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Issue 7652: Merge C version of decimal into py3k.
Patch Set: Created 7 years, 6 months ago
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1
2
3 (* Copyright (c) 2011 Stefan Krah. All rights reserved. *)
4
5
6 ========================================================================
7 Calculate (a * b) % p using the 80-bit x87 FPU
8 ========================================================================
9
10 A description of the algorithm can be found in the apfloat manual by
11 Tommila [1].
12
13 The proof follows an argument made by Granlund/Montgomery in [2].
14
15
16 Definitions and assumptions:
17 ----------------------------
18
19 The 80-bit extended precision format uses 64 bits for the significand:
20
21 (1) F = 64
22
23 The modulus is prime and less than 2**31:
24
25 (2) 2 <= p < 2**31
26
27 The factors are less than p:
28
29 (3) 0 <= a < p
30 (4) 0 <= b < p
31
32 The product a * b is less than 2**62 and is thus exact in 64 bits:
33
34 (5) n = a * b
35
36 The product can be represented in terms of quotient and remainder:
37
38 (6) n = q * p + r
39
40 Using (3), (4) and the fact that p is prime, the remainder is always
41 greater than zero:
42
43 (7) 0 <= q < p /\ 1 <= r < p
44
45
46 Strategy:
47 ---------
48
49 Precalculate the 80-bit long double inverse of p, with a maximum
50 relative error of 2**(1-F):
51
52 (8) pinv = (long double)1.0 / p
53
54 Calculate an estimate for q = floor(n/p). The multiplication has another
55 maximum relative error of 2**(1-F):
56
57 (9) qest = n * pinv
58
59 If we can show that q < qest < q+1, then trunc(qest) = q. It is then
60 easy to recover the remainder r. The complete algorithm is:
61
62 a) Set the control word to 64-bit precision and truncation mode.
63
64 b) n = a * b # Calculate exact product.
65
66 c) qest = n * pinv # Calculate estimate for the quotient.
67
68 d) q = (qest+2**63)-2**63 # Truncate qest to the exact quotient.
69
70 f) r = n - q * p # Calculate remainder.
71
72
73 Proof for q < qest < q+1:
74 -------------------------
75
76 Using the cumulative error, the error bounds for qest are:
77
78 n n * (1 + 2**(1-F))**2
79 (9) --------------------- <= qest <= ---------------------
80 p * (1 + 2**(1-F))**2 p
81
82
83 Lemma 1:
84 --------
85 n q * p + r
86 (10) q < --------------------- = ---------------------
87 p * (1 + 2**(1-F))**2 p * (1 + 2**(1-F))**2
88
89
90 Proof:
91 ~~~~~~
92
93 (I) q * p * (1 + 2**(1-F))**2 < q * p + r
94
95 (II) q * p * 2**(2-F) + q * p * 2**(2-2*F) < r
96
97 Using (1) and (7), it is sufficient to show that:
98
99 (III) q * p * 2**(-62) + q * p * 2**(-126) < 1 <= r
100
101 (III) can easily be verified by substituting the largest possible
102 values p = 2**31-1 and q = 2**31-2.
103
104 The critical cases occur when r = 1, n = m * p + 1. These cases
105 can be exhaustively verified with a test program.
106
107
108 Lemma 2:
109 --------
110
111 n * (1 + 2**(1-F))**2 (q * p + r) * (1 + 2**(1-F))**2
112 (11) --------------------- = ------------------------------- < q + 1
113 p p
114
115 Proof:
116 ~~~~~~
117
118 (I) (q * p + r) + (q * p + r) * 2**(2-F) + (q * p + r) * 2**(2-2*F) < q * p + p
119
120 (II) (q * p + r) * 2**(2-F) + (q * p + r) * 2**(2-2*F) < p - r
121
122 Using (1) and (7), it is sufficient to show that:
123
124 (III) (q * p + r) * 2**(-62) + (q * p + r) * 2**(-126) < 1 <= p - r
125
126 (III) can easily be verified by substituting the largest possible
127 values p = 2**31-1, q = 2**31-2 and r = 2**31-2.
128
129 The critical cases occur when r = (p - 1), n = m * p - 1. These cases
130 can be exhaustively verified with a test program.
131
132
133 [1] http://www.apfloat.org/apfloat/2.40/apfloat.pdf
134 [2] http://gmplib.org/~tege/divcnst-pldi94.pdf
135 [Section 7: "Use of floating point"]
136
137
138
139 (* Coq proof for (10) and (11) *)
140
141 Require Import ZArith.
142 Require Import QArith.
143 Require Import Qpower.
144 Require Import Qabs.
145 Require Import Psatz.
146
147 Open Scope Q_scope.
148
149
150 Ltac qreduce T :=
151 rewrite <- (Qred_correct (T)); simpl (Qred (T)).
152
153 Theorem Qlt_move_right :
154 forall x y z:Q, x + z < y <-> x < y - z.
155 Proof.
156 intros.
157 split.
158 intros.
159 psatzl Q.
160 intros.
161 psatzl Q.
162 Qed.
163
164 Theorem Qlt_mult_by_z :
165 forall x y z:Q, 0 < z -> (x < y <-> x * z < y * z).
166 Proof.
167 intros.
168 split.
169 intros.
170 apply Qmult_lt_compat_r. trivial. trivial.
171 intros.
172 rewrite <- (Qdiv_mult_l x z). rewrite <- (Qdiv_mult_l y z).
173 apply Qmult_lt_compat_r.
174 apply Qlt_shift_inv_l.
175 trivial. psatzl Q. trivial. psatzl Q. psatzl Q.
176 Qed.
177
178 Theorem Qle_mult_quad :
179 forall (a b c d:Q),
180 0 <= a -> a <= c ->
181 0 <= b -> b <= d ->
182 a * b <= c * d.
183 intros.
184 psatz Q.
185 Qed.
186
187
188 Theorem q_lt_qest:
189 forall (p q r:Q),
190 (0 < p) -> (p <= (2#1)^31 - 1) ->
191 (0 <= q) -> (q <= p - 1) ->
192 (1 <= r) -> (r <= p - 1) ->
193 q < (q * p + r) / (p * (1 + (2#1)^(-63))^2).
194 Proof.
195 intros.
196 rewrite Qlt_mult_by_z with (z := (p * (1 + (2#1)^(-63))^2)).
197
198 unfold Qdiv.
199 rewrite <- Qmult_assoc.
200 rewrite (Qmult_comm (/ (p * (1 + (2 # 1) ^ (-63)) ^ 2)) (p * (1 + (2 # 1) ^ (- 63)) ^ 2)).
201 rewrite Qmult_inv_r.
202 rewrite Qmult_1_r.
203
204 assert (q * (p * (1 + (2 # 1) ^ (-63)) ^ 2) == q * p + (q * p) * ((2 # 1) ^ (- 62) + (2 # 1) ^ (-126))).
205 qreduce ((1 + (2 # 1) ^ (-63)) ^ 2).
206 qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
207 ring_simplify.
208 reflexivity.
209 rewrite H5.
210
211 rewrite Qplus_comm.
212 rewrite Qlt_move_right.
213 ring_simplify (q * p + r - q * p).
214 qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
215
216 apply Qlt_le_trans with (y := 1).
217 rewrite Qlt_mult_by_z with (z := 85070591730234615865843651857942052864 # 1844 6744073709551617).
218 ring_simplify.
219
220 apply Qle_lt_trans with (y := ((2 # 1) ^ 31 - (2#1)) * ((2 # 1) ^ 31 - 1)).
221 apply Qle_mult_quad.
222 assumption. psatzl Q. psatzl Q. psatzl Q. psatzl Q. psatzl Q. assumption. psat zl Q. psatzl Q.
223 Qed.
224
225 Theorem qest_lt_qplus1:
226 forall (p q r:Q),
227 (0 < p) -> (p <= (2#1)^31 - 1) ->
228 (0 <= q) -> (q <= p - 1) ->
229 (1 <= r) -> (r <= p - 1) ->
230 ((q * p + r) * (1 + (2#1)^(-63))^2) / p < q + 1.
231 Proof.
232 intros.
233 rewrite Qlt_mult_by_z with (z := p).
234
235 unfold Qdiv.
236 rewrite <- Qmult_assoc.
237 rewrite (Qmult_comm (/ p) p).
238 rewrite Qmult_inv_r.
239 rewrite Qmult_1_r.
240
241 assert ((q * p + r) * (1 + (2 # 1) ^ (-63)) ^ 2 == q * p + r + (q * p + r) * ( (2 # 1) ^ (-62) + (2 # 1) ^ (-126))).
242 qreduce ((1 + (2 # 1) ^ (-63)) ^ 2).
243 qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
244 ring_simplify. reflexivity.
245 rewrite H5.
246
247 rewrite <- Qplus_assoc. rewrite <- Qplus_comm. rewrite Qlt_move_right.
248 ring_simplify ((q + 1) * p - q * p).
249
250 rewrite <- Qplus_comm. rewrite Qlt_move_right.
251
252 apply Qlt_le_trans with (y := 1).
253 qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
254
255 rewrite Qlt_mult_by_z with (z := 85070591730234615865843651857942052864 # 1844 6744073709551617).
256 ring_simplify.
257
258 ring_simplify in H0.
259 apply Qle_lt_trans with (y := (2147483646 # 1) * (2147483647 # 1) + (214748364 6 # 1)).
260
261 apply Qplus_le_compat.
262 apply Qle_mult_quad.
263 assumption. psatzl Q. auto with qarith. assumption. psatzl Q.
264 auto with qarith. auto with qarith.
265 psatzl Q. psatzl Q. assumption.
266 Qed.
267
268
269
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