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2011-09-02 18:13:33
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module smlbflot; % Basic support for bigfloat arithmetic. % Authors: S.L. Kameny and T. Sasaki. % Modified for binary bigfloat arithmetic by Iain Beckingham and Rainer % Schoepf. % Modified for double precision printing by Herbert Melenk. % Last change made Dec 8, 1992. exports abs!:, bfexplode0, bflerrmsg, bfprin!:, bftrim!:, bfzerop!:, conv!:mt, cut!:ep, cut!:mt, decimal2internal, decprec!:, difference!:, divide!:, equal!:, fl2bf, greaterp!:, incprec!:, leq!:, lessp!:, max!:, max!:, max2!:, min!:, min2!:, minus!:, minusp!:, order!:, plus!:, read!:num, round!:mt, round!:last, times!:; imports aconc, ashift, bfp!:, ceiling, conv!:bf2i, ep!:, eqcar, floor, geq, i2bf!:, leq, lshift, make!:ibf, msd!:, mt!:, neq, normbf, oddintp, preci!:, precision, prin2!*, rerror, retag, reversip; fluid '(!*bfspace !*fullprec !*nat !:prec!: !:bprec!: !:print!-prec!: !:upper!-sci!: !:lower!-sci!:); global '(!!nfpd !!nbfpd bften!* bfz!* fort_exponent); switch bfspace,fullprec; flag('(fort_exponent),'share); !*bfspace := nil; % !*fullprec := t; !:lower!-sci!: := 10; !:upper!-sci!: := 5; symbolic procedure bflerrmsg u; % Revised error message for BFLOAT module, using error, not rederr. error(0,{"Invalid argument to",u}); symbolic procedure bfzerop!: u; % This is possibly too restricted a definition. mt!: u = 0; symbolic procedure fl2bf x; (if zerop x then bfz!* else begin scalar s,r; integer d; if x<0 then <<x := -x; s := t>>; % convert x to an integer equivalent; r := normbf read!:num x; d := ep!: r+msd!: mt!: r; x := x*2.0**-d; x := x + 0.5/2**!:bprec!:; x := fix(x*2**!:bprec!:); return make!:ibf (if s then -x else x, d - !:bprec!:) end) where !:bprec!:=!!nbfpd; symbolic procedure bfprin!: u; % if preci!: u>!!nbfpd then bfprin0 u % else (bfprin0 u where !*bfspace=nil); bfprin0 u; symbolic procedure divide!-by!-power!-of!-ten (x, n); if n < 0 then bflerrmsg 'divide!-by!-power!-of!-ten else << while n > 0 do << if oddintp n then x := normbf divide!: (x, f, !:bprec!:); n := lshift (n, -1); f := normbf cut!:mt (times!: (f, f), !:bprec!:) >>; x >> where f := bften!*; symbolic procedure multiply!-by!-power!-of!-ten (x, n); if n < 0 then bflerrmsg 'multiply!-by!-power!-of!-ten else << while n > 0 do << if oddintp n then x := normbf times!: (x, f); n := lshift (n, -1); f := normbf cut!:mt (times!: (f, f), !:bprec!:) >>; normbf cut!:mt (x, !:bprec!:) >> where f := bften!*; global '(log2of10); symbolic procedure round!:dec (x, p); % % rounds bigfloat x to p decimal places % begin scalar setpr; integer m, ex; if null p then p := if !:print!-prec!: then !:print!-prec!: else !:prec!: - 2 else if p > precision 0 then setpr := precision p; x := round!:dec1 (x,p); m := car x; ex := cdr x; x := i2bf!: m; if ex < 0 then x := divide!-by!-power!-of!-ten (x, -ex) else if ex > 0 then x := multiply!-by!-power!-of!-ten (x, ex); if setpr then precision setpr; return round!:mt (x, ceiling (p * log2of10)) end; symbolic procedure round!:dec1 (x, p); % % rounds bigfloat x to p decimal places % returns pair (m . ex) of mantissa and exponent to base 10, % m having exactly p digits % performs all calculations at at least current precision, % but increases the precision of the calculations to log10(x) % if this is larger % if bfzerop!: x then cdr x else (begin scalar exact, lo, sign; integer ex, k, m, n, l; % % We need to calculate the number k so that 10^(k+1) > |x| >= 10^k % k = floor (log10 |x|) = floor (log2 |x| / log2of10); % We can easily compute n so that 2^(n+1) > |x| >= 2^n, % i.e., n = floor (log2 |x|), since this is just order!:(x). % Since n+1 > log2 |x| >= n, it follows that % floor ((n+1) / log2of10) >= k >= floor (n / log2of10) % I.e., if both bounds agree, we know k, otherwise we have to check. % if mt!: x < 0 then <<sign := t; x := minus!: x>>; n := order!: x; % % The division by log2of10 has to be done with precision larger than % the precision of n. In particular, log2of10 has to be calculated % to a larger precision. Instead of dividing by log2of10, we % multiply by log10of2. % l := msd!: abs n; <<lo := divide!: (!:log2 !:bprec!:, !:log10 !:bprec!:, !:bprec!:); k := conv!:bf2i times!: (i2bf!: n, lo); if k = conv!:bf2i times!: (i2bf!: (n+1), lo) then exact := t>> where !:bprec!: := max (!!nbfpd, l + 7); % % For the following calculation the precision must be increased by % the precision of n. The is necessary to ensure that the mantissa % is calculated correctly for large values of the exponent. This is % due to the fact that if we multiply the number x by 10^n its % precision will be decreased by n. % !:bprec!: := !:bprec!: + l; % % since conv!:bf2i rounds always towards 0, we must correct for n<0 % if n < 0 then k := k - 1; ex := k - p + 1; if ex < 0 then x := multiply!-by!-power!-of!-ten (x, -ex) else if ex > 0 then x := divide!-by!-power!-of!-ten (x, ex); if exact then nil else <<l := length explode conv!:bf2i x; if l < p then <<x := times!: (x, bften!*); ex := ex - 1>> else if l > p then <<x := divide!: (x, bften!*, !:bprec!:); ex := ex + 1>>>>; % % do rounding % x := plus!:(x, bfhalf!*); m := conv!:bf2i x; if length explode m > p then <<m := (m + 5) / 10; ex := ex + 1>>; if sign then m := -m; return (m . ex); end) where !:bprec!: := !:bprec!:; % symbolic procedure internal2decimal (x, p); % % converts bigfloat x to decimal format, with precision p % Result is a pair (m . e), so that x = m*10^e, with % m having exactly p decimal digits. % Calculation is done with the current precision, % but at least with p + 2. % % begin scalar setpr; % if null p then p := if !:print!-prec!: then !:print!-prec!: % else !:prec!: - 2 % else if p > precision 0 then setpr := precision p; % x := round!:dec1 (x,p); % if setpr then precision setpr; % return x % end; symbolic procedure bfprin0 u; begin scalar r; integer m, ex; r := round!:dec1 (u, if !:print!-prec!: then !:print!-prec!: else !:prec!: - 2); m := car r; ex := cdr r; bfprin0x (m, ex) end; symbolic procedure bfprin0x(m,ex); begin scalar lst; integer dotpos; lst := bfexplode0x(m,ex); ex := cadr lst; dotpos := caddr lst; lst := car lst; return bfprin!:lst (lst,ex,dotpos) end; symbolic procedure bfexplode0 u; % returns a list (lst ex dotpos) where % lst = list of characters in mantissa % (ie optional sign and digits) % ex = decimal exponent % dotpos = position of decimal point in lst % (note that the sign is counted) begin scalar r; integer m, ex; r := round!:dec1 (u,if !:print!-prec!: then !:print!-prec!: else !:prec!: - 2); m := car r; ex := cdr r; return bfexplode0x (m, ex) end; symbolic procedure bfexplode0x (m, ex); begin scalar lst, s; integer dotpos, l; if m<0 then <<s := t; m := -m>>; lst := explode m; l := length lst; if ex neq 0 and (l+ex < -!:lower!-sci!: or l+ex > !:upper!-sci!:) then <<dotpos := 1; ex := ex + l - 1>> else <<dotpos := l + ex; ex := 0; if dotpos > l - 1 % % add dotpos - l + 1 zeroes at the end % then lst := nconc!*(lst,nlist('!0,dotpos - l + 1)) else while dotpos < 1 do <<lst := '!0 . lst; dotpos := dotpos + 1>>>>; if s then <<lst := '!- . lst; dotpos := dotpos + 1>>; if null !*fullprec then <<lst := reversip lst; while lst and car lst eq '!0 and length lst > dotpos + 1 do lst := cdr lst; lst := reversip lst>>; return {lst, ex, dotpos} end; symbolic procedure bfprin!:lst (lst, ex, dotpos); begin scalar result,ee,w; integer j; ee:='E; if !*fort and liter(w:=reval fort_exponent) then ee:=w else w:=nil; if car lst eq '!- and dotpos = 1 then <<dotpos := 2; ex := ex - 1>>; if ex neq 0 then if car lst eq '!- then <<ex := ex + dotpos - 2; dotpos := 2>> else <<ex := ex + dotpos - 1; dotpos := 1>> else if dotpos = length lst then dotpos := -1; for each char in lst do << result := char . result; j := j + 1; dotpos := dotpos - 1; if j=5 then <<if !*nat and !*bfspace then result := '! . result; j := 0>>; if dotpos = 0 then <<result := '!. . result; j := j + 1>>; if j=5 then <<if !*nat and !*bfspace then result := '! . result; j := 0>>>>; if ex neq 0 or w then << if not (!*nat and !*bfspace) then result := ee . result else if j=0 then <<result := '! . ee . result; j := 2>> else if j=1 then <<result := '! . ee . '! . result; j := 4>> else if j=2 then <<result := '! . '! . ee . '! . result; j := 0>> else if j=3 then <<result := '! . ee . '! . result; j := 0>> else if j=4 then <<result := '! . ee . '! . result; j := 2>>; lst := if ex > 0 then '!+ . explode ex else explode ex; for each char in lst do << result := char . result; j := j + 1; if j=5 then <<if !*nat and !*bfspace then result := '! . result; j := 0>>>>>>; % if !*nat then for each char in reversip result do prin2!* char else prin2!* compress('!" . reversip('!" . result)) end; symbolic procedure scientific_notation n; begin scalar oldu,oldl; oldu := !:upper!-sci!:; oldl := !:lower!-sci!: + 1; if fixp n then << if n<0 then rerror(arith,1, {"Invalid argument to scientific_notation:",n}); !:lower!-sci!: := n - 1; !:upper!-sci!: := n; >> else if eqcar(n,'list) and length n=3 then if not (fixp cadr n and fixp caddr n) then rerror(arith,2, {"Invalid argument to scientific_notation:",n}) else <<!:upper!-sci!: := cadr n; !:lower!-sci!: := caddr n - 1 >>; return {'list,oldu,oldl} % Return previous range. end; flag('(scientific_notation), 'opfn); symbolic procedure order!: nmbr; % This function counts the order of a number "n". NMBR is a bigfloat % representation of "n". % **** ORDER(n)=k if 2**k <= ABS(n) < 2**(k+1) % **** when n is not 0, and ORDER(0)=0. % if mt!: nmbr = 0 then 0 else preci!: nmbr + ep!: nmbr - 1; symbolic smacro procedure decprec!:(nmbr, k); make!:ibf(ashift(mt!: nmbr,-k), ep!: nmbr + k); symbolic smacro procedure incprec!:(nmbr, k); make!:ibf(ashift(mt!: nmbr,k), ep!: nmbr - k); symbolic procedure conv!:mt(nmbr, k); % This function converts a number "n" to an equivalent number of % binary precision K by rounding "n" or adding "0"s to "n". % NMBR is a binary bigfloat representation of "n". % K is a positive integer. if bfp!: nmbr and fixp k and k > 0 then if (k := preci!: nmbr - k) = 0 then nmbr else if k < 0 then incprec!:(nmbr, -k) else round!:last(decprec!:(nmbr, k - 1)) else bflerrmsg 'conv!:mt; symbolic procedure round!:mt(nmbr, k); % This function rounds a number "n" at the (K+1)th place and returns % an equivalent number of binary precision K if the precision of "n" % is greater than K, else it returns the given number unchanged. % NMBR is a bigfloat representation of "n". K is a positive integer. if bfp!: nmbr and fixp k and k > 0 then if (k := preci!: nmbr - k - 1) < 0 then nmbr else if k = 0 then round!:last nmbr else round!:last decprec!:(nmbr, k) else bflerrmsg 'round!:mt; symbolic procedure round!:ep(nmbr, k); % This function rounds a number "n" and returns an % equivalent number having the exponent K if % the exponent of "n" is less than K, else % it returns the given number unchanged. % NMBR is a BINARY BIG-FLOAT representation of "n". % K is an integer (positive or negative). if bfp!: nmbr and fixp k then if (k := k - 1 - ep!: nmbr) < 0 then nmbr else if k = 0 then round!:last nmbr else round!:last decprec!:(nmbr, k) else bflerrmsg 'round!:ep$ symbolic procedure round!:last nmbr; % This function rounds a number "n" at its last place. % NMBR is a binary bigfloat representation of "n". << if m < 0 then << m := -m; s := t >>; m := if oddintp m then lshift (m, -1) + 1 else lshift (m, -1); if s then m := -m; make!:ibf (m, e) >> where m := mt!: nmbr, e := ep!: nmbr + 1, s := nil; symbolic procedure cut!:mt(nmbr,k); % This function returns a given number "n" unchanged % if its binary precision is not greater than K, else it % cuts off its mantissa at the (K+1)th place and % returns an equivalent number of precision K. % **** CAUTION! No rounding is made. % NMBR is a BINARY BIG-FLOAT representation of "n". % K is a positive integer. if bfp!: nmbr and fixp k and k > 0 then if (k := preci!: nmbr - k) <= 0 then nmbr else decprec!:(nmbr, k) else bflerrmsg 'cut!:mt$ symbolic procedure cut!:ep(nmbr, k); % This function returns a given number "n" unchanged % if its exponent is not less than K, else it % cuts off its mantissa and returns an equivalent % number of exponent K. % **** CAUTION! No rounding is made. % NMBR is a BINARY BIG-FLOAT representation of "n". % K is an integer (positive or negative). if bfp!: nmbr and fixp k then if (k := k - ep!: nmbr) <= 0 then nmbr else decprec!:(nmbr, k) else bflerrmsg 'cut!:ep$ symbolic procedure bftrim!: v; normbf round!:mt(v,!:bprec!: - 3); symbolic procedure decimal2internal (base10,exp10); if exp10 >= 0 then i2bf!: (base10 * 10**exp10) else divide!-by!-power!-of!-ten (i2bf!: base10, -exp10); symbolic procedure read!:num(n); % This function reads a number or a number-like entity N % and constructs a bigfloat representation of it. % N is an integer, a floating-point number, or a string % representing a number. % **** If the system does not accept or may incorrectly % **** accept the floating-point numbers, you can % **** input them as strings such as "1.234E-56", % **** "-78.90 D+12" , "+3456 B -78", or "901/234". % **** A rational number in a string form is converted % **** to a bigfloat of precision !:PREC!: if % **** !:PREC!: is not NIL, else the precision of % **** the result is set 170. % **** Some systems set the maximum size of strings. If % **** you want to input long numbers exceeding % **** such a maximum size, please use READ!:LNUM. if fixp n then make!:ibf(n, 0) else if not(numberp n or stringp n) then bflerrmsg 'read!:num else begin integer j,m,sign; scalar ch,u,v,l,appear!.,appear!/; j := m := 0; sign := 1; u := v := appear!. := appear!/ := nil; l := explode n; loop: ch := car l; if digit ch then << u := ch . u; j := j + 1 >> else if ch eq '!. then << appear!. := t; j := 0 >> else if ch eq '!/ then << appear!/ := t; v := u; u := nil >> else if ch eq '!- then sign := -1 else if ch memq '(!E !D !B !e !d !b) then go to jump; %JBM if l := cdr l then goto loop else goto make; jump: while l := cdr l do <<if digit(ch := car l) or ch eq '!- then v := ch . v >>; l := reverse v; if car l eq '!- then m := - compress cdr l else m:= compress l; make: u := reverse u; v := reverse v; if appear!/ then return conv!:r2bf(make!:ratnum(sign*compress v,compress u), if !:bprec!: then !:bprec!: else 170); if appear!. then j := - j else j := 0; if sign = 1 then u := compress u else u := - compress u; return round!:mt (decimal2internal (u, j + m), !:bprec!:) where !:bprec!: := if !:bprec!: then !:bprec!: else msd!: abs u end; symbolic procedure abs!: nmbr; % This function makes the absolute value of "n". N is a binary % bigfloat representation of "n". if mt!: nmbr > 0 then nmbr else make!:ibf(- mt!: nmbr, ep!: nmbr); symbolic procedure minus!: nmbr; % This function makes the minus number of "n". N is a binary % bigfloat representation of "n". make!:ibf(- mt!: nmbr, ep!: nmbr); symbolic procedure plus!:(n1,n2); begin scalar m1,m2,e1,e2,d; return if (m1 := mt!: n1)=0 then n2 else if (m2 := mt!: n2)=0 then n1 else if (d := (e1 := ep!: n1)-(e2 := ep!: n2))=0 then make!:ibf(m1+m2, e1) else if d>0 then make!:ibf(ashift(m1,d)+m2,e2) else make!:ibf(m1+ashift(m2,-d),e1) end; symbolic procedure difference!:(n1,n2); begin scalar m1,m2,e1,e2,d; return if (m1 := mt!: n1)=0 then minus!: n2 else if (m2 := mt!: n2)=0 then n1 else if (d := (e1 := ep!: n1)-(e2 := ep!: n2))=0 then make!:ibf(m1 - m2, e1) else if d>0 then make!:ibf(ashift(m1,d) - m2,e2) else make!:ibf(m1 - ashift(m2,-d),e1) end; symbolic procedure times!:(n1, n2); % This function calculates the product of "n1" and "n2". % N1 and N2 are bigfloat representations of "n1" and "n2". make!:ibf(mt!: n1 * mt!: n2, ep!: n1 + ep!: n2); symbolic procedure divide!:(n1,n2,k); % This function calculates the quotient of "n1" and "n2", with the % precision K, by rounding the ratio of "n1" and "n2" at the (K+1)th % place. N1 and N2 are bigfloat representations of "n1" and "n2". % K is any positive integer. begin n1 := conv!:mt(n1, k + preci!: n2 + 1); n1 := make!:ibf(mt!: n1 / mt!: n2, ep!: n1 - ep!: n2); return round!:mt(n1, k) end; symbolic procedure max2!:(a,b); % This function returns the larger of "n1" and "n2". % N1 and N2 are bigfloat representations of "n1" and "n2". if greaterp!:(a,b) then a else b; macro procedure max!: x; expand(cdr x,'max2!:); symbolic procedure min2!:(a,b); % This function returns the smaller of "n1" and "n2". % N1 and N2 are binary bigfloat representations of "n1" and "n2". if greaterp!:(a,b) then b else a; macro procedure min!: x; expand(cdr x,'min2!:); symbolic procedure greaterp!:(a,b); % this function calculates the a > b, but avoids % generating large numbers if magnitude difference is large. if ep!: a=ep!: b then mt!: a>mt!: b else (((if d=0 then ma>mb else ((if d>p2 then ma>0 else if d<-p2 then mb<0 else if d>0 then ashift(ma,d)>mb else ma>ashift(mb,-d)) where p2=2*!:bprec!:)) where d=ep!: a - ep!: b, ma=mt!: a, mb=mt!: b) where a= normbf a, b=normbf b); symbolic procedure equal!:(a,b); %tests bfloats for a=b rapidly without generating digits. %SK zerop mt!: a and zerop mt!: b or ep!:(a := normbf a)=ep!:(b := normbf b) and mt!: a=mt!: b; symbolic procedure lessp!:(n1, n2); % This function returns T if "n1" < "n2" else returns NIL. % N1 and N2 are bigfloat representations of "n1" and "n2". greaterp!:(n2, n1); symbolic procedure leq!:(n1, n2); % This function returns T if "n1" <= "n2" else returns NIL. % N1 and N2 are bigfloat representations of "n1" and "n2". not greaterp!:(n1, n2); symbolic procedure minusp!: x; % This function returns T if "x"<0 else returns NIL. % X is any Lisp entity. bfp!: x and mt!: x < 0; symbolic procedure make!:ratnum(nm,dn); % This function constructs an internal representation % of a rational number composed of the numerator % NM and the denominator DN. % NM and DN are any integers (positive or negative). % **** Four routines in this section are temporary. % **** That is, if your system has own routines % **** for rational number arithmetic, you can % **** accommodate our system to yours only by % **** redefining these four routines. if zerop dn then rerror(arith,3,"Zero divisor in make:ratnum") else if dn > 0 then '!:ratnum!: . (nm . dn) else '!:ratnum!: . (-nm . -dn); symbolic procedure ratnump!:(x); % This function returns T if X is a rational number % representation, else NIL. % X is any Lisp entity. eqcar(x,'!:ratnum!:); %JBM Change to EQCAR. symbolic smacro procedure numr!: rnmbr; % This function selects the numerator of a rational number "n". % RNMBR is a rational number representation of "n". cadr rnmbr; symbolic smacro procedure denm!: rnmbr; % This function selects the denominator of a rational number "n". % RNMBR is a rational number representation of "n". cddr rnmbr; symbolic procedure conv!:r2bf(rnmbr,k); % This function converts a rational number RNMBR to a bigfloat of % precision K, i.e., a bigfloat representation with a given % precision. RNMBR is a rational number representation. K is a % positive integer. if ratnump!: rnmbr and fixp k and k > 0 then divide!:(make!:ibf( numr!: rnmbr, 0), make!:ibf( denm!: rnmbr, 0),k) else bflerrmsg 'conv!:r2bf; endmodule; end;