This is an automated email from the git hooks/post-receive script. It was generated because a ref change was pushed to the repository containing the project "OCT: A portable Lisp implementation for quad-double precision floats". The branch, master has been updated via 6127a79e0d6300bc7ac7754190fbd559d00e7401 (commit) from 94d15953c5116999ac15377515b6c7e5ae430ea7 (commit) Those revisions listed above that are new to this repository have not appeared on any other notification email; so we list those revisions in full, below. - Log ----------------------------------------------------------------- commit 6127a79e0d6300bc7ac7754190fbd559d00e7401 Author: Raymond Toy <toy.raymond@gmail.com> Date: Sun Nov 24 10:41:27 2013 -0800 Fix ticket:5 o Implement better pi reduction and also always do pi reduction. o Add test for sin(pi) and cos(2^120), to test pi reduction. o Accuracy for erfc is only 198 bits. diff --git a/qd-fun.lisp b/qd-fun.lisp index d131a7a..91c0b32 100644 --- a/qd-fun.lisp +++ b/qd-fun.lisp @@ -696,38 +696,46 @@ is the cosine of A" (values mod f) (values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+)))))) -(defun rem-pi/2-int (qd) - ;; Compute qd rem pi/2 = k*pi/2+y. So we compute k + y*2/pi = - ;; qd*2/pi. - ;; - ;; First convert qd to 2^e*I. We already have 2/pi in the form - ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the - ;; integer and fractional parts of this. For the integer part we - ;; only need it modulo 4, because of the periodicity. For the - ;; fractional part, we only need 212 (or so bits of fraction). - ;; - ;; FIXME: But we don't really need to compute all the bits of I*J. - ;; In the product, we really only need the 2 bits to the left of the - ;; binary point, and then 212 bits to the right. This doesn't - ;; require doing the full multiplication. - (multiple-value-bind (i e s) +(defun split-prod-2/pi (qd) + ;; Compute k and f such that k + f = qd*2/pi, where k is an intege + ;; and f < 1. Since we do not need more than the the least + ;; two bits from k, only the least two bits of k are returned. + (multiple-value-bind (i e) (integer-decode-qd qd) + ;; First convert qd to 2^e*I. We already have 2/pi in the form + ;; 2^-1584*J. Then qd*2/pi = 2^(e-1584)*I*J. Extract out the + ;; integer and fractional parts of this. For the integer part we + ;; only need it modulo 4, because of the periodicity. For the + ;; fractional bits, we return the actual fraction as a ratio. + ;; + ;; FIXME: We compute the entire product I*J, but we only need the + ;; 2 bits to the left of the binary point. But we need many of + ;; the bits to the right of the binary point becaue we have no + ;; lower bound on the difference between a quad-double and pi. (let* ((exp (- e (integer-length +2/pi-bits+))) - (prod (* (* s i) +2/pi-bits+)) - (mod (ldb (byte 2 (- exp)) prod)) - ;; A quad-double has about 212 bits, but we add another 53 - ;; (5 doubles) for some extra accuracty. - (qd-bits 265) - (frac (ldb (byte qd-bits (- (- exp) qd-bits)) prod)) - (f (mul-qd (scale-float-qd (rational-to-qd frac) (- qd-bits)) - +qd-pi/2+))) - ;; We want the remainder part to be <= pi/4 because the trig - ;; functions want that. So if the fraction is too big, adjust - ;; it, and mod value. - (if (qd-<= (abs-qd f) +qd-pi/4+) - (values mod f) - (values (mod (1+ mod) 4) (sub-qd f +qd-pi/2+)))))) + (binary-point (- exp)) + (prod (* i +2/pi-bits+)) + (int (ldb (byte 2 binary-point) prod)) + (frac (/ (ldb (byte binary-point 0) prod) + (ash 1 binary-point)))) + (values int frac)))) +(defun rem-pi/2-int (qd) + "Compute qd rem pi/2 = k + f, where k is an integer and |f| < + pi/4. Two values are returned: k mod 4 and f." + (if (minusp-qd qd) + (multiple-value-bind (k f) + (rem-pi/2-int (neg-qd qd)) + (values (mod (- k) 4) (neg-qd f))) + (multiple-value-bind (k f) + (split-prod-2/pi qd) + ;; The trig functions want the arg to be less than pi/4. That + ;; is f < 1/2. Thus adjust k and f accordingly. + (when (> f 1/2) + (setf k (mod (1+ k) 4)) + (setf f (- f 1))) + (values k (mul-qd +qd-pi/2+ (rational-to-qd f)))))) + (defun rem-pi/2-int-b (qd) (declare (type %quad-double qd)) (multiple-value-bind (i e s) @@ -785,6 +793,13 @@ is the cosine of A" (sub-qd a (mul-qd-d +qd-pi/2+ (float quot 1d0)))))) (t (rem-pi/2-int a)))) + +(defun rem-pi/2 (a) + ;; If the number is small enough, we don't need to use the full + ;; precision algorithm to compute the remainder. The value of 1024 + ;; here is rather arbitrary. We should do an analysis to figure + ;; where the breakpoint should be. + (rem-pi/2-int a)) (defun sin-qd (a) diff --git a/rt-tests.lisp b/rt-tests.lisp index caf568f..69f6ee7 100644 --- a/rt-tests.lisp +++ b/rt-tests.lisp @@ -187,6 +187,18 @@ (check-accuracy 212 val true)) nil) +;;; sin +(rt:deftest oct.sin.pi + (not (zerop (sin +pi+))) + t) + +;;; cos +(rt:deftest oct.cos.big + (let* ((val (cos (scale-float #q1 120))) + (err (abs (- val -0.9258790228548379d0)))) + (<= err 5.2d-17)) + t) + ;;; Tests of atan where we know the analytical result (rt:deftest oct.atan.1 (let* ((arg (/ (sqrt #q3))) @@ -1731,6 +1743,6 @@ nil) (rt:deftest erfc - (check-accuracy 210 (erfc #q-4) + (check-accuracy 198 (erfc #q-4) #q1.9999999845827420997199811478403265131159514278547464108088316570950057869589732) nil) ----------------------------------------------------------------------- Summary of changes: qd-fun.lisp | 73 ++++++++++++++++++++++++++++++++++----------------------- rt-tests.lisp | 14 ++++++++++- 2 files changed, 57 insertions(+), 30 deletions(-) hooks/post-receive -- OCT: A portable Lisp implementation for quad-double precision floats