance, and it may be marginally better for real-time --
85 -- applications with hard deadlines. --
86 -- --
87 ------------------------------------------------------------------------------
88
89 with Ada.Unchecked_Conversion;
90
91 with System.Random_Seed;
92
93 with Interfaces; use Interfaces;
94
95 use Ada;
96
97 package body System.Random_Numbers with
98 SPARK_Mode => Off
99 is
100 Image_Numeral_Length : constant := Max_Image_Width / N;
101
102 subtype Image_String is String (1 .. Max_Image_Width);
103
104 ----------------------------
105 -- Algorithmic Parameters --
106 ----------------------------
107
108 Lower_Mask : constant := 2**31 - 1;
109 Upper_Mask : constant := 2**31;
110
111 Matrix_A : constant array (State_Val range 0 .. 1) of State_Val
112 := (0, 16#9908b0df#);
113 -- The twist transformation is represented by a matrix of the form
114 --
115 -- [ 0 I(31) ]
116 -- [ _a ]
117 --
118 -- where 0 is a 31x31 block of 0s, I(31) is the 31x31 identity matrix and
119 -- _a is a particular bit row-vector, represented here by a 32-bit integer.
120 -- If integer x represents a row vector of bits (with x(0), the units bit,
121 -- last), then
122 -- x * A = [0 x(31..1)] xor Matrix_A(x(0)).
123
124 U : constant := 11;
125 S : constant := 7;
126 B_Mask : constant := 16#9d2c5680#;
127 T : constant := 15;
128 C_Mask : constant := 16#efc60000#;
129 L : constant := 18;
130 -- The tempering shifts and bit masks, in the order applied
131
132 Seed0 : constant := 5489;
133 -- Default seed, used to initialize the state vector when Reset not called
134
135 Seed1 : constant := 19650218;
136 -- Seed used to initialize the state vector when calling Reset with an
137 -- initialization vector.
138
139 Mult0 : constant := 1812433253;
140 -- Multiplier for a modified linear congruential generator used to
141 -- initialize the state vector when calling Reset with a single integer
142 -- seed.
143
144 Mult1 : constant := 1664525;
145 Mult2 : constant := 1566083941;
146 -- Multipliers for two modified linear congruential generators used to
147 -- initialize the state vector when calling Reset with an initialization
148 -- vector.
149
150 -----------------------
151 -- Local Subprograms --
152 -----------------------
153
154 procedure Init (Gen : Generator; Initiator : Unsigned_32);
155 -- Perform a default initialization of the state of Gen. The resulting
156 -- state is identical for identical values of Initiator.
157
158 procedure Insert_Image
159 (S : in out Image_String;
160 Index : Integer;
161 V : State_Val);
162 -- Insert image of V into S, in the Index'th 11-character substring
163
164 function Extract_Value (S : String; Index : Integer) return State_Val;
165 -- Treat S as a sequence of 11-character decimal numerals and return
166 -- the result of converting numeral #Index (numbering from 0)
167
168 function To_Unsigned is
169 new Unchecked_Conversion (Integer_32, Unsigned_32);
170 function To_Unsigned is
171 new Unchecked_Conversion (Integer_64, Unsigned_64);
172
173 ------------
174 -- Random --
175 ------------
176
177 function Random (Gen : Generator) return Unsigned_32 is
178 G : Generator renames Gen.Writable.Self.all;
179 Y : State_Val;
180 I : Integer; -- should avoid use of identifier I ???
181
182 begin
183 I := G.I;
184
185 if I < N - M then
186 Y := (G.S (I) and Upper_Mask) or (G.S (I + 1) and Lower_Mask);
187 Y := G.S (I + M) xor Shift_Right (Y, 1) xor Matrix_A (Y and 1);
188 I := I + 1;
189
190 elsif I < N - 1 then
191 Y := (G.S (I) and Upper_Mask) or (G.S (I + 1) and Lower_Mask);
192 Y := G.S (I + (M - N))
193 xor Shift_Right (Y, 1)
194 xor Matrix_A (Y and 1);
195 I := I + 1;
196
197 elsif I = N - 1 then
198 Y := (G.S (I) and Upper_Mask) or (G.S (0) and Lower_Mask);
199 Y := G.S (M - 1) xor Shift_Right (Y, 1) xor Matrix_A (Y and 1);
200 I := 0;
201
202 else
203 Init (G, Seed0);
204 return Random (Gen);
205 end if;
206
207 G.S (G.I) := Y;
208 G.I := I;
209
210 Y := Y xor Shift_Right (Y, U);
211 Y := Y xor (Shift_Left (Y, S) and B_Mask);
212 Y := Y xor (Shift_Left (Y, T) and C_Mask);
213 Y := Y xor Shift_Right (Y, L);
214
215 return Y;
216 end Random;
217
218 generic
219 type Unsigned is mod <>;
220 type Real is digits <>;
221 with function Random (G : Generator) return Unsigned is <>;
222 function Random_Float_Template (Gen : Generator) return Real;
223 pragma Inline (Random_Float_Template);
224 -- Template for a random-number generator implementation that delivers
225 -- values of type Real in the range [0 .. 1], using values from Gen,
226 -- assuming that Unsigned is large enough to hold the bits of a mantissa
227 -- for type Real.
228
229 ---------------------------
230 -- Random_Float_Template --
231 ---------------------------
232
233 function Random_Float_Template (Gen : Generator) return Real is
234
235 pragma Compile_Time_Error
236 (Unsigned'Last <= 2**(Real'Machine_Mantissa - 1),
237 "insufficiently large modular type used to hold mantissa");
238
239 begin
240 -- This code generates random floating-point numbers from unsigned
241 -- integers. Assuming that Real'Machine_Radix = 2, it can deliver all
242 -- machine values of type Real (as implied by Real'Machine_Mantissa and
243 -- Real'Machine_Emin), which is not true of the standard method (to
244 -- which we fall back for nonbinary radix): computing Real(<random
245 -- integer>) / (<max random integer>+1). To do so, we first extract an
246 -- (M-1)-bit significand (where M is Real'Machine_Mantissa), and then
247 -- decide on a normalized exponent by repeated coin flips, decrementing
248 -- from 0 as long as we flip heads (1 bits). This process yields the
249 -- proper geometric distribution for the exponent: in a uniformly
250 -- distributed set of floating-point numbers, 1/2 of them will be in
251 -- (0.5, 1], 1/4 will be in (0.25, 0.5], and so forth. It makes a
252 -- further adjustment at binade boundaries (see comments below) to give
253 -- the effect of selecting a uniformly distributed real deviate in
254 -- [0..1] and then rounding to the nearest representable floating-point
255 -- number. The algorithm attempts to be stingy with random integers. In
256 -- the worst case, it can consume roughly -Real'Machine_Emin/32 32-bit
257 -- integers, but this case occurs with probability around
258 -- 2**Machine_Emin, and the expected number of calls to integer-valued
259 -- Random is 1. For another discussion of the issues addressed by this
260 -- process, see Allen Downey's unpublished paper at
261 -- http://allendowney.com/research/rand/downey07randfloat.pdf.
262
263 if Real'Machine_Radix /= 2 then
264 return Real'Machine
265 (Real (Unsigned'(Random (Gen))) * 2.0**(-Unsigned'Size));
266
267 else
268 declare
269 type Bit_Count is range 0 .. 4;
270
271 subtype T is Real'Base;
272
273 Trailing_Ones : constant array (Unsigned_32 range 0 .. 15)
274 of Bit_Count :=
275 (2#00000# => 0, 2#00001# => 1, 2#00010# => 0, 2#00011# => 2,
276 2#00100# => 0, 2#00101# => 1, 2#00110# => 0, 2#00111# => 3,
277 2#01000# => 0, 2#01001# => 1, 2#01010# => 0, 2#01011# => 2,
278 2#01100# => 0, 2#01101# => 1, 2#01110# => 0, 2#01111# => 4);
279
280 Pow_Tab : constant array (Bit_Count range 0 .. 3) of Real
281 := (0 => 2.0**(0 - T'Machine_Mantissa),
282 1 => 2.0**(-1 - T'Machine_Mantissa),
283 2 => 2.0**(-2 - T'Machine_Mantissa),
284 3 => 2.0**(-3 - T'Machine_Mantissa));
285
286 Extra_Bits : constant Natural :=
287 (Unsigned'Size - T'Machine_Mantissa + 1);
288 -- Random bits left over after selecting mantissa
289
290 Mantissa : Unsigned;
291
292 X : Real; -- Scaled mantissa
293 R : Unsigned_32; -- Supply of random bits
294 R_Bits : Natural; -- Number of bits left in R
295 K : Bit_Count; -- Next decrement to exponent
296
297 begin
298 Mantissa := Random (Gen) / 2**Extra_Bits;
299 R := Unsigned_32 (Mantissa mod 2**Extra_Bits);
300 R_Bits := Extra_Bits;
301 X := Real (2**(T'Machine_Mantissa - 1) + Mantissa); -- Exact
302
303 if Extra_Bits < 4 and then R < 2 ** Extra_Bits - 1 then
304
305 -- We got lucky and got a zero in our few extra bits
306
307 K := Trailing_Ones (R);
308
309 else
310 Find_Zero : loop
311
312 -- R has R_Bits unprocessed random bits, a multiple of 4.
313 -- X needs to be halved for each trailing one bit. The
314 -- process stops as soon as a 0 bit is found. If R_Bits
315 -- becomes zero, reload R.
316
317 -- Process 4 bits at a time for speed: the two iterations
318 -- on average with three tests each was still too slow,
319 -- probably because the branches are not predictable.
320 -- This loop now will only execute once 94% of the cases,
321 -- doing more bits at a time will not help.
322
323 while R_Bits >= 4 loop
324 K := Trailing_Ones (R mod 16);
325
326 exit Find_Zero when K < 4; -- Exits 94% of the time
327
328 R_Bits := R_Bits - 4;
329 X := X / 16.0;
330 R := R / 16;
331 end loop;
332
333 -- Do not allow us to loop endlessly even in the (very
334 -- unlikely) case that Random (Gen) keeps yielding all ones.
335
336 exit Find_Zero when X = 0.0;
337 R := Random (Gen);
338 R_Bits := 32;
339 end loop Find_Zero;
340 end if;
341
342 -- K has the count of trailing ones not reflected yet in X. The
343 -- following multiplication takes care of that, as well as the
344 -- correction to move the radix point to the left of the mantissa.
345 -- Doing it at the end avoids repeated rounding errors in the
346 -- exceedingly unlikely case of ever having a subnormal result.
347
348 X := X * Pow_Tab (K);
349
350 -- The smallest value in each binade is rounded to by 0.75 of
351 -- the span of real numbers as its next larger neighbor, and
352 -- 1.0 is rounded to by half of the span of real numbers as its
353 -- next smaller neighbor. To account for this, when we encounter
354 -- the smallest number in a binade, we substitute the smallest
355 -- value in the next larger binade with probability 1/2.
356
357 if Mantissa = 0 and then Unsigned_32'(Random (Gen)) mod 2 = 0 then
358 X := 2.0 * X;
359 end if;
360
361 return X;
362 end;
363 end if;
364 end Random_Float_Template;
365
366 ------------
367 -- Random --
368 ------------
369
370 function Random (Gen : Generator) return Float is
371 function F is new Random_Float_Template (Unsigned_32, Float);
372 begin
373 return F (Gen);
374 end Random;
375
376 function Random (Gen : Generator) return Long_Float is
377 function F is new Random_Float_Template (Unsigned_64, Long_Float);
378 begin
379 return F (Gen);
380 end Random;
381
382 function Random (Gen : Generator) return Unsigned_64 is
383 begin
384 return Shift_Left (Unsigned_64 (Unsigned_32'(Random (Gen))), 32)
385 or Unsigned_64 (Unsigned_32'(Random (Gen)));
386 end Random;
387
388 ---------------------
389 -- Random_Discrete --
390 ---------------------
391
392 function Random_Discrete
393 (Gen : Generator;
394 Min : Result_Subtype := Default_Min;
395 Max : Result_Subtype := Result_Subtype'Last) return Result_Subtype
396 is
397 begin
398 if Max = Min then
399 return Max;
400
401 elsif Max < Min then
402 raise Constraint_Error;
403
404 elsif Result_Subtype'Base'Size > 32 then
405 declare
406 -- In the 64-bit case, we have to be careful, since not all 64-bit
407 -- unsigned values are representable in GNAT's root_integer type.
408 -- Ignore different-size warnings here since GNAT's handling
409 -- is correct.
410
411 pragma Warnings ("Z");
412 function Conv_To_Unsigned is
413 new Unchecked_Conversion (Result_Subtype'Base, Unsigned_64);
414 function Conv_To_Result is
415 new Unchecked_Conversion (Unsigned_64, Result_Subtype'Base);
416 pragma Warnings ("z");
417
418 N : constant Unsigned_64 :=
419 Conv_To_Unsigned (Max) - Conv_To_Unsigned (Min) + 1;
420
421 X, Slop : Unsigned_64;
422
423 begin
424 if N = 0 then
425 return Conv_To_Result (Conv_To_Unsigned (Min) + Random (Gen));
426
427 else
428 Slop := Unsigned_64'Last rem N + 1;
429
430 loop
431 X := Random (Gen);
432 exit when Slop = N or else X <= Unsigned_64'Last - Slop;
433 end loop;
434
435 return Conv_To_Result (Conv_To_Unsigned (Min) + X rem N);
436 end if;
437 end;
438
439 elsif Result_Subtype'Pos (Max) - Result_Subtype'Pos (Min) =
440 2 ** 32 - 1
441 then
442 return Result_Subtype'Val
443 (Result_Subtype'Pos (Min) + Unsigned_32'Pos (Random (Gen)));
444 else
445 declare
446 N : constant Unsigned_32 :=
447 Unsigned_32 (Result_Subtype'Pos (Max) -
448 Result_Subtype'Pos (Min) + 1);
449 Slop : constant Unsigned_32 := Unsigned_32'Last rem N + 1;
450 X : Unsigned_32;
451
452 begin
453 loop
454 X := Random (Gen);
455 exit when Slop = N or else X <= Unsigned_32'Last - Slop;
456 end loop;
457
458 return
459 Result_Subtype'Val
460 (Result_Subtype'Pos (Min) + Unsigned_32'Pos (X rem N));
461 end;
462 end if;
463 end Random_Discrete;
464
465 ------------------
466 -- Random_Float --
467 ------------------
468
469 function Random_Float (Gen : Generator) return Result_Subtype is
470 begin
471 if Result_Subtype'Base'Digits > Float'Digits then
472 return Result_Subtype'Machine (Result_Subtype
473 (Long_Float'(Random (Gen))));
474 else
475 return Result_Subtype'Machine (Result_Subtype
476 (Float'(Random (Gen))));
477 end if;
478 end Random_Float;
479
480 -----------
481 -- Reset --
482 -----------
483
484 procedure Reset (Gen : Generator) is
485 begin
486 Init (Gen, Unsigned_32'Mod (Random_Seed.Get_Seed));
487 end Reset;
488
489 procedure Reset (Gen : Generator; Initiator : Integer_32) is
490 begin
491 Init (Gen, To_Unsigned (Initiator));
492 end Reset;
493
494 procedure Reset (Gen : Generator; Initiator : Unsigned_32) is
495 begin
496 Init (Gen, Initiator);
497 end Reset;
498
499 procedure Reset (Gen : Generator; Initiator : Integer) is
500 begin
501 -- This is probably an unnecessary precaution against future change, but
502 -- since the test is a static expression, no extra code is involved.
503
504 if Integer'Size <= 32 then
505 Init (Gen, To_Unsigned (Integer_32 (Initiator)));
506
507 else
508 declare
509 Initiator1 : constant Unsigned_64 :=
510 To_Unsigned (Integer_64 (Initiator));
511 Init0 : constant Unsigned_32 :=
512 Unsigned_32 (Initiator1 mod 2 ** 32);
513 Init1 : constant Unsigned_32 :=
514 Unsigned_32 (Shift_Right (Initiator1, 32));
515 begin
516 Reset (Gen, Initialization_Vector'(Init0, Init1));
517 end;
518 end if;
519 end Reset;
520
521 procedure Reset (Gen : Generator; Initiator : Initialization_Vector) is
522 G : Generator renames Gen.Writable.Self.all;
523 I, J : Integer;
524
525 begin
526 Init (G, Seed1);
527 I := 1;
528 J := 0;
529
530 if Initiator'Length > 0 then
531 for K in reverse 1 .. Integer'Max (N, Initiator'Length) loop
532 G.S (I) :=
533 (G.S (I) xor ((G.S (I - 1)
534 xor Shift_Right (G.S (I - 1), 30)) * Mult1))
535 + Initiator (J + Initiator'First) + Unsigned_32 (J);
536
537 I := I + 1;
538 J := J + 1;
539
540 if I >= N then
541 G.S (0) := G.S (N - 1);
542 I := 1;
543 end if;
544
545 if J >= Initiator'Length then
546 J := 0;
547 end if;
548 end loop;
549 end if;
550
551 for K in reverse 1 .. N - 1 loop
552 G.S (I) :=
553 (G.S (I) xor ((G.S (I - 1)
554 xor Shift_Right (G.S (I - 1), 30)) * Mult2))
555 - Unsigned_32 (I);
556 I := I + 1;
557
558 if I >= N then
559 G.S (0) := G.S (N - 1);
560 I := 1;
561 end if;
562 end loop;
563
564 G.S (0) := Upper_Mask;
565 end Reset;
566
567 procedure Reset (Gen : Generator; From_State : Generator) is
568 G : Generator renames Gen.Writable.Self.all;
569 begin
570 G.S := From_State.S;
571 G.I := From_State.I;
572 end Reset;
573
574 procedure Reset (Gen : Generator; From_State : State) is
575 G : Generator renames Gen.Writable.Self.all;
576 begin
577 G.I := 0;
578 G.S := From_State;
579 end Reset;
580
581 procedure Reset (Gen : Generator; From_Image : String) is
582 G : Generator renames Gen.Writable.Self.all;
583 begin
584 G.I := 0;
585
586 for J in 0 .. N - 1 loop
587 G.S (J) := Extract_Value (From_Image, J);
588 end loop;
589 end Reset;
590
591 ----------
592 -- Save --
593 ----------
594
595 procedure Save (Gen : Generator; To_State : out State) is
596 Gen2 : Generator;
597
598 begin
599 if Gen.I = N then
600 Init (Gen2, 5489);
601 To_State := Gen2.S;
602
603 else
604 To_State (0 .. N - 1 - Gen.I) := Gen.S (Gen.I .. N - 1);
605 To_State (N - Gen.I .. N - 1) := Gen.S (0 .. Gen.I - 1);
606 end if;
607 end Save;
608
609 -----------
610 -- Image --
611 -----------
612
613 function Image (Of_State : State) return String is
614 Result : Image_String;
615
616 begin
617 Result := (others => ' ');
618
619 for J in Of_State'Range loop
620 Insert_Image (Result, J, Of_State (J));
621 end loop;
622
623 return Result;
624 end Image;
625
626 function Image (Gen : Generator) return String is
627 Result : Image_String;
628
629 begin
630 Result := (others => ' ');
631 for J in 0 .. N - 1 loop
632 Insert_Image (Result, J, Gen.S ((J + Gen.I) mod N));
633 end loop;
634
635 return Result;
636 end Image;
637
638 -----------
639 -- Value --
640 -----------
641
642 function Value (Coded_State : String) return State is
643 Gen : Generator;
644 S : State;
645 begin
646 Reset (Gen, Coded_State);
647 Save (Gen, S);
648 return S;
649 end Value;
650
651 ----------
652 -- Init --
653 ----------
654
655 procedure Init (Gen : Generator; Initiator : Unsigned_32) is
656 G : Generator renames Gen.Writable.Self.all;
657 begin
658 G.S (0) := Initiator;
659
660 for I in 1 .. N - 1 loop
661 G.S (I) :=
662 (G.S (I - 1) xor Shift_Right (G.S (I - 1), 30)) * Mult0
663 + Unsigned_32 (I);
664 end loop;
665
666 G.I := 0;
667 end Init;
668
669 ------------------
670 -- Insert_Image --
671 ------------------
672
673 procedure Insert_Image
674 (S : in out Image_String;
675 Index : Integer;
676 V : State_Val)
677 is
678 Value : constant String := State_Val'Image (V);
679 begin
680 S (Index * 11 + 1 .. Index * 11 + Value'Length) := Value;
681 end Insert_Image;
682
683 -------------------
684 -- Extract_Value --
685 -------------------
686
687 function Extract_Value (S : String; Index : Integer) return State_Val is
688 Start : constant Integer := S'First + Index * Image_Numeral_Length;
689 begin
690 return State_Val'Value (S (Start .. Start + Image_Numeral_Length - 1));
691 end Extract_Value;
692
693 end System.Random_Numbers;