# mochi/mochiweb

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 %% @copyright 2007 Mochi Media, Inc. %% @author Bob Ippolito %% @doc Useful numeric algorithms for floats that cover some deficiencies %% in the math module. More interesting is digits/1, which implements %% the algorithm from: %% http://www.cs.indiana.edu/~burger/fp/index.html %% See also "Printing Floating-Point Numbers Quickly and Accurately" %% in Proceedings of the SIGPLAN '96 Conference on Programming Language %% Design and Implementation. -module(mochinum). -author("Bob Ippolito "). -export([digits/1, frexp/1, int_pow/2, int_ceil/1]). %% IEEE 754 Float exponent bias -define(FLOAT_BIAS, 1022). -define(MIN_EXP, -1074). -define(BIG_POW, 4503599627370496). %% External API %% @spec digits(number()) -> string() %% @doc Returns a string that accurately represents the given integer or float %% using a conservative amount of digits. Great for generating %% human-readable output, or compact ASCII serializations for floats. digits(N) when is_integer(N) -> integer_to_list(N); digits(0.0) -> "0.0"; digits(Float) -> {Frac1, Exp1} = frexp_int(Float), [Place0 | Digits0] = digits1(Float, Exp1, Frac1), {Place, Digits} = transform_digits(Place0, Digits0), R = insert_decimal(Place, Digits), case Float < 0 of true -> [\$- | R]; _ -> R end. %% @spec frexp(F::float()) -> {Frac::float(), Exp::float()} %% @doc Return the fractional and exponent part of an IEEE 754 double, %% equivalent to the libc function of the same name. %% F = Frac * pow(2, Exp). frexp(F) -> frexp1(unpack(F)). %% @spec int_pow(X::integer(), N::integer()) -> Y::integer() %% @doc Moderately efficient way to exponentiate integers. %% int_pow(10, 2) = 100. int_pow(_X, 0) -> 1; int_pow(X, N) when N > 0 -> int_pow(X, N, 1). %% @spec int_ceil(F::float()) -> integer() %% @doc Return the ceiling of F as an integer. The ceiling is defined as %% F when F == trunc(F); %% trunc(F) when F < 0; %% trunc(F) + 1 when F > 0. int_ceil(X) -> T = trunc(X), case (X - T) of Pos when Pos > 0 -> T + 1; _ -> T end. %% Internal API int_pow(X, N, R) when N < 2 -> R * X; int_pow(X, N, R) -> int_pow(X * X, N bsr 1, case N band 1 of 1 -> R * X; 0 -> R end). insert_decimal(0, S) -> "0." ++ S; insert_decimal(Place, S) when Place > 0 -> L = length(S), case Place - L of 0 -> S ++ ".0"; N when N < 0 -> {S0, S1} = lists:split(L + N, S), S0 ++ "." ++ S1; N when N < 6 -> %% More places than digits S ++ lists:duplicate(N, \$0) ++ ".0"; _ -> insert_decimal_exp(Place, S) end; insert_decimal(Place, S) when Place > -6 -> "0." ++ lists:duplicate(abs(Place), \$0) ++ S; insert_decimal(Place, S) -> insert_decimal_exp(Place, S). insert_decimal_exp(Place, S) -> [C | S0] = S, S1 = case S0 of [] -> "0"; _ -> S0 end, Exp = case Place < 0 of true -> "e-"; false -> "e+" end, [C] ++ "." ++ S1 ++ Exp ++ integer_to_list(abs(Place - 1)). digits1(Float, Exp, Frac) -> Round = ((Frac band 1) =:= 0), case Exp >= 0 of true -> BExp = 1 bsl Exp, case (Frac =/= ?BIG_POW) of true -> scale((Frac * BExp * 2), 2, BExp, BExp, Round, Round, Float); false -> scale((Frac * BExp * 4), 4, (BExp * 2), BExp, Round, Round, Float) end; false -> case (Exp =:= ?MIN_EXP) orelse (Frac =/= ?BIG_POW) of true -> scale((Frac * 2), 1 bsl (1 - Exp), 1, 1, Round, Round, Float); false -> scale((Frac * 4), 1 bsl (2 - Exp), 2, 1, Round, Round, Float) end end. scale(R, S, MPlus, MMinus, LowOk, HighOk, Float) -> Est = int_ceil(math:log10(abs(Float)) - 1.0e-10), %% Note that the scheme implementation uses a 326 element look-up table %% for int_pow(10, N) where we do not. case Est >= 0 of true -> fixup(R, S * int_pow(10, Est), MPlus, MMinus, Est, LowOk, HighOk); false -> Scale = int_pow(10, -Est), fixup(R * Scale, S, MPlus * Scale, MMinus * Scale, Est, LowOk, HighOk) end. fixup(R, S, MPlus, MMinus, K, LowOk, HighOk) -> TooLow = case HighOk of true -> (R + MPlus) >= S; false -> (R + MPlus) > S end, case TooLow of true -> [(K + 1) | generate(R, S, MPlus, MMinus, LowOk, HighOk)]; false -> [K | generate(R * 10, S, MPlus * 10, MMinus * 10, LowOk, HighOk)] end. generate(R0, S, MPlus, MMinus, LowOk, HighOk) -> D = R0 div S, R = R0 rem S, TC1 = case LowOk of true -> R =< MMinus; false -> R < MMinus end, TC2 = case HighOk of true -> (R + MPlus) >= S; false -> (R + MPlus) > S end, case TC1 of false -> case TC2 of false -> [D | generate(R * 10, S, MPlus * 10, MMinus * 10, LowOk, HighOk)]; true -> [D + 1] end; true -> case TC2 of false -> [D]; true -> case R * 2 < S of true -> [D]; false -> [D + 1] end end end. unpack(Float) -> <> = <>, {Sign, Exp, Frac}. frexp1({_Sign, 0, 0}) -> {0.0, 0}; frexp1({Sign, 0, Frac}) -> Exp = log2floor(Frac), <> = <>, {Frac1, -(?FLOAT_BIAS) - 52 + Exp}; frexp1({Sign, Exp, Frac}) -> <> = <>, {Frac1, Exp - ?FLOAT_BIAS}. log2floor(Int) -> log2floor(Int, 0). log2floor(0, N) -> N; log2floor(Int, N) -> log2floor(Int bsr 1, 1 + N). transform_digits(Place, [0 | Rest]) -> transform_digits(Place, Rest); transform_digits(Place, Digits) -> {Place, [\$0 + D || D <- Digits]}. frexp_int(F) -> case unpack(F) of {_Sign, 0, Frac} -> {Frac, ?MIN_EXP}; {_Sign, Exp, Frac} -> {Frac + (1 bsl 52), Exp - 53 - ?FLOAT_BIAS} end. %% %% Tests %% -ifdef(TEST). -include_lib("eunit/include/eunit.hrl"). int_ceil_test() -> ?assertEqual(1, int_ceil(0.0001)), ?assertEqual(0, int_ceil(0.0)), ?assertEqual(1, int_ceil(0.99)), ?assertEqual(1, int_ceil(1.0)), ?assertEqual(-1, int_ceil(-1.5)), ?assertEqual(-2, int_ceil(-2.0)), ok. int_pow_test() -> ?assertEqual(1, int_pow(1, 1)), ?assertEqual(1, int_pow(1, 0)), ?assertEqual(1, int_pow(10, 0)), ?assertEqual(10, int_pow(10, 1)), ?assertEqual(100, int_pow(10, 2)), ?assertEqual(1000, int_pow(10, 3)), ok. digits_test() -> ?assertEqual("0", digits(0)), ?assertEqual("0.0", digits(0.0)), ?assertEqual("1.0", digits(1.0)), ?assertEqual("-1.0", digits(-1.0)), ?assertEqual("0.1", digits(0.1)), ?assertEqual("0.01", digits(0.01)), ?assertEqual("0.001", digits(0.001)), ?assertEqual("1.0e+6", digits(1000000.0)), ?assertEqual("0.5", digits(0.5)), ?assertEqual("4503599627370496.0", digits(4503599627370496.0)), %% small denormalized number %% 4.94065645841246544177e-324 =:= 5.0e-324 <> = <<0,0,0,0,0,0,0,1>>, ?assertEqual("5.0e-324", digits(SmallDenorm)), ?assertEqual(SmallDenorm, list_to_float(digits(SmallDenorm))), %% large denormalized number %% 2.22507385850720088902e-308 <> = <<0,15,255,255,255,255,255,255>>, ?assertEqual("2.225073858507201e-308", digits(BigDenorm)), ?assertEqual(BigDenorm, list_to_float(digits(BigDenorm))), %% small normalized number %% 2.22507385850720138309e-308 <> = <<0,16,0,0,0,0,0,0>>, ?assertEqual("2.2250738585072014e-308", digits(SmallNorm)), ?assertEqual(SmallNorm, list_to_float(digits(SmallNorm))), %% large normalized number %% 1.79769313486231570815e+308 <> = <<127,239,255,255,255,255,255,255>>, ?assertEqual("1.7976931348623157e+308", digits(LargeNorm)), ?assertEqual(LargeNorm, list_to_float(digits(LargeNorm))), %% issue #10 - mochinum:frexp(math:pow(2, -1074)). ?assertEqual("5.0e-324", digits(math:pow(2, -1074))), ok. frexp_test() -> %% zero ?assertEqual({0.0, 0}, frexp(0.0)), %% one ?assertEqual({0.5, 1}, frexp(1.0)), %% negative one ?assertEqual({-0.5, 1}, frexp(-1.0)), %% small denormalized number %% 4.94065645841246544177e-324 <> = <<0,0,0,0,0,0,0,1>>, ?assertEqual({0.5, -1073}, frexp(SmallDenorm)), %% large denormalized number %% 2.22507385850720088902e-308 <> = <<0,15,255,255,255,255,255,255>>, ?assertEqual( {0.99999999999999978, -1022}, frexp(BigDenorm)), %% small normalized number %% 2.22507385850720138309e-308 <> = <<0,16,0,0,0,0,0,0>>, ?assertEqual({0.5, -1021}, frexp(SmallNorm)), %% large normalized number %% 1.79769313486231570815e+308 <> = <<127,239,255,255,255,255,255,255>>, ?assertEqual( {0.99999999999999989, 1024}, frexp(LargeNorm)), %% issue #10 - mochinum:frexp(math:pow(2, -1074)). ?assertEqual( {0.5, -1073}, frexp(math:pow(2, -1074))), ok. -endif.
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