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-%% @copyright 2007 Mochi Media, Inc.
-%% @author Bob Ippolito <bob@mochimedia.com>
-
-%% @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 <bob@mochimedia.com>").
--export([digits/1, frexp/1, int_pow/2, int_ceil/1, test/0]).
-
-%% 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) ->
- {Frac, Exp} = frexp(Float),
- Exp1 = Exp - 53,
- Frac1 = trunc(abs(Frac) * (1 bsl 53)),
- [Place | Digits] = digits1(Float, Exp1, Frac1),
- R = insert_decimal(Place, [$0 + D || D <- Digits]),
- case Float < 0 of
- true ->
- [$- | 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 &lt; 0;
-%% trunc(F) + 1 when F &gt; 0.
-int_ceil(X) ->
- T = trunc(X),
- case (X - T) of
- Neg when Neg < 0 -> T;
- 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:1, Exp:11, Frac:52>> = <<Float:64/float>>,
- {Sign, Exp, Frac}.
-
-frexp1({_Sign, 0, 0}) ->
- {0.0, 0};
-frexp1({Sign, 0, Frac}) ->
- Exp = log2floor(Frac),
- <<Frac1:64/float>> = <<Sign:1, ?FLOAT_BIAS:11, (Frac-1):52>>,
- {Frac1, -(?FLOAT_BIAS) - 52 + Exp};
-frexp1({Sign, Exp, Frac}) ->
- <<Frac1:64/float>> = <<Sign:1, ?FLOAT_BIAS:11, Frac:52>>,
- {Frac1, Exp - ?FLOAT_BIAS}.
-
-log2floor(Int) ->
- log2floor(Int, 0).
-
-log2floor(0, N) ->
- N;
-log2floor(Int, N) ->
- log2floor(Int bsr 1, 1 + N).
-
-
-test() ->
- ok = test_frexp(),
- ok = test_int_ceil(),
- ok = test_int_pow(),
- ok = test_digits(),
- ok.
-
-test_int_ceil() ->
- 1 = int_ceil(0.0001),
- 0 = int_ceil(0.0),
- 1 = int_ceil(0.99),
- 1 = int_ceil(1.0),
- -1 = int_ceil(-1.5),
- -2 = int_ceil(-2.0),
- ok.
-
-test_int_pow() ->
- 1 = int_pow(1, 1),
- 1 = int_pow(1, 0),
- 1 = int_pow(10, 0),
- 10 = int_pow(10, 1),
- 100 = int_pow(10, 2),
- 1000 = int_pow(10, 3),
- ok.
-
-test_digits() ->
- "0" = digits(0),
- "0.0" = digits(0.0),
- "1.0" = digits(1.0),
- "-1.0" = digits(-1.0),
- "0.1" = digits(0.1),
- "0.01" = digits(0.01),
- "0.001" = digits(0.001),
- ok.
-
-test_frexp() ->
- %% zero
- {0.0, 0} = frexp(0.0),
- %% one
- {0.5, 1} = frexp(1.0),
- %% negative one
- {-0.5, 1} = frexp(-1.0),
- %% small denormalized number
- %% 4.94065645841246544177e-324
- <<SmallDenorm/float>> = <<0,0,0,0,0,0,0,1>>,
- {0.5, -1073} = frexp(SmallDenorm),
- %% large denormalized number
- %% 2.22507385850720088902e-308
- <<BigDenorm/float>> = <<0,15,255,255,255,255,255,255>>,
- {0.99999999999999978, -1022} = frexp(BigDenorm),
- %% small normalized number
- %% 2.22507385850720138309e-308
- <<SmallNorm/float>> = <<0,16,0,0,0,0,0,0>>,
- {0.5, -1021} = frexp(SmallNorm),
- %% large normalized number
- %% 1.79769313486231570815e+308
- <<LargeNorm/float>> = <<127,239,255,255,255,255,255,255>>,
- {0.99999999999999989, 1024} = frexp(LargeNorm),
- ok.
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