summaryrefslogtreecommitdiff
path: root/apps/mochiweb/src/mochinum.erl
blob: 6a866042bb115f200b732da70c2a9b898a699083 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
%% @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.