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+
+
+(* Copyright (c) 2011 Stefan Krah. All rights reserved. *)
+
+
+========================================================================
+ Calculate (a * b) % p using the 80-bit x87 FPU
+========================================================================
+
+A description of the algorithm can be found in the apfloat manual by
+Tommila [1].
+
+The proof follows an argument made by Granlund/Montgomery in [2].
+
+
+Definitions and assumptions:
+----------------------------
+
+The 80-bit extended precision format uses 64 bits for the significand:
+
+ (1) F = 64
+
+The modulus is prime and less than 2**31:
+
+ (2) 2 <= p < 2**31
+
+The factors are less than p:
+
+ (3) 0 <= a < p
+ (4) 0 <= b < p
+
+The product a * b is less than 2**62 and is thus exact in 64 bits:
+
+ (5) n = a * b
+
+The product can be represented in terms of quotient and remainder:
+
+ (6) n = q * p + r
+
+Using (3), (4) and the fact that p is prime, the remainder is always
+greater than zero:
+
+ (7) 0 <= q < p /\ 1 <= r < p
+
+
+Strategy:
+---------
+
+Precalculate the 80-bit long double inverse of p, with a maximum
+relative error of 2**(1-F):
+
+ (8) pinv = (long double)1.0 / p
+
+Calculate an estimate for q = floor(n/p). The multiplication has another
+maximum relative error of 2**(1-F):
+
+ (9) qest = n * pinv
+
+If we can show that q < qest < q+1, then trunc(qest) = q. It is then
+easy to recover the remainder r. The complete algorithm is:
+
+ a) Set the control word to 64-bit precision and truncation mode.
+
+ b) n = a * b # Calculate exact product.
+
+ c) qest = n * pinv # Calculate estimate for the quotient.
+
+ d) q = (qest+2**63)-2**63 # Truncate qest to the exact quotient.
+
+ f) r = n - q * p # Calculate remainder.
+
+
+Proof for q < qest < q+1:
+-------------------------
+
+Using the cumulative error, the error bounds for qest are:
+
+ n n * (1 + 2**(1-F))**2
+ (9) --------------------- <= qest <= ---------------------
+ p * (1 + 2**(1-F))**2 p
+
+
+ Lemma 1:
+ --------
+ n q * p + r
+ (10) q < --------------------- = ---------------------
+ p * (1 + 2**(1-F))**2 p * (1 + 2**(1-F))**2
+
+
+ Proof:
+ ~~~~~~
+
+ (I) q * p * (1 + 2**(1-F))**2 < q * p + r
+
+ (II) q * p * 2**(2-F) + q * p * 2**(2-2*F) < r
+
+ Using (1) and (7), it is sufficient to show that:
+
+ (III) q * p * 2**(-62) + q * p * 2**(-126) < 1 <= r
+
+ (III) can easily be verified by substituting the largest possible
+ values p = 2**31-1 and q = 2**31-2.
+
+ The critical cases occur when r = 1, n = m * p + 1. These cases
+ can be exhaustively verified with a test program.
+
+
+ Lemma 2:
+ --------
+
+ n * (1 + 2**(1-F))**2 (q * p + r) * (1 + 2**(1-F))**2
+ (11) --------------------- = ------------------------------- < q + 1
+ p p
+
+ Proof:
+ ~~~~~~
+
+ (I) (q * p + r) + (q * p + r) * 2**(2-F) + (q * p + r) * 2**(2-2*F) < q * p + p
+
+ (II) (q * p + r) * 2**(2-F) + (q * p + r) * 2**(2-2*F) < p - r
+
+ Using (1) and (7), it is sufficient to show that:
+
+ (III) (q * p + r) * 2**(-62) + (q * p + r) * 2**(-126) < 1 <= p - r
+
+ (III) can easily be verified by substituting the largest possible
+ values p = 2**31-1, q = 2**31-2 and r = 2**31-2.
+
+ The critical cases occur when r = (p - 1), n = m * p - 1. These cases
+ can be exhaustively verified with a test program.
+
+
+[1] http://www.apfloat.org/apfloat/2.40/apfloat.pdf
+[2] http://gmplib.org/~tege/divcnst-pldi94.pdf
+ [Section 7: "Use of floating point"]
+
+
+
+(* Coq proof for (10) and (11) *)
+
+Require Import ZArith.
+Require Import QArith.
+Require Import Qpower.
+Require Import Qabs.
+Require Import Psatz.
+
+Open Scope Q_scope.
+
+
+Ltac qreduce T :=
+ rewrite <- (Qred_correct (T)); simpl (Qred (T)).
+
+Theorem Qlt_move_right :
+ forall x y z:Q, x + z < y <-> x < y - z.
+Proof.
+ intros.
+ split.
+ intros.
+ psatzl Q.
+ intros.
+ psatzl Q.
+Qed.
+
+Theorem Qlt_mult_by_z :
+ forall x y z:Q, 0 < z -> (x < y <-> x * z < y * z).
+Proof.
+ intros.
+ split.
+ intros.
+ apply Qmult_lt_compat_r. trivial. trivial.
+ intros.
+ rewrite <- (Qdiv_mult_l x z). rewrite <- (Qdiv_mult_l y z).
+ apply Qmult_lt_compat_r.
+ apply Qlt_shift_inv_l.
+ trivial. psatzl Q. trivial. psatzl Q. psatzl Q.
+Qed.
+
+Theorem Qle_mult_quad :
+ forall (a b c d:Q),
+ 0 <= a -> a <= c ->
+ 0 <= b -> b <= d ->
+ a * b <= c * d.
+ intros.
+ psatz Q.
+Qed.
+
+
+Theorem q_lt_qest:
+ forall (p q r:Q),
+ (0 < p) -> (p <= (2#1)^31 - 1) ->
+ (0 <= q) -> (q <= p - 1) ->
+ (1 <= r) -> (r <= p - 1) ->
+ q < (q * p + r) / (p * (1 + (2#1)^(-63))^2).
+Proof.
+ intros.
+ rewrite Qlt_mult_by_z with (z := (p * (1 + (2#1)^(-63))^2)).
+
+ unfold Qdiv.
+ rewrite <- Qmult_assoc.
+ rewrite (Qmult_comm (/ (p * (1 + (2 # 1) ^ (-63)) ^ 2)) (p * (1 + (2 # 1) ^ (-63)) ^ 2)).
+ rewrite Qmult_inv_r.
+ rewrite Qmult_1_r.
+
+ assert (q * (p * (1 + (2 # 1) ^ (-63)) ^ 2) == q * p + (q * p) * ((2 # 1) ^ (-62) + (2 # 1) ^ (-126))).
+ qreduce ((1 + (2 # 1) ^ (-63)) ^ 2).
+ qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
+ ring_simplify.
+ reflexivity.
+ rewrite H5.
+
+ rewrite Qplus_comm.
+ rewrite Qlt_move_right.
+ ring_simplify (q * p + r - q * p).
+ qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
+
+ apply Qlt_le_trans with (y := 1).
+ rewrite Qlt_mult_by_z with (z := 85070591730234615865843651857942052864 # 18446744073709551617).
+ ring_simplify.
+
+ apply Qle_lt_trans with (y := ((2 # 1) ^ 31 - (2#1)) * ((2 # 1) ^ 31 - 1)).
+ apply Qle_mult_quad.
+ assumption. psatzl Q. psatzl Q. psatzl Q. psatzl Q. psatzl Q. assumption. psatzl Q. psatzl Q.
+Qed.
+
+Theorem qest_lt_qplus1:
+ forall (p q r:Q),
+ (0 < p) -> (p <= (2#1)^31 - 1) ->
+ (0 <= q) -> (q <= p - 1) ->
+ (1 <= r) -> (r <= p - 1) ->
+ ((q * p + r) * (1 + (2#1)^(-63))^2) / p < q + 1.
+Proof.
+ intros.
+ rewrite Qlt_mult_by_z with (z := p).
+
+ unfold Qdiv.
+ rewrite <- Qmult_assoc.
+ rewrite (Qmult_comm (/ p) p).
+ rewrite Qmult_inv_r.
+ rewrite Qmult_1_r.
+
+ assert ((q * p + r) * (1 + (2 # 1) ^ (-63)) ^ 2 == q * p + r + (q * p + r) * ((2 # 1) ^ (-62) + (2 # 1) ^ (-126))).
+ qreduce ((1 + (2 # 1) ^ (-63)) ^ 2).
+ qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
+ ring_simplify. reflexivity.
+ rewrite H5.
+
+ rewrite <- Qplus_assoc. rewrite <- Qplus_comm. rewrite Qlt_move_right.
+ ring_simplify ((q + 1) * p - q * p).
+
+ rewrite <- Qplus_comm. rewrite Qlt_move_right.
+
+ apply Qlt_le_trans with (y := 1).
+ qreduce ((2 # 1) ^ (-62) + (2 # 1) ^ (-126)).
+
+ rewrite Qlt_mult_by_z with (z := 85070591730234615865843651857942052864 # 18446744073709551617).
+ ring_simplify.
+
+ ring_simplify in H0.
+ apply Qle_lt_trans with (y := (2147483646 # 1) * (2147483647 # 1) + (2147483646 # 1)).
+
+ apply Qplus_le_compat.
+ apply Qle_mult_quad.
+ assumption. psatzl Q. auto with qarith. assumption. psatzl Q.
+ auto with qarith. auto with qarith.
+ psatzl Q. psatzl Q. assumption.
+Qed.
+
+
+
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