DIRECTORY Ieee USING [CVExtended, ExponentBias, SingleReal, Unpack], Real USING [Extended, Fix, FixC, FixI, RealError, RealException], RealFns; RealFnsImpl: CEDAR PROGRAM IMPORTS Ieee, Real EXPORTS RealFns = BEGIN Ln2: REAL = 0.693147181; LogBase2ofE: REAL _ 1/Ln2; smallnumber: REAL = 0.00000000005; PI: REAL = 3.14159265; twoPI: REAL _ PI*2; PI3ovr2: REAL _ 3*PI/2; PIovr2: REAL = 1.570796327; PI3ovr8: REAL = 1.1780972; PIovr4: REAL = .785398163; PIovr8: REAL = .392699; radiansPerDegree: REAL _ PI/180; degreesPerRadian: REAL _ 180/PI; rec2pi: REAL _ 1.0/twoPI; rec360: REAL _ 1.0/360.0; fourOverPI: REAL _ 4.0/PI; Exp: PUBLIC PROC [x: REAL] RETURNS [REAL] = { r, f: REAL; k: INTEGER; fl: Ieee.SingleReal _ LOOPHOLE[1.0, Ieee.SingleReal]; SELECT ABS[x] FROM < 3e-4 => RETURN [1+x]; < 3e-3 => RETURN [1+x+x*x*0.5]; ENDCASE; x _ x*LogBase2ofE; --exponent of 2 (instead of e) IF x < -Ieee.ExponentBias THEN RETURN [0.0]; IF x > Ieee.ExponentBias THEN ERROR Real.RealError[]; k _ Real.FixI[x]; --integer part x _ x - k; --fractional part x _ x*Ln2; --now result = 2^k * exp(x) and 0<=x 3e-3 => {}; > 3e-4 => RETURN [xm1 - 0.5*xm1*xm1]; ENDCASE => RETURN [xm1]; fl _ LOOPHOLE[x, Ieee.SingleReal]; m _ fl.exp - (Ieee.ExponentBias - 1); fl.exp _ Ieee.ExponentBias - 1; --now 1/2 <= z < 1 x _ LOOPHOLE[fl, REAL]; IF x = 0.5 THEN RETURN[Ln2*(m-1)]; OneMinusX _ 1 - x; x _ m*Ln2; OneMinusXPowerRep _ 1; FOR i: CARDINAL _ 1, i+1 DO PreviousVal: REAL _ x; OneMinusXPowerRep _ OneMinusXPowerRep*OneMinusX; x _ x - OneMinusXPowerRep/i; IF x = PreviousVal THEN EXIT; ENDLOOP; RETURN[x]; }; Bomb: PROC [x: REAL] = { [] _ SIGNAL Real.RealException[flags: [invalidOperation: TRUE], vp: NEW[Real.Extended _ Ieee.CVExtended[Ieee.Unpack[x]]]]; ERROR Real.RealError[]; }; SquareReal: TYPE = MACHINE DEPENDENT RECORD [ m2: CARDINAL, sign: BOOL, expD2: [0..128), index: [0..32), m1: [0..8)]; SqRt: PUBLIC PROC [a: REAL] RETURNS [b: REAL _ 0.0] = { SELECT TRUE FROM a <= 0.0 => { IF a < 0.0 THEN Bomb[a]; b _ 0.0; }; LOOPHOLE[a, Ieee.SingleReal].exp = 0 => b _ SqRt[a * (LONG[20000B] * LONG[20000B])] / 20000B; ENDCASE => { aFmt: SquareReal _ LOOPHOLE[a, SquareReal]; bFmt: Ieee.SingleReal _ LOOPHOLE[sqGuesses[aFmt.index], Ieee.SingleReal]; bFmt.exp _ bFmt.exp + aFmt.expD2 - 63; b _ LOOPHOLE[bFmt, REAL]; b _ LOOPHOLE[LOOPHOLE[b + a/b, LONG CARDINAL] - 40000000B, REAL]; b _ LOOPHOLE[LOOPHOLE[b + a/b, LONG CARDINAL] - 40000000B, REAL]; }; }; sqGuesses: ARRAY[0..32) OF REAL _ [ LOOPHOLE[7715741440B, REAL], -- 0.7178211 LOOPHOLE[7717240700B, REAL], LOOPHOLE[7720514140B, REAL], LOOPHOLE[7721745140B, REAL], LOOPHOLE[7723155200B, REAL], LOOPHOLE[7724346000B, REAL], LOOPHOLE[7725517500B, REAL], LOOPHOLE[7726654000B, REAL], -- 0.8568115 LOOPHOLE[7727773400B, REAL], -- 0.8748627 LOOPHOLE[7731077100B, REAL], LOOPHOLE[7732167300B, REAL], LOOPHOLE[7733245000B, REAL], LOOPHOLE[7734310400B, REAL], LOOPHOLE[7735342500B, REAL], LOOPHOLE[7736363400B, REAL], LOOPHOLE[7737373700B, REAL], -- 0.9920616 LOOPHOLE[7740370240B, REAL], -- 1.015156 LOOPHOLE[7741351440B, REAL], LOOPHOLE[7742314540B, REAL], LOOPHOLE[7743243000B, REAL], LOOPHOLE[7744155200B, REAL], LOOPHOLE[7745054400B, REAL], LOOPHOLE[7745741300B, REAL], LOOPHOLE[7746614600B, REAL], -- 1.211716 LOOPHOLE[7747457040B, REAL], -- 1.237247 LOOPHOLE[7750310640B, REAL], LOOPHOLE[7751132500B, REAL], LOOPHOLE[7751745000B, REAL], LOOPHOLE[7752550100B, REAL], LOOPHOLE[7753344400B, REAL], LOOPHOLE[7754132600B, REAL], LOOPHOLE[7754712500B, REAL] -- 1.402992 ]; Root: PUBLIC PROC [index, arg: REAL] RETURNS [REAL] = { RETURN[Exp[Ln[arg]/index]]; }; Power: PUBLIC PROC [base, exponent: REAL] RETURNS [REAL] = { IF base = 0 THEN IF exponent = 0 THEN RETURN[1] ELSE RETURN[0]; RETURN[Exp[exponent*Ln[base]]]; }; Sin: PUBLIC PROC [radians: REAL] RETURNS [sin: REAL] = { rem: REAL _ ABS[radians]; IF rem > twoPI THEN rem _ GetWithinTwoPI[rem]; { SELECT Real.FixC[rem * fourOverPI] FROM 0 => {GO TO posSin}; 1 => {rem _ PIovr2 - rem; GO TO posCos}; 2 => {rem _ rem - PIovr2; GO TO posCos}; 3 => {rem _ PI - rem; GO TO posSin}; 4 => {rem _ rem - PI; GO TO negSin}; 5 => {rem _ PI3ovr2 - rem; GO TO negCos}; 6 => {rem _ rem - PI3ovr2; GO TO negCos}; 7 => {rem _ twoPI - rem; GO TO negSin}; ENDCASE => RETURN [0.0]; EXITS posSin => {sin _ SinFirstOctant[rem]; IF radians < 0.0 THEN GO TO neg}; negSin => {sin _ SinFirstOctant[rem]; IF radians > 0.0 THEN GO TO neg}; posCos => {sin _ CosFirstOctant[rem]; IF radians < 0.0 THEN GO TO neg}; negCos => {sin _ CosFirstOctant[rem]; IF radians > 0.0 THEN GO TO neg}; }; EXITS neg => sin _ - sin; }; Cos: PUBLIC PROC [radians: REAL] RETURNS [cos: REAL] = { rem: REAL _ ABS[radians]; IF rem > twoPI THEN rem _ GetWithinTwoPI[rem]; { SELECT Real.FixC[rem * fourOverPI] FROM 0 => {GO TO posCos}; 1 => {rem _ PIovr2 - rem; GO TO posSin}; 2 => {rem _ rem - PIovr2; GO TO negSin}; 3 => {rem _ PI - rem; GO TO negCos}; 4 => {rem _ rem - PI; GO TO negCos}; 5 => {rem _ PI3ovr2 - rem; GO TO negSin}; 6 => {rem _ rem - PI3ovr2; GO TO posSin}; 7 => {rem _ twoPI - rem; GO TO posCos}; ENDCASE => cos _ 1.0; EXITS posSin => {cos _ SinFirstOctant[rem]}; negSin => {cos _ SinFirstOctant[rem]; cos _ -cos}; posCos => {cos _ CosFirstOctant[rem]}; negCos => {cos _ CosFirstOctant[rem]; cos _ -cos}; }; }; SinDeg: PUBLIC PROC [degrees: REAL] RETURNS [sin: REAL] = { IF ABS[degrees] > 360 THEN degrees _ GetWithin360[degrees]; sin _ Sin[degrees*radiansPerDegree]; }; CosDeg: PUBLIC PROC [degrees: REAL] RETURNS [cos: REAL] = { degrees _ ABS[degrees]; IF degrees > 360 THEN degrees _ GetWithin360[degrees]; cos _ Cos[degrees*radiansPerDegree]; }; Tan: PUBLIC PROC [radians: REAL] RETURNS [tan: REAL] = { IF ABS[radians] > twoPI THEN radians _ GetWithinTwoPI[radians]; tan _ Sin[radians] / Cos[radians]; }; TanDeg: PUBLIC PROC [degrees: REAL] RETURNS [tan: REAL] = { IF ABS[degrees] > 360 THEN degrees _ GetWithin360[degrees]; tan _ SinDeg[degrees] / CosDeg[degrees]; }; tanPI16: REAL = .1989123673; x2: REAL _ 1/(.414213562); x22: REAL _ x2*x2 + 1; tan3PI16: REAL = .668178638; x4: REAL = 1.0; x44: REAL _ x4*x4 + 1; tan5PI16: REAL = 1.496605763; x6: REAL _ 1/(2.41421356); x66: REAL _ x6*x6 + 1; tan7PI16: REAL = 5.02733949; p0: REAL = .999999998; p1: REAL = -.333331726; p2: REAL = .1997952738; p3: REAL = -.134450639; ArcTan: PUBLIC PROC [y, x: REAL] RETURNS [radians: REAL] = { s, v, t, t2, c, q: REAL; IF ABS[x] <= ABS[smallnumber] THEN IF ABS[y] <= ABS[smallnumber] THEN RETURN[0] ELSE IF y < 0 THEN RETURN[-PIovr2] ELSE RETURN[PIovr2]; IF x < 0 THEN IF y < 0 THEN {q _ -PI; s _ 1} ELSE {q _ PI; s _ -1} ELSE IF y < 0 THEN {q _ 0; s _ -1} ELSE {q _ 0; s _ 1}; v _ ABS[y/x]; SELECT v FROM < tanPI16 => {t _ v; c _ 0}; < tan3PI16 => {t _ x2 - x22/(x2 + v); c _ PIovr8}; < tan5PI16 => {t _ x4 - x44/(x4 + v); c _ PIovr4}; < tan7PI16 => {t _ x6 - x66/(x6 + v); c _ PI3ovr8}; ENDCASE => {t _ -1/v; c _ PIovr2}; t2 _ t*t; radians _ s*(t*(p0 + t2*(p1 + t2*(p2 + t2*p3))) + c) + q; }; ArcTanDeg: PUBLIC PROC [y, x: REAL] RETURNS [degrees: REAL] = { degrees _ ArcTan[y, x]*degreesPerRadian; }; AlmostZero: PUBLIC PROC [x: REAL, distance: INTEGER [-126..127]] RETURNS [BOOL] = { RETURN[LOOPHOLE[x, Ieee.SingleReal].exp < (distance + Ieee.ExponentBias)]; }; AlmostEqual: PUBLIC PROC [y, x: REAL, distance: INTEGER [-126..0]] RETURNS [BOOL] = { fl: Ieee.SingleReal _ LOOPHOLE[(x - y)]; oldexp: INTEGER _ MAX[ LOOPHOLE[x, Ieee.SingleReal].exp, LOOPHOLE[y, Ieee.SingleReal].exp]; RETURN[fl.exp < (distance + oldexp)]; }; fact9recip: REAL _ 1.0/(INT[1]*2*3*4*5*6*7*8*9); fact8recip: REAL _ 1.0/(INT[1]*2*3*4*5*6*7*8); fact7recip: REAL _ 1.0/(INT[1]*2*3*4*5*6*7); fact6recip: REAL _ 1.0/(INT[1]*2*3*4*5*6); fact5recip: REAL _ 1.0/(INT[1]*2*3*4*5); fact4recip: REAL _ 1.0/(INT[1]*2*3*4); fact3recip: REAL _ 1.0/(INT[1]*2*3); fact2recip: REAL _ 1.0/(INT[1]*2); minEps: REAL _ 1.0 - LOOPHOLE[LOOPHOLE[1.0, LONG CARDINAL]-1, REAL]; sinCosEps: REAL _ SqRt[minEps]; SinFirstOctant: PROC [radians: REAL] RETURNS [REAL] = { IF radians <= sinCosEps THEN RETURN [radians] ELSE { r2: REAL ~ radians*radians; RETURN [(((fact9recip*r2-fact7recip)*r2+fact5recip)*r2-fact3recip)*r2*radians+radians]; }; }; CosFirstOctant: PROC [radians: REAL] RETURNS [REAL] = { IF radians <= sinCosEps THEN RETURN [1.0] ELSE { r2: REAL ~ radians*radians; RETURN [((((fact8recip*r2-fact6recip)*r2+fact4recip)*r2-fact2recip)*r2+1)]; }; }; PIc1: REAL _ 6.28125; -- 402.0/64 (exact representation near twoPI) PIc2: REAL = 1.9353072e-3; -- 2*PI - c1 GetWithinTwoPI: PROC [radians: REAL] RETURNS [REAL] = { xn: REAL _ Real.Fix[radians * rec2pi]; RETURN [(radians - xn*PIc1) - xn*PIc2]; }; GetWithin360: PROC [degrees: REAL] RETURNS [REAL] = { xn: REAL _ Real.Fix[degrees * rec360]; RETURN [degrees - xn*360]; }; END. September 16, 1980 3:13 PM, Stewart; Bug in Exp September 28, 1980 8:28 PM, Stewart; Add AlmostEqual and AlmostZero, format November 7, 1980 3:33 PM, Stewart; fix AlmostZero August 27, 1982 1:07 pm, Stewart; CEDAR October 24, 1984 4:37:17 pm PDT, Plass; New impls for Sin, Cos, SinDeg, CosDeg November 2, 1984, Atkinson; Faster & more accurate SqRt, Sin, Cos, SinDeg, CosDeg fRealFnsImpl.mesa Copyright c 1984, 1985 by Xerox Corporation. All rights reserved. Stewart on August 27, 1982 1:07 pm Paul Rovner on May 4, 1983 10:13 am Levin, August 8, 1983 4:39 pm Plass, October 24, 1984 4:26:48 pm PDT Russ Atkinson (RRA), May 29, 1985 1:16:23 am PDT NOTE: The compiler is bloody incompetent about floating point! In particular, it cannot deal with constant folding. Therefore, various constants are given as variables to avoid excess computation. Please make these revert to constants when the compiler gets smarter. (RRA) Some useful constants compute e^x using continued fractions ACM Algorithm 229, Morelock, James C., "Elementary functions by continued fractions" To keep the error small near 0.0, use this Taylor's series Expand the continued fraction loop a few times Note, if LoopCount > 3, then use the following comments f _ (4*(LoopCount-3) - 2) + r/f; f _ (4*(LoopCount-4) - 2) + r/f; unscaled result is (f+x)/(f-x) scale: multipy by 2^k by adding k to the exponent logxy = Ln(y)/Ln(x) To keep the error small near 1.0, use this Taylor's series if x = z * 2^m then Ln(x) = m * Ln2 + Ln(z). Here, 1/2 <= z < 1 special case for limit condition (x a power of 2) This declaration obviously depends on IEEE format 32-bit numbers (Ieee.SingleReal). The idea is that the index field has the low bit of the exponent and the high 4 bits of the mantissa, which is used to index a table of initial guesses at the square root, each guess is roughly halfway between the lowest and highest root with the given index, which allows convergence with only two iterations. I don't know where the table originated. RRA: The idea is to lookup a square root guess from the table, adjusting for the exponent, then perform a few iterations to get within a reasonable error. Someday we will get ambitious and share the table with IeeeFloatB. The guess is only valid when we have a normalized number, so we force it to be normalized, then correct for the shift. The guess is only valid when we have a normalized number, so we force it to be normalized, then correct for the shift. Get the initial guess based on the top 4 bits of the mantissa and the bottom bit of the exponent. Adjust the guess exponent based on adding half of the initial exponent (adjusted for the offset inherent in Ieee format). The result is initially the guess This is really b _ (b + a/b)/2, but much faster This is really b _ (b + a/b)/2, but much faster yth root of x = e^(Ln(x)/y) x^y = e^(y*Ln(x)) We need to have rem IN [0..twoPI] Now select the appropriate formula given the octant. We need to have rem IN [0..twoPI] Now select the appropriate formula given the octant. We get better results if we compute in the degrees domain for a while. We get better results if we compute in the degrees domain for a while. Constants for ArcTan This function is good to 8.7 decimal places and is taken from: Computer Approximations, John F. Hart et.al. pp129(top 2nd col). Helpful internal functions Coefficients for the taylor series for sin and cos. roughly 5.96e-8, the minimum relative error between two different reals RRA: IF radians <= sinEps THEN SinFirstOctant[radians] = radians AND CosFirstOctant[radians] = 1.0 to the precision of our number system. The contribution of the smaller term of two consecutive terms in the Sin or Cos polynomials is down by the square from the larger (and a factor of at least two). So sinCosEps roughly equals 1.726e-4. Testing for this case improves accuracy at small values. This version of Sin is only guaranteed to be accurate within the first octant: [0.0..PI/4], which is approximately [0.0..0.785+]. When radians = PI/4 we have the maximum contribution of the term after the fact9recip term, for a truncation error of roughly 1.76e-9, which is within minEps*Sin[radians] (= 4.2e-8). This version of Cos is only guaranteed to be accurate within the first octant: [0.0..PI/4]. When radians = PI/4 we have the maximum contribution of the terms after the fact8recip term, for a truncation error of roughly 2.46e-8, which is within minEps*Cos[PI/4] (= 4.2e-8). Two constants used to add N*PI with more than normal precision. Let the returned value be R. Then radians <= 0 J radians S xn*twoPI + R & -twoPI <= R <= 0 radians >= 0 J radians S xn*twoPI + R & 0 <= R <= twoPI Let the returned value be R. Then degrees <= 0 J degrees S xn*360 + R & -360 <= R <= 0 degrees >= 0 J degrees S xn*360 + R & 0 <= R <= 360 Э╩┴ШcodeЪЬЩKЪЬ ╧mЬ7ЩBKЪЬ"Щ"KЪЬ#Щ#KЪЬЩKЪЬ&Щ&KЪЬ0Щ0ЧKШЪ╧k Ш KЪЬЮЬ0Ш:KЪЬЮЬ7ШAKШKШЧKЪ╧bЬЮЬЮЬЮЬЮЬ ЮШEШKЪЯЬСЩХKШЧЪЬЩKЪЬЮЬШKЪЬ ЮЬ ШKЪЬ ЮЬШ"KЪЮЬЮЬШKЪЬЮЬЮЬШKЪЬ ЮЬЮЬШKЪЬЮЬШKЪЬ ЮЬ ШKЪЬЮЬШKЪЬЮЬШKЪЬЮЬЮЬШ KЪЬЮЬЮЬШ KЪЬЮЬ ШKЪЬЮЬ ШKЪЬЮЬЮЬШЧKШЪ╧nЬЮЬЮЬЮЬЮЬЮЬШ-ЪЬ%Щ%KЪЬTЩTЧKЪЬЮЬШKЪЬЮЬШKЪЬЮЬШ5KШKЪЬ:Щ:ЪЮЬЮЬЮШKЪЬ ЮЬШKЪЬ ЮЬШKЪЮЬШЧKШKЪЬ╧cШ1KЪЮЬЮЬЮЬШ,KЪЮЬЮЬЮЬШ5KШKЪЬбШ KЪЬбШKШKЪЬб(Ш3KШKШШKЪЬ.Щ.KЪЬЮЬШKШKЪЬ Ш KЪЬ Ш KЪЬ Ш KШЪЬ7Щ7KЪЬ Щ KЪЬ Щ ЧKШЧKШKЪЬЩKЪЬШKЪЬ1Щ1KЪЬШKЪЮЬЮЬЮЬШKШKШЧЪаЬЮЬЮЬ ЮЬЮЬЮЬШ5KЪЬЩKЪЮЬШKШKШЧЪаЬЮЬЮЬЮЬЮЬЮЬШ,KЪЬШKЪЬЮЬШKЪЬЮЬШKЪЬЮЬШ#KЪЮЬЮЬ ШKШKЪЬ:Щ:ЪЮЬЮЬЮШKЪЬ Ш KЪЬ ЮЬШ%KЪЮЬЮЬШKШЧKЪЬ@Щ@KЪЬЮЬШ"KЪЬ%Ш%KЪЬ бШ2KЪЬЮЬЮЬШKШKЪЬ1Щ1KЪЮЬ ЮЬЮЬШ"KШKШKШ KШЪЮЬЮЬ ЮШKЪЬ ЮЬШKШ0KШKЪЮЬЮЬЮЬШKЪЮЬШKШЧKЪЮЬШ KШKШЧЪаЬЮЬЮЬШKЪЬЮЬ.ЮЬЮЬ3ШzKЪЮЬШKШKШЧЪ ЬЮЬЮЬЮ ЬЮЬШ-KЪЬЮЬШ KЪЬЮЬ/Ш9KШKЪЬ╡Щ╡KШЧЪ аЬЮЬЮЬЮЬЮЬШ7KЪЮЬ█Щ▐ЪЮЬЮЬЮШЪЬ Ш KЪЬvЩvKЪЮЬ ЮЬ ШKЪЬШKШЧЪЮЬШ'KЪЬvЩvKЪЬЮЬЮЬШ5ЧЪЮЬШKЪЬЮЬШ+ЪЬЮЬ)ШIKЪЬaЩaЧШ&KЪЬyЩyЧЪЬЮЬЮЬШKЪЬ!Щ!ЧЪЬЮЬЮЬ ЮЬЮЬЮЬШAKЪЬ/Щ/ЧЪЬЮЬЮЬ ЮЬЮЬЮЬШAKЪЬ/Щ/ЧKШЧЧKШKШЧЪЬЮЬЮЬЮЬШ#KЪЮЬЮЬбШ)KЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬбШ)KШKЪЮЬЮЬбШ)KЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬбШ)KШKЪЮЬЮЬбШ(KЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬбШ(KШKЪЮЬЮЬбШ(KЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬШKЪЮЬЮЬбШ'KЪЬШKШЧЪаЬЮЬЮЬЮЬЮЬЮЬШ7KЪЬЩKЪЮЬШKШKШЧЪаЬЮЬЮЬЮЬЮЬЮЬШЩ>KЪЬ@Щ@ЧKЪЬЮЬШЪЮЬЮЬЮЬЮШ"ЪЮЬЮЬЮЬ ШKЪЮЬЮЬШKЪЮЬЮЬЮЬЮЬ ЮЬЮЬ Ш7ЧЧЪЮЬШKЪЮЬЮЬЮЬЮЬ ЮЬЮЬ Ш9KЪЮЬЮЬЮЬЮЬШ7ЧKЪЬЮЬШ ЪЮЬЮШ KЪЬШKЪЬ2Ш2KЪЬ2Ш2KЪЬ3Ш3KЪЮЬШ$ЧKШ KШ9KШKШЧЪа ЬЮЬЮЬЮЬЮЬЮЬШ?KЪЬ(Ш(KЪЬШKШЧЪа ЬЮЬЮЬЮЬЮЬЮЬЮЬШSKЪЮЬЮЬ;ШJKШKШЧЪаЬЮЬЮЬЮЬЮЬЮЬЮЬШUKЪЬЮЬ Ш(ЪЬЮЬЮЬШKЪЮЬЮЬШDЧKЪЮЬШ%KШKШЧЪЬЩKШЩ3KЪЬЮЬЮЬШ0KЪЬЮЬЮЬШ.KЪЬЮЬЮЬШ,KЪЬЮЬЮЬШ*KЪЬЮЬЮЬ Ш(KЪЬЮЬЮЬШ&KЪЬЮЬЮЬ Ш$KЪЬЮЬЮЬШ"KШЧЪ ЬЮЬ ЮЬЮЬЮЬЮЬЮЬШDKЪЬGЩGЧЪЬЮЬШKЪЮЬЮЬЮЬ#ЮЬ╔ЩНЧKШЪ аЬЮЬЮЬЮЬЮЬШ7KЪЬ╣Щ╣ЪЮЬЮЬЮЬЮЬШ4KЪЬЮЬШKЪЮЬQШWKШЧKЪЬШKШЧЪ аЬЮЬЮЬЮЬЮЬШ7KЪЬUЮЬЮЬТЮЬЩСЪЮЬЮЬЮЬЮЬШ0KЪЬЮЬШKЪЮЬEШKKШЧKЪЬШKШЧЪЬ?Щ?KЪЬЮЬ б-ШDKЪЬЮЬбШ'KШЧЪ аЬЮЬЮЬЮЬЮЬШ7ЪЬ"Щ"KЪЬ ЭЬ ЭЬЭЬЩ8KЪЬ ЭЬ ЭЬЭЬЩ7ЧKЪЬЮЬШ&KЪЮЬ!Ш'KШKШЧЪ аЬЮЬЮЬЮЬЮЬШ5ЪЬ"Щ"KЪЬ ЭЬ ЭЬЭЬЩ4KЪЬ ЭЬ ЭЬЭЬЩ3ЧKЪЬЮЬШ&KЪЮЬШKШЧKШЧЪЮЬШKШЧKЪЬЮЬШ0KЪЬЮЬ1ШLKЪЬЮЬШ2KЪЬ"ЮШ'KШNKШQЧЕЧ%▓G┘